Soft Numerical Computing in Uncertain Dynamic Systems 0128228555, 9780128228555

Table of contents :
Front Matter
Copyright
Dedication
Preface
Introduction
Introduction
Introduction to uncertain dynamic systems
History
Structure of the book
References
Uncertain sets
Short introduction to this chapter
Textual short outline
Measures
Measurable space
Examples
Uncertain sets and variables
Examples
Zigzag uncertain variable
Experimental uncertain variables
Membership function
Fuzzy numbers and their properties
Definition of a fuzzy number
Level-wise form of a fuzzy number
Definition of a fuzzy number in level-wise form
Definition of a fuzzy number in level-wise form
A singleton fuzzy number
Definition of a fuzzy number in parametric form
Nonlinear fuzzy number
Trapezoidal fuzzy number
Triangular fuzzy number
Operations on level-wise form of fuzzy numbers
Summation
Example
Multiplication
Difference
Hukuhara difference
Example
Example
Generalized Hukuhara difference
The level-wise form of generalized difference
Some properties of gH-difference
Example
Partial ordering
Some properties of partial ordering
Absolute value of a fuzzy number
Some properties of partial ordering in gH-difference
Approximately generalized Hukuhara difference
Some properties of g-difference
Example
Example
Generalized division
Some properties of division
Examples
Approximately generalized division
Example
Piece-wise membership function
Some properties of addition and scalar product on fuzzy numbers
Definition-singleton fuzzy number
Advanced uncertainties and their properties
Pseudo-octagonal sets
Z-process
Definition-Z-process
Example
Example
Computations on Z-numbers
Summation of two Z-numbers
Difference of two Z-numbers
Multiplication of two Z-numbers
Division of two Z-numbers
Level-wise form of a Z-number
High membership degree does have high reliability
Definition-Level-wise form of a standard Z-number
Summation in level-wise form
Scalar multiplication in level-wise form
Hukuhara difference in level-wise form
Generalized Hukuhara difference in level-wise form
Some properties of generalized Hukuhara
References
Further reading
Soft computing with uncertain sets
Introduction
Expected value
Distance of two fuzzy numbers
P-distance
Hausdorff Distance
Limit of fuzzy number valued functions
Definition-Fuzzy set valued function
Definition-Fuzzy number valued function
Definition-The limit of Fuzzy number valued function
Theorem-Limit of summation of functions
Theorem-Limit of difference of functions
Theorem-Limit of multiplication
Other properties of limit
Fuzzy Riemann integral operator
Some properties of fuzzy Riemann integral
Differential operator
Definition-gH-differentiability
Example
Example
Definition-gH-differentiability in level-wise form
Definition-Switching points of gH-differentiability
Example
Proposition-Summation in gH-differentiability
Proposition-Difference in gH-differentiability
Proposition-Production in gH-differentiability
Proposition-Composition of gH-differentiability
Proposition-Minimum and maximum
Definition-Continuous fuzzy number valued function
Proposition
Proposition-Cauchy's fuzzy mean value theorem
Corollary-Fuzzy mean value theorem
Proposition-Increasing and decreasing function
Proposition-Integral of gH-differentiability
Proposition-Switching points in integration
High order differentiability
Extended integral relation
Part-by-part integration
Taylor expansion
Example
gH-partial differentiability
Example
Another simple example
Level-wise form of gH-partial differentiability
Switching point in gH-partial differentiability
Example
Higher order of gH-partial differentiability
Integral relation in gH-partial differentiability
Multivariate fuzzy chain rule in gH-partial differentiability
The fuzzy Laplace transform operator
Example
Definition-Absolutely convergence
First translation theorem
Second translation theorem
Laplace transform on the derivative
Derivative theorem
High order derivation theorem
Fuzzy improper integral
Definition-Uniform convergence
Theorem-Interchanging integrals
Theorem-Integral and derivative
Fourier transform operator
Definition-Fuzzy Fourier transform
Example-Fuzzy Fourier transform
Definition-Fuzzy inverse Fourier transform
Theorem-Existence
Theorem-Linearity property
Theorem-Fourier transform of gH-derivative
References
Continuous numerical solutions of uncertain differential equations
Introduction
Uncertain differential equations
Definition-Uncertain process as a canonical Liu process
Definition-Liu integral of an uncertain process
Theorem-Chain rule
Theorem-Integration by parts
Definition-Uncertain differential equation
Remark
Fuzzy differential equations
Theorem-Existence and uniqueness
Fuzzy differential equations-Variation of constants
Theorem-Existence of the solution
Length function
Definition-Length function
Nonlinear property
Theorem-Nonlinear property of fuzzy functions
Remark
Remark-Differentiability and length
Theorem-Nonlinear property of fuzzy functions
Theorem-Derivative of integral equation
The length function-Fuzzy differential equations
Fuzzy differential equations-Laplace transform
Fuzzy differential equations-Second order
Fuzzy differential equations-Variational iteration method
Fuzzy differential equations-Legendre differential equation
Definition-Power series with fuzzy coefficients
Some properties of fuzzy series
Fuzzy calculated operations
Fuzzy power series method for solving Legendre's equation
Linear systems of fuzzy differential equations
Homogeneous fuzzy linear differential systems
Nonhomogeneous fuzzy linear differential systems
Reduction of a second order fuzzy differential equations to a system of first order equations
Z-differential equations
References
Discrete numerical solutions of uncertain differential equations
Introduction
Fuzzy Euler method
Analysis of the fuzzy Euler method
Local truncation error and consistency
Global truncation error and convergence
Theorem-Convergence
Stability
Fuzzy modified Euler method
Analysis of the fuzzy modified Euler method
Local truncation error and consistency
Global truncation error and convergence
Theorem-Convergence
Stability of the modified fuzzy Euler method
Fuzzy Euler method for fuzzy hybrid differential equations
Fuzzy Euler method for fuzzy impulsive differential equations
Error analysis
Stability
Fuzzy predictor and corrector methods
Definition-Fuzzy explicit method
Definition-Fuzzy implicit method
Fuzzy explicit three steps method
Fuzzy implicit two steps method
Fuzzy predictor and corrector three steps methods
Numerical solution of fuzzy nth-order differential equations
References
Further reading
Numerical solutions of uncertain fractional differential equations
Introduction
Fuzzy Riemann-Liouville Derivative-Fuzzy RL Derivative
Note-Combination Property
Level-Wise form of Fuzzy Riemann-Liouville Integral Operators
The RL Fractional Integral Operator
The Fuzzy Riemann-Liouville Derivative Operators
Fuzzy Caputo Fractional Derivative
Caputo gH-Differentiability
Caputo-Katugampola gH-Fractional Derivative
Fuzzy Fractional Differential Equations-Caputo-Katugampola Derivative
Definition-Fuzzy Fractional Differential Equations
Existence and Uniqueness of the Solution
Theorem-Existence and Uniqueness in Real Fractional Differential Equation
Theorem-Existence and Uniqueness in Fuzzy Fractional Differential Equation
Some Properties of the Mittag-Leffler Function
Fuzzy Generalized TaylorS Expansion
Fuzzy Fractional Euler Method
References
Numerical solutions of uncertain partial differential equations
Introduction
Partial ordering
Continuity
Minimum and maximum
Production in partial gH-differentiability
Fuzzy integrating factor
The fuzzy heat equation
Theorem-Fuzzy maximum principle
Theorem-Existence
Analytical solution of the fuzzy heat equation
The fundamental solution of the fuzzy heat equation
Fuzzy Fourier transform
Fuzzy inverse Fourier transform
Fourier transform of gH-derivative
Fuzzy finite difference method for solving the fuzzy Poisson's equation
Theorem-Uniqueness
Error analysis
References
Index
A
B
C
D
E
F
G
H
I
L
M
N
P
R
S
T
U
V
W
Z

Citation preview

Soft Numerical Computing in Uncertain Dynamic Systems

Soft Numerical Computing in Uncertain Dynamic Systems Prof. Tofigh Allahviranloo Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey

Prof. Witold Pedrycz Department of Electrical and Computer Engineering, University of Alberta, Canada

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom © 2020 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-822855-5

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Dedication

This book is dedicated to the scientific society.

Preface

This book contains a wealth of useful information to identify dynamic systems and differential equations under uncertainty. By studying this book, you can get acquainted with various kinds of ambiguous information and learn how to use it efficiently in dynamic systems. A careful examination of fuzzy differential equations from a numerical point of view has been carried out. For this purpose, numerical and semianalytical methods have been investigated to solve these equations. The development of solutions in the presence of complex data has been proposed. In addition, a thorough error analysis of solutions has been delivered. Interesting applications of these systems in engineering and biology have also been reported. Tofigh Allahviranloo Istanbul Witold Pedrycz Edmonton

1.1

Chapter 1

Introduction Introduction

Uncertainty is an intrinsic component of knowledge, and yet in an age of controversial expertise, many fear their audience’s reaction if they publicly communicate their uncertainty about what they know. Experimental research is widely dispersed in many disciplines. These interdisciplinary overview structures and summaries of current practice and research in the field are a combination of a statistical and psychological perspective. They inform us of a framework for communicating uncertainty in which we identify three uncertainties—facts, numbers, and science—and two levels of uncertainty: direct and indirect. Examining current practices provides a scale of nine expressions of direct uncertainty. We discuss attempts to decode indirect uncertainty in terms of the quality of the underlying evidence. We review the limited literature on the impact of epistemological uncertainty on cognition, affect, trust, and decision-making. While there is evidence that epistemic uncertainty does not necessarily have a negative effect on the audience, its impact can vary between individuals and formats of communication. Case studies in economic statistics and climate change illustrate the framework for action. In this 1 Soft Numerical Computing in Uncertain Dynamic Systems. https://doi.org/10.1016/B978-0-12-822855-5.00001-X © 2020 Elsevier Inc. All rights reserved.

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Soft Numerical Computing in Uncertain Dynamic Systems

important, by consulting with regard to future researchers we conclude but so far neglected field. When we know that we actually live in uncertainty, we must admit that uncertainty lives with us and may be involved with any of our real-life problems. When the mind is in a state of uncertainty, it gives the slightest motivation to each side. Either you allow uncertainty to stop, hinder, or paralyze you, or you use it to inspire you to take up opportunities, seek out excitement, and achieve growth. Eventually, it all comes down to perspective and focus. Choose to focus on growth instead of stagnation. Choose to focus on possibilities instead of limitations. Choose to focus on elevating instead of staying the same. It is very important to know that we cannot distinguish the exact answers to many of our real questions. This attitude of mind—this attitude of uncertainty—is critical to the scientists, and it is the attitude of the mind that the students must first acquire. Instead of trying to fight uncertainty, our duty is to figure out how we can embrace it. Because in uncertainty there is also the growth, the possibilities, and the life you want. The students should know that uncertainty creates an environment for us to grow our ability and to learn more. Richard P. Feynman

The mathematical truth has a validity independent of place, personality, or human authority. Proving a mathematical theorem may have been a very romantic part of the mathematician’s personal life, but one cannot expect to use it to discover his/ her race, gender, or temperament itself. Truth cannot be explained or evaluated only in terms of false or true. Sometimes it is almost right, or sometimes it is probably wrong. Most of the time, the answer is given by personal feelings, for example, “I hope it’s true” or “I think so.” This is why a mathematician should consider the linguistic propositions in mathematical logic, which is an entrance for uncertainty in mathematics. In fact, uncertainty has a history in human civilization, and humanity has long been thinking about controlling and exploiting this kind of information. The theory of probability can be considered as one of the cases of uncertainty. Dice gambling was the beginning of what is now called the theory of probability. In the 16th century, there was no way to quantify luck. If someone rolled two sixes during a game of dice, people thought it was just good luck. That means we can measure an event and exactly how lucky or unlucky we are—to work. One of the most ancient and obscure concepts is the word “luck.” Gambling and dice have had important roles in developing the theory of probability. In the 15th century, Cardano (2016) was one of the most well-known figures in formal algebraic activity. In The Game of Chance, he gave his first analysis of the rules of luck, and solved such problems numerically. In 1657, Huygens wrote the first book on probability, On the Calculation of Chances. This book arguably marks the real birth of probability. The theory of probability was started mathematically by Blaise Pascal

Introduction

3

and Pierre de Fermat in the 17th century, who sought to solve mathematical problems in certain gambling issues. Its translation was done in 1714 (Huygens, 1714). The concept of “expected value” is now a key part of economics and finance: by calculating the expected value of an investment, we can work out how much it is worth to each party. This is defined as the proportion of times in a game such as a coin-tossing such that each side would win on average if the game were repeatedly played to completion. Since the 17th century, the theory of probability has been continuously formulated and applied in various disciplines. Today, it is important in most areas of engineering and tool management, and is even used in medicine, ethics, law, and other areas. In classical logic, the values “true” and “false” or the numbers “zero” and “one” are the values for making a decision in binary logic. However, these will not work well in multivalued logic with a set of degrees of truth. Another aspect of uncertainty appears in multivalued logic. This is nonclassical logic similar to classical logic in that it embodies the truth-function principle; that is, the truth of a compound sentence is determined by the truth values of its component sentences. Multivalued logic uses its truth degrees as technical tools for choosing particular suitable applications. It is a challenging philosophical problem to discuss the (possible, nontechnical) nature of such “truth degrees” or “truth values.” In this logic, there are propositional variables together with connectives, truth degree constants, and object constants, in the case of propositional languages and functional symbols, as well as quantifiers. The first person to speak of ambiguity was Planck (1937). He published an article on the analysis of logic called ambiguity in the journal of Science. He did not mention a fuzzy word, but in fact explained the fuzzy logic that, of course, was provided by the universe; science and philosophy were ignored (Robitaille, 2007). Eventually, in 1965, Professor L.A. Zadeh, by using fuzziness as one type of uncertainty, developed a new way to accept this idea (Zadeh, 1965). He published an article entitled “Fuzzy Sets” in the journal Information and Control, which used a new fuzzy logic for the sets. Zadeh considered the fuzzy name for these sets to divert it from binary logic. The emergence of this science opened up a new way of solving problems with vague values and reality-based modeling, and scientists used it to model ambiguity as part of the system. As a practical example, in the literature of artificial intelligence and uncertainty modeling, there has long been a misunderstanding about the role of fuzzy set theory and multivalued logic. The recurrent question is whether there is a mathematical meaning to the performance of a compound calculation and the validity of the rules between exceptions. This confusion, despite some early philosophers’ warnings, encompasses early developments in probable logic. Regarding this fact, three main points can be identified. First, it shows that the root of the differences lies in the uneasy confusion between the hierarchy of beliefs and what logicians call “degrees of truth.” The latter are

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usually composites, while the former is not. This claim is first shown by spurring the argument of noncompeting belief embedded in the standard proposition account. It seems that a copy of all or nothing of the theory is not possible. This framework is then expanded to discuss the case of fuzzy logic vs the grading theory. Next, it has been shown that any belief in the idea that combinability is accepted is at worst a fall into a Boolean truth task and at best a weak tool. Finally, some claims about the possibility of the theory of possibility being combinatory are rejected, thus illuminating the pervasive confusion between axioms of possibility theory and the underlying fuzzy set connections. Section 1.1 provides an introduction to uncertain dynamic systems.

1.1.1

INTRODUCTION

TO UNCERTAIN DYNAMIC SYSTEMS

The modeling of these systems is used to describe and predict the interaction over time between several uncertain components of a phenomenon that is viewed as a system. This focuses on the mechanism of how uncertain components and systems evolve over time. While many communication researchers have linked theories to uncertain dynamic processes that unfold over time, few have applied mathematical modeling techniques and none has applied soft numerical computing languages to represent formally the theories proposed. The uncertain dynamic systems allow communication researchers to explore the gap between conceptual and linguistic phenomena as dynamic and formally studied phenomena. Uncertain dynamic systems in many academic fields have a rich history stemming from uncertain mathematics and physics, prior to their entrance into the life, social, and behavioral sciences. In the field of communication, they have already been used for a number of important research questions. Communication researchers are aware of their many benefits as data collection and analytical techniques become more accessible. Dynamic systems can be of use not only in analyzing data and testing hypotheses, but also in designing and developing theories. The main issue here is the definition of an uncertain dynamic system. Why is it called a dynamic system, and what is its main concept? An examination of each of the so-called components of dynamic system modeling can help clarify its meaning. The dynamic component shows that time is incorporated as a fundamental element of the model. In traditional static models, time is often overlooked, and this may impair the influence of the variables studied. In dynamic models, time is applied to the underlying data structure and understanding of how a fundamental process is formed. In some uncertain dynamic system models, uncertain and accurate data are organized by time as a sequence of repeated observations of time-varying variables called time-series data. The data is assumed to be serial dependent, meaning that individual data points are not conceived independently of each other, but each data value is affected by previous values.

Introduction

5

Generally speaking, in dynamic models, time variable always plays a crucial role in the distribution of response time in model formulation and prediction. Dynamic system models assume that the current states of a system are dependent on past states. This is generally reflected in formal models using differential operators or differential equations with system feedback or lag terms. Within a dynamic system, the interactions are treated in an orderly manner, following rules that can be identified and defined. While systems are orderly, they behave in complex ways that make it difficult for researchers to describe them with natural verbal language. Changing one component can lead to a larger change throughout the system. A central question of any dynamic system is how it maintains stability or resistance to change over time. Systems vary in their stability, and the following factors may affect this stability: – how likely the system is to change states; – the amount of expected variability within the system across time; and – how it responds to small disturbances. The uncertain dynamic system modeling component indicates that dynamic relationships between uncertain components of a system are executed as formal mathematical equations with uncertainty and/or using soft computing languages. Broadly speaking, modeling methods can be distinguished as mathematical/computational and statistical. In general, mathematical models express theoretical structures and relationships as explicit mathematical relationships, and make specific and detailed claims. This is in contrast to the more common theological theories in the social and behavioral sciences, which can provide rich descriptions of phenomena to generate hypotheses in which may be ambiguous and able to make equally accurate and specific testable predictions such as formal models. It is clear that computational models are based on mathematical models. They typically implement mathematical models using computer resources and rely on computers to develop, simulate, and test complex systems. Mathematical models differ from statistical models, which are more familiar to communication researchers. Statistical models are general analytical tools that researchers use to test specific hypotheses. The models themselves may provide support or a theoretical prediction, but the statistical model is not a formal representation of the theory itself. In comparison, a mathematical or computational model is a formal representation of the theory’s arguments, and how the model works directly shows what the theory predicts, and can be used for testing against experimental data. Mathematical/computational models and statistical models can be seen as the ends of a spectrum, and a variety of models fall between these two types. More specifically, uncertain dynamic systems models commonly have the following three main elements to describe how a system generally behaves.

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Soft Numerical Computing in Uncertain Dynamic Systems

– First, the state of the system in an uncertain environment, which represents all the system information at a determined moment in time. In this case, time and the state-space in uncertain dynamic system models may be either discrete or continuous. If time is indexed as a series of equally spaced points with unique sequential values, then it is discrete. In contrast, uncertain differential equations are used when time is treated as continuous. In this case, the variables are not indexed using specific units of time in the same manner. In general, the defined uncertain differential equation for the continuous case of time has an uncertain continuous solution that can be determined and illustrated in a discrete-time series. – Second, the uncertainty state-space of the system, which represents all possible system states that can occur. In this case, the models may be either linear or nonlinear in the relationships they situate between variables. That is, the relationships between model uncertain parameters are explained by either linear or nonlinear functions, and even in linear cases, the model may not produce a linear output. This is because the behavior of the system predicted by the linear model can be nonlinear due to the influence of previous system states. – Third, the state-transition function, which describes how the uncertainty state of the system changes over time. Here, the models may be stochastic or as any other cases of uncertainty. In each model, a future uncertain system state can be fully determined with complete uncertain knowledge of current system states. In a stochastic model, a future uncertain system cannot be exactly predicted, but only probabilistically determined. This formally reflects some fundamental uncertainty, imprecision, or unpredictability of a system by including an uncertain random variable or variables. The coefficients in the model can be considered as either time-invariant or time-variant. In other words, this suggests that as a process unfolds over time, the formal relationship between components in a system either does or does not change. As far as we know, many systems in the applied sciences like medicine, economics, management, engineering, finance, and some parts of psychology work with data combined with uncertainty and ambiguities like those mentioned in the above data. The concept of uncertainty is highlighted and has long been of use, particularly in the most important areas of decision-making that are involved with our lives and problems. This is because without the right and true making of a decision, life may go in the wrong direction. As an additional illustration, a logical decision should be made in the field of indeterminacy or in the real-life environment that is formed and combined by undetermined concepts and data. Modeling, analyzing, and solving real-life problems are the key challenges facing us. For instance, in medical science most of a patient’s problems are dynamic, because they are changing over time and their behaviors are dependent on time

Introduction

7

passing. For example, nowadays cancer is a common disease and as time passes, cancer grows. Modeling and solving cancer-related problems, and analyzing and simulating solutions, are very helpful and important tasks.

1.1.2

HISTORY

Dynamic systems conceptually have long been part of some theories of social scientists. James’ (1890) “stream of consciousness” emphasizes the dynamic changes in attention and thinking, while Levine’s field theory (1939) explicitly uses the concepts of physics in describing dynamic interactions between the person and their environment. Early research in cybernetics has shown in detail how systems can be self-regulated through feedback, and many examples are explicitly derived from human behavior (Weiner, 1961 cited in Latta and Patten, 1978). While cybernetic ideas have been presented mathematically, many social scientists only emphasized the broader conceptual framework. Some reasons why dynamic systems models have lagged behind social and behavioral science and theories are the lack of research and resources, and the absence of performing the appropriate data analyses in relation to capable computer technologies. Before 1950, computer simulations of dynamic processes started with system dynamics modeling and, during the 1980s, when sufficiently powerful computer technologies became more available, modeling efforts in the social and behavioral sciences began to grow in popularity. Certainly, cognitive models based on simulated and real experimental data are now increasingly popular and the whole field of cognitive science is moving toward a dynamic research paradigm. Further information about uncertain information can be found in the book by Allahviranloo (2020). A book entitled Uncertain Dynamic Systems, authored by Schweppe (1973), is the only one that deals with subject matter similar to the one mentioned here. The book concerns dynamics, estimation theory, statistical hypothesis testing, and system analysis. This book is designed to help the reader make certain statements about dynamic systems when uncertainty exists in a system’s input and output, and in the nature of the system itself. The methods of interest are applicable to large-scale, multivariable systems. The problems of and interactions among modeling, analysis, and design are considered. Emphasis is placed on techniques that can be implemented on a computer. In general, there is a huge difference between the abovementioned book and this book, because in our book, the numerical simulations of the solutions obtained by numerical methods are discussed in detail. Uncertain Dynamic Systems does not discuss the numerical solutions, convergence, and consistency, although it mentions theoretical analysis of uncertain dynamic systems.

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Soft Numerical Computing in Uncertain Dynamic Systems

Another relevant book is Extremal Fuzzy Dynamic Systems, written by Sirbiladze (2013). Sirbiladze presents a new approach to the study of dynamic systems with weak structures. The book’s approach differs from those of other publications by considering time as a source of fuzzy uncertainty in dynamic systems. Extremal Fuzzy Dynamic Systems progresses systematically by covering the theoretical aspects first before tackling the applications. In the application section, a software library is described, which contains discrete EFDS identification methods elaborated during fundamental research of the book. The uncertainty mentioned in Extremal Fuzzy Dynamic Systems is very limited, and involves only fuzzy sets; the models are fuzzy models with fuzzy inputs and outputs. In addition, the author does not discuss several other types of uncertainty and numerical simulations that we consider in this volume. To the best of our knowledge and at the time of writing, there is no other book with our book title and subject matter, making this volume completely original. Fuzzy Dynamic Systems was published by de Barros et al. (2017). It presents an introduction to fuzzy dynamic systems, both continuous and discrete. To study the dynamic case, the concepts of fuzzy derivative and fuzzy integral are presented. Several kinds of derivatives are explored, and several types of fuzzy differential equations are consequently studied. The discrete case is studied by means of an interactive process. All cases are illustrated using the Malthusian model. It is known that uncertainty can appear as fuzziness; the topic of fuzzy dynamic systems has recently gained more interest and many researchers have worked on the topic both theoretically and numerically. A system of fuzzy differential equations is utilized for the purpose of modeling problems in the fields of engineering, biology, and physics among others. Differential equations have been unable to predict all the possible forms of arrhythmia, leading to failure to detect numerous disturbances by cardiologists. As an application of fuzzy differential equations in biomedical science, fuzzy differential equations are promising in terms of anticipating the nuances of each category of arrhythmia (different run lengths of premature ventricular contraction), contributing to the efficient diagnosis of heart abnormalities. Another aspect of a dynamic system can occur as fuzzy fractional differential equations. It differentiates different materials and processes in many applied sciences like electrical circuits, biology, biomechanics, electrochemistry, electromagnetic processes, and others, and is widely recognized to predict accurately by using fractional differential operators in accordance with their memory and hereditary properties. For complex phenomena, the modeling and its results in diverse widespread fields of science and engineering are very complicated, and to achieve accuracy, the only powerful tool suitable is fractional calculus. This is not only a very important and productive topic, but also represents a new point of view on how to construct and apply a certain type of nonlocal operator to realworld problems.

Introduction

1.1.3

STRUCTURE

9

OF THE BOOK

The structure of this monograph is planned as follows. Chapter 2 covers the concepts of uncertain sets. In this chapter, we define, introduce, and explain types of uncertainties. The concept of uncertainty can come in many forms, such as an unspecified set, the expected amount or interval of data, random data, or a combination of all of these. A different perspective can be described as a membership function called fuzzy data and a combination of membership functions, probability, and distribution. In Chapter 3, operators like distance, derivative, and ranking on fuzzy sets are considered. One of the most important operations on fuzzy sets is the difference. It may seem strange that we call it difference and not by subtract; this is because the subtraction of two sets of ordered pairs or functions cannot be defined properly, and here we define it by the Hukuhara difference. This concept is a key operator to define the derivative of a fuzzy function. In advance, the combination of distributive function and fuzzy membership function and its combination with other uncertain sets are defined as advanced combined uncertain sets. Soft computing on the last type of uncertainty is a very different and new discussion topic. Sometimes this operation does not have a formal form of computing and should be done by expert systems. We will therefore need to discuss uncertain inferences and logical decisions to reach the results for the calculations. Chapter 4 covers the concept of the continuous numerical solution, which means that the solution of the numerical method approximates the exact solution functionally. For instance, the Taylor method is one method that works with an arbitrary order of differentiability under uncertainty. Another functional numerical method is variational iteration. To explain this method, we need to discuss the weakly and strongly uncertain nonlinear equation cases. Variational iteration is the most effective and convenient method for both weakly and strongly uncertain nonlinear equations. This method has been shown to solve effectively, easily, and accurately a large class of nonlinear problems with components converging rapidly to accurate solutions. Chapter 5 discusses Euler’s method and other finite difference methods, such as Runge–Kutta’s method, and predictor and corrector methods to solve the problem numerically. It is more appropriate to use the prefix “uncertain” for all of these methods. Because the modeling of the methods under uncertainty will be almost different. Since the definition of difference has two cases, the derivative of an uncertain function is defined in two cases as well. Now considering the order of differentiability or differential equations with high order, the cases for the uncertain differential equations have several types. In each of several cases, we discuss the details for convergence theorems and consistency, and stability as well. Stability is one of the most important concepts in dynamic systems, especially in uncertain cases. Several types of stability are considered in detail.

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Soft Numerical Computing in Uncertain Dynamic Systems

Chapter 6 discusses fractional operators, which have been much highlighted over the past decade, and which have many applications in real-world problems. Different materials and processes in many applied sciences like electrical circuits, biology, biomechanics, electrochemistry, electromagnetic processes, and others are widely recognized to be predicted effectively by using fractional differential operators in accordance with their memory and hereditary properties. For complex phenomena, the modeling and its results in diverse widespread fields of science and engineering are very complicated, and to achieve accuracy, the only powerful tool suitable is fractional calculus. This is not only a very important and productive topic, but also represents a new point of view on how to construct and apply a certain type of nonlocal operator to real-world problems. Since the uncertainty in our real environment and data plays an important role, this causes us to discuss the uncertainty in the mentioned topics. Finally, in Chapter 7, we discuss partial differential equations with uncertainty. As far as we know, many real-world problems in engineering science relate to partial differential equations, such as wave equations, heat equations, heat and mass transferring, etc. All of the related equations work with uncertain information like fuzzy, interval, and any others that we mentioned earlier. Since most of the numerical methods for solving these types of equations are finite difference methods, we will investigate further new uncertain finite difference methods for solving these equations. These methods are connected to some linear systems and in this case, we will be involved with uncertain linear systems. As similar to other equations, the numerical solutions of these equations need to prove convergence, stability, and consistency. We will therefore consider these properties theoretically as well. On this topic, the operator of a derivative as a partial derivative can be an uncertain operator, and this is why, when starting this subject, we have to illustrate the uncertain operators of the partial derivative.

References Allahviranloo, T., 2020. Uncertain Information and Linear Systems, Studies in Systems, Decision and Control. Springer. Cardano, G., 2016. The Book of Game of Chance: The 16th-Century Treatise on Probability. Mathematical Association of America (Chapter 1). de Barros, L.C., Bassanezi, R.C., Lodwick, W.A., 2017. A First Course in Fuzzy Logic, Fuzzy Dynamical Systems, and Biomathematics—Theory and Applications. Studies in Fuzziness and Soft Computing, vol. 347. Springer. ISBN 978-3-662-53322-2, pp. 1–295. Huygens, C., 1714. The Value of Chances in Games of Fortune. English translation. https://math.dartmouth. edu/doyle/docs/huygens/huygens.pdf. James, W., 1890. The Principles of Psychology. Henry Holt, New York, NY.

Introduction

11

Latta, R.M., Patten, R.L., 1978. A test of Weiner’s attribution theory inertial motivation hypothesis. J. Pers. 46, 383–399. Robitaille, P.-M., 2007. Max Karl Ernst Ludwig Planck: (1858–1947). Prog. Phys 4, 117–121. Schweppe, F.C., 1973. Uncertain Dynamic Systems. Prentice Hall, Englewood Cliffs, NJ. Sirbiladze, G., 2013. Extremal fuzzy dynamic systems—theory and applications. In: IFSR International Series in Systems Science and Systems Engineering, Springer. Zadeh, L.A., 1965. Fuzzy sets. Inf. Control. 8 (3), 338–353.

2.1

Chapter 2

Uncertain sets Short introduction to this chapter

Let us start with a quote from Richard P. Feynman, the American physicist: I think that when we know that we actually do live in uncertainty, then we ought to admit it; it is of great value to realize that we do not know the answers to different questions. This attitude of mind—this attitude of uncertainty—is vital to the scientist, and it is this attitude of mind which the student must first acquire. In this chapter, several types of uncertainties are discussed. First, an uncertain set is defined, and it is considered as a membership function. Second, a special case of the membership function appears as an interval; it is then defined and explained. Under some conditions, a fuzzy set can be defined as uncertain data. After explaining these data, a combination of membership function and distributive function as Z-sets will be discussed. All these uncertain data are evaluated completely and calculations on them are evaluated properly.

13 Soft Numerical Computing in Uncertain Dynamic Systems. https://doi.org/10.1016/B978-0-12-822855-5.00002-1 © 2020 Elsevier Inc. All rights reserved.

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Soft Numerical Computing in Uncertain Dynamic Systems

2.2

Textual short outline

Any uncertainty space is formed by uncertain variables and an uncertain variable to be completed by measurable space. To define a measurable space, we need to define a measurable set. A measurable set is an extended set with measures. It means any member of the set has a measure. The measures for the members come from a measurable function. The measures have their own rules and these rules define an algebra. We therefore have to work with a space of measurable functions and these can be called membership functions, because normally the measures can play the role of memberships for the members of the set. To continue the discussion, distributive functions, sigma algebra, and Borel sets should be discussed.

2.3

Measures

As we mentioned, to define the measure we have to define a measurable space. For defining a measurable space we need to have a measurable set. Mathematically and essentially, the uncertainty theory is an alternative theory of measure in a measurable space, and this is why uncertainty theory should be discussed on a measurable space. In this section we are not going to discuss the preliminary definitions in-depth, because more and sufficient results can be found in many other books that discuss them thoroughly (Allahviranloo, 2020; Liu, 2015).

2.3.1

MEASURABLE

SPACE

A set, , with a nonempty collection, ℂ, of subsets of , ð, ℂÞ, is called a measurable space if it satisfies the following two conditions: 1. For A, B  ℂ then A  B ¼ A \ Bc  ℂ. ∞ S 2. For any {Ai}∞ Ai ℂ. i¼1  ℂ then i¼1

The members of this collection ℂ are called measurable sets. Based on the definition, several properties of the measurable sets are immediately obvious: 1. The empty set ø  ℂ is a measurable set and ð, ℂÞ is a measurable space, because it satisfies two of the conditions. 2. If A and B are two measurable sets then A \ B, A [ B and A  B are measurable sets. The proofs are clear. First, suppose we want to prove that A  B is a measurable set. Since A,B, then based on the first condition A  B  ℂ and it is exactly the first condition of the definition. Now, to prove that A \ B  ℂ, we

Uncertain sets

15

use A \ B ¼ A  (A  B) and clearly it is already proved. Finally, for proving the union A [ B, using the second property of the definition on the sequence A, B, ∅ , ∅ , … our assertion is finalized. Tn 3. For {Ai}∞ i¼1 measurable sets the i¼1 Ai is a measurable set. It is clear because A1 \ A2 \ A3 \ … ¼ A1  {(A1  A2) [ (A1  A3) [ (A1  A4) [ …} and {(A1  Ai)}∞ i¼1, and their unions are measurable, therefore the proof is completed. Note: Measurable sets are closed under taking countable intersections and unions operators.

2.3.2

EXAMPLES

The spaces ð, ℂ ¼ f∅gÞ, ð, ℂ ¼ fAll subsetsof gÞ: Now, let ð, ℂÞ be a measurable space. A measure on this space consists of a nonempty subset, M, of ℂ, together with a mapping μ : M ! R + satisfying the following two conditions: 1. For any AM and ℂ 3 B  A, we have BM: ∞ S 2. Let A ¼ Ai for all disjoint fAi g∞ i¼1 M, then: i¼1

AM⟺

∞ X

μðAi Þ ! μðAÞ

i¼1

Each type of measurable space is defined by its own measure: probability space is defined by probability measure, uncertainty space is defined by uncertainty measure, and fuzzy space can be defined by fuzzy measure. In fact, the measures play the role of membership for the member of a set belonging to ℂ of the measurable space. Suppose that ð, ℂÞ is an uncertain space and μ is the degree of uncertainty of members of a set M: This concept points out that the members of every set that belongs to uncertainty space have a membership degree in the set. These membership degrees satisfy the two abovementioned properties.

2.4

Uncertain sets and variables

Uncertain variables are tools to model the data in an indeterminacy field that cannot be predicted exactly, such as throwing coins, tossing dice, playing poker, stock pricing, marketing and market demand, and lifetime. What is the answer to this question, which person is beautiful? Apparently, the answer depends on one’s personality and varies by different ones. It is clear that we have an uncertain variable regarding beautiful people. The membership of everyone in this set is called a membership degree. The membership degree can be considered as a real number from the interval [0, 1].

16

Soft Numerical Computing in Uncertain Dynamic Systems 1

0.5

–2

Fig. 2.1

–1.5

–1

–0.5

0

0

0.5

1

1.5

2

Gaussian membership function on [a1, a2] ¼ [2, 2].

Indeed, an uncertain set is formed by ordered pairs such that the second component is the membership degree of the first one. Graphically, these ordered pairs lie on the function and this function is called the membership function. Sometimes the uncertain set takes the role of an uncertain variable. It depends on the situation. For instance, to consider an uncertain function requires an uncertain variable, and is indeed a measurable set and has a membership function. In the following, several uncertain sets and variables are plotted and modeled mathematically. •

Gaussian form membership function on [a1, a2] (Fig. 2.1) 8 < 0, x  a1 2 f1 ðxÞ ¼ ex , a1  x  a2 : 0, x  a2

•

Triangular form membership function on [a1, a2] 8 0, > > > x  a 1 > > , > > < a  a1 f 2 ðxÞ ¼ 1, > > > a2  x , > > > > : a2  a 0,

x  a1 a1  x < a x¼a a < x  a2 x  a2

If we suppose that a1 ¼ 2,a ¼ 1,a2 ¼ 2, the figure is Fig. 2.2. •

Trapezoidal form membership function on [a1, a2] 8 0, > > > x  a1 > > , > > < a  a1 f2 ðxÞ ¼ 1, > > a x > > 2 , > > > : a2  a 0,

x  a1 a 1  x < a0 a0  x  a 00

00

a < x  a2 x  a2

Uncertain sets

17

1

0.5

–2

–1.5

–1

–0.5

0

0

0.5

1

1.5

2

Fig. 2.2 Triangular membership function.

1

0.5

–2

–1.5

–1

–0.5

0

0

0.5

1

1.5

2

Fig. 2.3 Trapezoidal membership function.

If we suppose that a1 ¼  2, a0 ¼  1, a00 ¼ 1, a2 ¼ 2, the figure is Fig. 2.3. Consider the following function. In this figure x(t)  [0, 1]  R+; the variable x is an uncertain variable and can be chosen from uncertainty space (Fig. 2.4).

2.4.1

EXAMPLES x1 ðtÞ ¼ t, x2 ðtÞ ¼ ktn , k,nR + , x3 ðtÞ ¼ exp ðtÞ, …

2.4.1.1 Zigzag uncertain variable This example is the shape of zigzag uncertain variables. Suppose we are talking about the tallness of five children, everyone shorter than 1 m is not called as a tall child, but all children with tallness between 1 and 1.20 m can be called tall. The children with tallness between 1.20 and 1.30 are called almost tall and the rest more than 1.30 m are called completely tall. The children with tallness 1.20–1.30 are almost tall and the rest more than 1.30 m are completely tall (Fig. 2.5). 8 0, t1 > > > t  40 > > < , 1 < t  1:20 30 xðtÞ ¼ t  60 > > > , 1:20 < t  1:30 > > : 10 1, 1:30 < t

18

Soft Numerical Computing in Uncertain Dynamic Systems

0.8

x

x(t) 0.6

0.4

0.2

0 –0.4

Fig. 2.4

–0.2

0

0.2

t 0.4

0.6

0.8

1

Uncertain variable.

1

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

Fig. 2.5

The set of tall children as an uncertain variable.

2.4.2

EXPERIMENTAL

1.2

1.4

UNCERTAIN VARIABLES

The following membership function is for an experimental uncertain variable with the following figure. 8 t  t1 < 0, xðtÞ ¼ θ, t ¼ ti + θðti + 1  ti Þ : 1, tn < t where 1  i  n  1, 0 < θ  1 (Fig. 2.6).

Uncertain sets

19

x(t) 1

q

0 t1

ti

ti+1

tn

Fig. 2.6 An experimental uncertain variable.

1 0.8 0.6

1

x(t) = 0.4

1 + exp(

p (e – t) 3d

)

e = 1, d = 0.5

0.2 0 –0.4 –0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Fig. 2.7 An experimental uncertain variable.

Or another case of an experimental uncertain variable (Fig. 2.7): xðtÞ ¼

1   , e, δR + xR π ðe  t Þ 1 + exp pffiffiffi 3δ

Note. We now have enough information about the uncertainty and uncertain sets and variables. One of the uncertainty cases is fuzzy sets. Like other uncertain sets, the fuzzy set is a set of ordered pairs (the member and its membership). It is a membership function as well and its structures and forms are the same as the structures and forms of uncertain sets (see Figs. 2.1–2.3).

2.4.3

MEMBERSHIP

FUNCTION

For a fuzzy set (uncertain set), A, the membership of its elements, x  A, has a membership degree, A(x)  [0, 1]. If the degree is 0, the member does not belong to the set and if the degree is 1, the member belongs to the set completely.

20

Soft Numerical Computing in Uncertain Dynamic Systems

This is why any real set is a special case of an uncertain set and has the characteristic degree and function. In this case, the range of the membership degree is {0, 1}. It is obvious than the set A is a fuzzy set and A ¼ fðx, AðxÞÞ j AðxÞ½0, 1, xAg This set is called the membership function. In this concept, if x belongs to the uncertain set with the membership degree A(x), at the same time it does not belong to the set with 1  A(x) membership degree. In the following section, we discuss fuzzy numbers and their properties.

2.5

Fuzzy numbers and their properties

First of all, we should know why we need a fuzzy number and why it is important: because we need to make computations and rank the fuzzy sets. This is why fuzzy sets must have some additional properties. We shall now explain their graphical and mathematical forms and computations. First of all, we should define a fuzzy set as a fuzzy number.

2.5.1

DEFINITION

OF A FUZZY NUMBER

A fuzzy membership function A : R ! [0, 1] is called a fuzzy number if it has the following conditions: 1. A is normal. This means there is at least a real member x0 such that A(x0) ¼ 1. 2. A is fuzzy convex. This means for two arbitrary real points x1, x2 and λ  [0, 1], we have: Aðλx1 + ð1  λÞx2 Þ  min fAðx1 Þ, Aðx2 Þg 3. A, is upper-semicontinuous. This means that if we increase its value at a certain point x0 to f(x0) + E (for some positive constant E), then the result is uppersemicontinuous; if we decrease its value to f(x0)  E, then the result is lowersemicontinuous. 4. The closure of the set Supp(A) ¼ {x  R j A(x) > 0}, as a support set, is a compact set. As an example, the triangular, trapezoidal fuzzy sets are fuzzy numbers. The definition of a fuzzy number can be defined as other forms: parametric and level-wise.

2.5.2

LEVEL-WISE

FORM OF A FUZZY NUMBER

The level-wise membership function is, in fact, an inverse function of the membership function that proposes an interval-valued function. In fact, any level in the

Uncertain sets

21

r 1 A(x)

r – level

0 Al (r) = a

Au (r) = b

A–1(r)

xÎR

Fig. 2.8 Level-wise form of a fuzzy set.

vertical axis gives us an interval in the horizontal axis. For example, consider one of the following triangular membership functions (Fig. 2.8). In this figure, all real numbers in the interval [a, b] have a degree of membership greater than or equal to the value of “r-level” in the fuzzy set A, i.e., 8x½a, b, AðxÞ  r, 0  r  1 Then the r-cut or r-level set of the membership function can be defined as follows: A1 ðr Þ ¼ fxRj AðxÞ  r g ¼ ½a, b ¼ ½Al ðr Þ, Au ðr Þ≔A½r , 0  r  1 Based on the inequality property in the set, A[r], A(x) r, it is clear that the membership function, A(x), can be obtained by: AðxÞ ¼ supf0  r  1 jxA1 ðr Þg, xR This means that there is a one-to-one map between two functions: the membership function A1(r) and the level-wise membership function A(x). Fig. 2.9 shows that for each interval there is a degree or level, and vice versa. In fact, it can be claimed that: [ Domain of AðxÞ ¼ ½Al ðr Þ, Au ðr Þ 0r1

Now, in general, a fuzzy set A in level-wise form can be shown as follows: A½r  ¼ ½Al ðr Þ, Au ðr Þ, 0  r  1 Another form of the definition in Section 2.5.1 can be defined as follows.

22

Soft Numerical Computing in Uncertain Dynamic Systems

rn–2

rn–1

rn

Al (rn)

Au (rn)

r3

r2 r1 Al (r1)

Au (r1)

Fig. 2.9

One-to-one corresponding.

2.5.3

DEFINITION

OF A FUZZY NUMBER IN LEVEL-WISE FORM

A fuzzy membership function A : R ! [0, 1] is called a fuzzy number if its level-wise form A[r] ¼ [Al(r), Au(r)] is a compact interval for any 0  r  1. Another form of Definition 2.5.3 is called a stacking theorem and can be defined as follows.

2.5.4

DEFINITION

OF A FUZZY NUMBER IN LEVEL-WISE FORM

Sufficient and necessary conditions for A1(r) ¼ A[r] to be a level-wise membership function of a fuzzy number are as follows (Fig. 2.10): i. (Nesting property) For any two r-levels, r1, r2 If r1  r2 then A[r1]  A[r2] r – levels 1 A(x) r rn

r2 r1

0 A[r] A[rn] A[r2] A[r1]

Fig. 2.10

Two conditions in Definition in Section 2.5.4.

Uncertain sets

23

ii. For any monotone increasing sequence of levels, 0 < r1 < r2 < … < rn < 1, ðif Þ frn gn ↗r,then A½rn  ! A½r, for any 0  r  1

2.5.4.1 A singleton fuzzy number A real number “a” is called a singleton fuzzy number if: a½r  ¼ ½al ðr Þ, au ðr Þ ¼ ½a, a It means in the membership function the membership degree at “a” is 1 and at other values is zero. al(r) ¼ au(r) ¼ a. Based on this definition, we have the same for fuzzy zero number or origin and al(r) ¼ au(r) ¼ 0.

2.5.5

DEFINITION

OF A FUZZY NUMBER IN PARAMETRIC FORM

Any fuzzy number, A, has the parametric form A(r) ¼ (Al(r), Au(r)) for any 0  r  1, if and only if: i. Al(r)  Au(r); ii. Al(r) is an increasing and left continuous function on (0, 1] and right continuous at 0 with respect to r; and iii. Au(r) is a decreasing and left continuous function on (0, 1] and right continuous at 0 with respect to r. Note that, in items (ii) and (iii), both functions can be bounded. In both forms of fuzzy numbers—level-wise form and parametric form—both functions, lower, Al(r), and upper, Au(r), are the same. But the differences can be listed as follows: 1. In level-wise form, the values of both functions for any arbitrary but fixed r are real numbers. But in parametric form, they take the role of a function with respect to r. 2. In level-wise form, the level is an interval for any arbitrary but fixed r. However, in parametric form, it is a couple of functions with respect to r. Now we are going to introduce other forms of a fuzzy number in linear, nonlinear cases. The following figure shows that it does not matter which fuzzy number we consider for analyzing. Indeed, a membership function that corresponds to a fuzzy number is a piece-wise function. For example:

24

Soft Numerical Computing in Uncertain Dynamic Systems 8 0x1 < x, A1 ðxÞ ¼ 2  x, 1  x  2 : 0, otherwise  1  jxj,  1  x  1 A2 ðxÞ ¼ 0, otherwise  jxj,  1  x  1 A3 ð x Þ ¼ 0, otherwise  pffiffiffi x, 1x1 A4 ðxÞ ¼ 1, otherwise

In general form, the function can be shown 8 x  a > , L > > ba > < 1,   AðxÞ ¼ dx > , R > > > : dc 0,

as follows: axb bxc cxd otherwise

where L, R : [0, 1] ! [0, 1] are two nondecreasing shape functions and Rð0Þ ¼ Lð0Þ ¼ 0, Rð1Þ ¼ Lð1Þ ¼ 1: To obtain the level-wise or parametric forms of the fuzzy number in the general form, the relations are: Al ðr Þ ¼ a + ðb  aÞL1 ðr Þ, Au ðr Þ ¼ d  ðd  cÞR1 ðr Þ, r½0, 1 Clearly, if the functions L and R are linear, then we will have trapezoidal and triangular membership functions as fuzzy numbers. If they are nonlinear, then they will appear as curves that look trapezoidal or triangular. The general form of a fuzzy number in linear or nonlinear cases can be shown in Fig. 2.11.

2.5.6

NONLINEAR

FUZZY NUMBER

For instance, in the following cases, the fuzzy numbers do not have any linear lower and upper functions. LðxÞ ¼

1 1 , RðxÞ ¼ , α ¼ 3, β ¼ 2, m ¼ 4 2 1+x 1 + 2|x|

Uncertain sets

25

A(x) 1 L

x – m1 + a a

R

m2 – b + x b

r-level A = (m1, m2,a, b) A = (a, b, c,d) 0

a a

b := m1

c := m2

Al (r)

Fig. 2.11

d

b

Au (r)

General form of a fuzzy number.

Then the membership function is: 8   4x 1 > > L ¼ >   , > > 3 4 x 2 > > > 1+ > < 3   Að x Þ ¼ x4 1 >   ¼ R > >  x  4 , 2 > >   > 1 + 2 > > 2  > : 0,

x4

4x otherwise

The figure of this membership function is as shown in Fig. 2.12. Generally, the r-level set of a fuzzy set A is:

A½r  ¼ ½Al ðr Þ, Au ðr Þ ¼ a + ðb  aÞL1 ðr Þ, d  ðd  cÞR1 ðr Þ Note that members of an r-level set as an interval are included in the membership function with membership degree or uncertain measure, r. A(x)

1 L(x) =

1 1 + x2

0

Fig. 2.12

Nonlinear fuzzy number.

R(x) =

4

1 1 + 2|x|

x

26

2.5.7

Soft Numerical Computing in Uncertain Dynamic Systems

TRAPEZOIDAL

FUZZY NUMBER

The general level-wise or parametric forms are: Al ðr Þ ¼ b  ðb  aÞL1 ðr Þ, Au ðr Þ ¼ c + ðd  cÞR1 ðr Þ, r½0, 1 where, in the linear case, the left and right functions can be replaced by linear functions like R(r) ¼ L(r) ¼ r or others. Now suppose that b  a ¼ α, d  c ¼ β, b ¼ m1, c ¼ m2 then the membership function will be: 8 m  x 1 > m1  α  x  m1 > > L  α , > > < x  m2 , m2  x  m2 + β AðxÞ ¼ R β > > > > 1, m1  x  m2 > : 0, otherwise where α, β are left and right spreads, and m1, m2 are cores of a trapezoidal fuzzy number. Al ðr Þ ¼ m1  αL1 ðr Þ, Au ðr Þ ¼ m2 + βR1 ðr Þ, r½0, 1 The formal formats to show this number can be written as: A ¼ ða, b, c, dÞ≔A ¼ ðm1 , m2 , α, βÞ≔A½r  ¼ ½Al ðr Þ, Au ðr Þ≔Aðr Þ ¼ ðAl ðr Þ, Au ðr ÞÞ Note that these formats are only for presenting the fuzzy number and each one has its own properties and calculations. The calculations on them will be explained in the next sections. The r-level set of a trapezoidal fuzzy set A ¼ (a, b, c, d) is: A½r  ¼ ½Al ðr Þ, Au ðr Þ ¼ ½b + ðb  aÞðr  1Þ, c + ðd  cÞð1  r Þ

2.5.8

TRIANGULAR

FUZZY NUMBER

The only difference of triangular fuzzy number from trapezoidal is the number of the cores in membership functions. In the trapezoidal, if the cores are the same, i.e., m1 ¼ m2 ¼ m, then it will be a triangular fuzzy number. The general level-wise or parametric forms are: Al ðr Þ ¼ b  ðb  aÞL1 ðr Þ, Au ðr Þ ¼ b + ðc  bÞR1 ðr Þ, r½0, 1

Uncertain sets

27

where in the linear case, the left and right functions can be replaced by linear functions like R(r) ¼ L(r) ¼ r or others. Now suppose that b  a ¼ α, c  b ¼ β, b ¼ m then the membership function will be as follows: 8 x  m + α > L , mαxm > > <  α  Að x Þ ¼ R m + β  x , mxm+β > > β > : 0, otherwise where α, β are left and right spreads, and m is the core of a triangular fuzzy number. Al ðr Þ ¼ m  αL1 ðr Þ, Au ðr Þ ¼ m + βR1 ðr Þ, r½0, 1 The formal formats to show this number can be written as follows: A ¼ ða, b, cÞ≔A ¼ ðm, α, βÞ≔A½r  ¼ ½Al ðr Þ, Au ðr Þ≔Aðr Þ ¼ ðAl ðr Þ, Au ðr ÞÞ The r-level set of a triangular fuzzy set A ¼ (a, b, c) is: A½r  ¼ ½Al ðr Þ, Au ðr Þ ¼ ½b + ðb  aÞðr  1Þ, b + ðc  bÞð1  r Þ For both formats of the r-level sets of triangular and trapezoidal fuzzy numbers, the lower and upper functions can be obtained practically as follows. Consider the following figure of fuzzy set A. First, the line segment between two points A and B is defined as: y  yA yB  yA 1 ¼ ) y ¼ ðx  m + αÞ x  xA xB  xA α Then, after finding the equation of the line, we have this system: 1 y ¼ ðx  m + αÞ & y ¼ r α So: 1 r ¼ ðx  m + αÞ α The inverse is a reflective function on x ¼ r and it is as follows: A l ð r Þ ¼ x ¼ m + α ð r  1Þ The same procedure will be true for points B and C. Then it will be obtained as follows: y  y B yC  yB 1 ¼ ) y ¼ ðm  x Þ + 1 x  x B xC  xB β

28

Soft Numerical Computing in Uncertain Dynamic Systems

B(m, 1)

y=r

Al (r)

m

Au (r) C(m – b,0)

A(m – a,0)

Fig. 2.13

Fuzzy number in triangular form.

Again we will have: 1 r ¼ ðm  x Þ + 1 β and the inverse function will be Au(r) (Fig. 2.13): Au ð r Þ ¼ m + β ð 1  r Þ

2.5.9

OPERATIONS

ON LEVEL-WISE FORM OF FUZZY NUMBERS

In this section, the main calculations on fuzzy sets in parametric or level-wise form are going to be defined and discussed. Since the concept of the difference is different and needs more attention, so this operation will be discussed more, because here the subtraction has the meaning of difference between two sets or two functions. Before the discussion on the operations, we need some explanations. It should be noted that the fuzziness will be growing under these operators. It means, for two arbitrary fuzzy numbers A and B, and a continuous measurable function like f  { , ⊙ , }, the fuzziness of A and fuzziness of B are less than or equal to the fuzziness of f(A, B). One of the concepts of the fuzziness is the diameter of a fuzzy number. The diameter of an interval in level-wise form of a fuzzy number can be known as fuzziness of A in any level of r. FuzzðAr Þ ¼ diamð½Al ðr Þ, Au ðr ÞÞ ¼ Au ðr Þ  Al ðr Þ, 0  r  1 Now it is proven that:   FuzzðAr Þ  Fuzz f ðA, BÞr & FuzzðBr Þ  Fuzz f ðA, BÞr

Uncertain sets

29

and also: C ¼ f ðA, BÞ ) C½r  ¼ f ðA½r , B½r Þ This means that if the function f is continuous and measurable, then: if A B ¼ C then A½r  + B½r  ¼ C½r , if A⊙B ¼ C then A½r  + B½r  ¼ C½r , if A B ¼ C then A½r  + B½r  ¼ C½r : Now we can discuss the operators separately.

2.5.9.1 Summation For any two arbitrary fuzzy numbers A and B and any arbitrary but fixed 0  r  1, if A B ¼ C, we have: C½r  ¼ ½Cl ðr Þ, Cu ðr Þ ¼ A½r  + B½r  ¼ ½Al ðr Þ, Au ðr Þ + ½Bl ðr Þ, Bu ðr Þ Then: Cl ðr Þ ¼ Al ðr Þ + Bl ðr Þ, Cu ðr Þ ¼ Au ðr Þ + Bu ðr Þ 2.5.9.1.1 Example Suppose that:

A ¼ ½r  1, 1  r , B ¼ r, 2  r 2 , To compute the summation A B ¼ C we need the following: Al ðr Þ ¼ r  1,Au ðr Þ ¼ 1  r, Bl ðr Þ ¼ r,Bu ðr Þ ¼ 2  r 2 So, Cl(r) ¼ 2r  1, Cu(r) ¼ 3  r  r2. Fig. 2.14 shows the summation of two fuzzy numbers in the example.

r 1

Au (r) + Bu (r)

Al (r) + Bl (r)

0.5 Al (r)

Au (r)

Bl (r)

Bu (r)

0 –1

Fig. 2.14

–0.5

Summation.

0

0.5

1

1.5

2

2.5

3

30

Soft Numerical Computing in Uncertain Dynamic Systems

As can be seen, we have: Au ðr Þ  Al ðr Þ ¼ FuzzðAr Þ  FuzzðA BÞr ¼ Cu ðr Þ  Cl ðr Þ and Bu ðr Þ  Bl ðr Þ ¼ FuzzðBr Þ  FuzzðA BÞr ¼ Cu ðr Þ  Cl ðr Þ for all levels.

2.5.9.2 Multiplication For any two arbitrary fuzzy numbers A and B and any arbitrary but fixed 0  r  1, if A ⊙ B ¼ C, we have: C½r  ¼ ½Cl ðr Þ, Cu ðr Þ ¼ A½r   B½r  ¼ ½Al ðr Þ, Au ðr Þ  ½Bl ðr Þ, Bu ðr Þ Then: Cl ðr Þ ¼ min fAl ðr Þ  Bl ðr Þ, Al ðr Þ  Bu ðr Þ, Au ðr Þ  Bl ðr Þ, Au ðr Þ  Bu ðr Þg, Cu ðr Þ ¼ max fAl ðr Þ  Bl ðr Þ, Al ðr Þ  Bu ðr Þ, Au ðr Þ  Bl ðr Þ, Au ðr Þ  Bu ðr Þg In the previous example:

   Cl ðr Þ ¼ min ðr  1Þ  r, ðr  1Þ  2  r 2 , ð1  r Þ  r, ð1  r Þ  2  r 2 ,

   Cu ðr Þ ¼ max ðr  1Þ  r, ðr  1Þ  2  r 2 , ð1  r Þ  r, ð1  r Þ  2  r 2 We can see that:   C l ð r Þ ¼ ð r  1 Þ 2  r 2 , Cu ð r Þ ¼ ð 1  r Þ 2  r 2 Fig. 2.15 shows the multiplication of two fuzzy numbers in the example. For more illustration, see Allahviranloo (2020).

2.5.9.3 Difference The difference of two fuzzy numbers is indeed the difference of two membership functions. Two different differences in the sense of standard and nonstandard cases will be discussed in level-wise form. r

1 Bl (r)

Cl (r)

Bu (r) Cu (r)

0.5

Au (r)

Al (r) 0 –2

Fig. 2.15

–1.5

Multiplication.

–1

–0.5

0

0.5

1

1.5

2

Uncertain sets

31

First of all, we should consider the multiplying a membership function by a scalar in level-wise form. Suppose that λ  R is a scalar. Then in triple form of fuzzy number:  ð λ a, λ b, λ cÞ, λ  0 λ ⊙ A ¼ λ  ða, b, cÞ ¼ ð λ c, λ b, λ aÞ, λ < 0 and in level-wise form of fuzzy number:  ½λAl ðr Þ, λAu ðr Þ, λ  0 λA½r  ¼ ½λAu ðr Þ, λAl ðr Þ, λ < 0 For any 0  r  1. The concept of scalar multiplication is the same as the multiplication of the scalar to each member of the interval. It means: A½r  ¼ ½Al ðr Þ, Au ðr Þ ¼ fzt j zt ¼ Al ðr Þ + tðAu ðr Þ  Al ðr ÞÞ,0  t  1g For any 0  r  1. So, in r-level:  λA½r  ¼ fλzt j 0  t  1g ¼

½λAl ðr Þ, λAu ðr Þ, λ  0 ½λAu ðr Þ, λAl ðr Þ, λ < 0

For instance, if we consider one of the previous fuzzy numbers like:

A½r  ¼ ½Al ðr Þ, Au ðr Þ ¼ r, 2  r 2 and λ ¼  1, then:



ð1ÞA½r  ¼ ½Au ðr Þ,  Al ðr Þ ¼ r2  2,  r

For more illustration, see Fig. 2.16. For using this scalar multiplication in the definition of the difference of two fuzzy numbers, let A[r] ¼ [Al(r), Au(r)] and B[r] ¼ [B(r), Bu(r)] be two fuzzy numbers in level-wise form. In this case the difference is defined as follows:

r

(–1)A

A

r – level

–Au(r)

Fig. 2.16

–Al(r)

Multiplication of (1).

Al(r)

Au(r)

32

Soft Numerical Computing in Uncertain Dynamic Systems A½r   B½r  ¼ A½r  + ð1ÞB½r  ¼ ¼ ½Al ðr Þ, Au ðr Þ + ð1Þ ½Bl ðr Þ, Bu ðr Þ ¼ ½Al ðr Þ, Au ðr Þ + ½Bu ðr Þ,  Bl ðr Þ ¼ ½Al ðr Þ  Bu ðr Þ, Bu ðr Þ  Al ðr Þ

Or in the format of convex combination, for any arbitrary but fixed 0  r  1:

 A½r  ¼ z0t  z0t ¼ Al ðr Þ + tðAu ðr Þ  Al ðr ÞÞ, 0  t  1g

 B½r  ¼ z00t  z00t ¼ Bl ðr Þ + tðBu ðr Þ  Bl ðr ÞÞ,0  t  1g 

A½r   B½r  ¼ z0t  z00t  0  t  1g ¼ ½Al ðr Þ  Bu ðr Þ, Bu ðr Þ  Al ðr Þ In this definition: A ð1Þ ⊙ A 6¼ 0, because, based on the definition, the result is a symmetric interval centered at zero and it is always a nonzero interval. A½r   A½r  ¼ ½Al ðr Þ  Au ðr Þ, Au ðr Þ  Al ðr Þ 6¼ 0 ¼ ½0, 0 Note that the symmetric interval centered at zero is called a zero interval. This comes from the concept of equivalency class. If we consider an equivalency class of zero as a set of all symmetric intervals centered at zero, then all of the members of the class are known as a zero interval. Moreover:  FuzzðAr Þ  Fuzz ðλ ⊙ AÞr , 0  r  1:

2.5.9.4 Hukuhara difference Suppose that A and B are two fuzzy numbers in level-wise form. The Hukuhara difference of A H B is defined as follows: 9C;A H B ¼ C , A ¼ B C It is clear that the existence of the difference is conditional and depends on the existence of fuzzy number C. Note. For the existence of H-difference, A, B, and C must all be fuzzy numbers. This means that if the fuzzy set B can be transformed by C then it will fall into A. Now considering A ¼ B C, and the level-wise form of both sides of the equation, we have: A½r  ¼ B½r  + C½r 

Uncertain sets

33

½Al ðr Þ, Au ðr Þ ¼ ½Bl ðr Þ, Bu ðr Þ + ½Cl ðr Þ, Cu ðr Þ ¼ ½Bl ðr Þ + Cl ðr Þ, Bu ðr Þ + Cu ðr Þ Al ðr Þ ¼ Bl ðr Þ + Cl ðr Þ,Au ðr Þ ¼ Bu ðr Þ + Cu ðr Þ Finally: Cl ðr Þ ¼ Al ðr Þ  Bl ðr Þ,Cu ðr Þ ¼ Au ðr Þ  Bu ðr Þ The level-wise form of the Hukuhara difference, or H-difference, is defined as subtractions of two endpoints of two intervals, respectively.

ðA H BÞl ðr Þ, ðA H BÞu ðr Þ ¼ ½Cl ðr Þ, Cu ðr Þ ¼ ½Al ðr Þ  Bl ðr Þ, Au ðr Þ  Bu ðr Þ Note that the difference: A H B 6¼ A ð1Þ⊙ B, because in level-wise form the differences between intervals in both sides are not the same. ðA H BÞ½r  ¼ ½Al ðr Þ  Bl ðr Þ, Au ðr Þ  Bu ðr Þ 6¼ ½Al ðr Þ  Bu ðr Þ, Bu ðr Þ  Al ðr Þ ¼ ðA ð1Þ ⊙ BÞ½r  2.5.9.4.1 Example Consider the following two fuzzy numbers in parametric forms: A½r  ¼ ½Al ðr Þ, Au ðr Þ ¼ ½2r, 4  2r ,B½r  ¼ ½Bl ðr Þ, Bu ðr Þ ¼ ½r  1, 1  r , Now to obtain A H B ¼ C: Cl ðr Þ ¼ r + 1, Cu ðr Þ ¼ 3  r So the Hukuhara difference in parametric form is: C½r  ¼ ½r + 1, 3  r  Fig. 2.17 shows the H-difference. As can be seen in the figure, and based on the definition of H-difference: r

1 Al (r)

Bu (r)

Bl (r)

Au (r)

Cl (r) –1

Fig. 2.17

0

A H B.

0

1

Cu (r) 2

3

4

34

Soft Numerical Computing in Uncertain Dynamic Systems

Au ðr Þ  Al ðr Þ ¼ FuzzðAr Þ  FuzzðA H BÞr ¼ Cu ðr Þ  Cl ðr Þ and Bu ðr Þ  Bl ðr Þ ¼ FuzzðBr Þ  Au ðr Þ  Al ðr Þ ¼ FuzzðAr Þ because the difference C is a shift for extending of B to be into A. We shall now find some sufficient conditions for the existence of H-difference. To this purpose, let us consider: Ar ¼ ðA1,r , A2,r , A3, r Þ, Br ¼ ðB1,r , B2, r , B3, r Þ, Cr ¼ ðC1, r , C2,r , C3, r Þ where: A1, r  A2,r  A3, r , B1,r  B2, r  B3, r , and B1, r  B2, r  B3,r are representation of fuzzy numbers in triple forms for each level r. This means in each level the intervals are satisfying the conditions of a real interval. Lemma The sufficient condition for the existence of the H-difference A H B ¼ C is: FuzzðBr Þ ¼ B3, r  B1, r  min fA2, r  A1, r , A3, r  A2,r g To show A ¼ B C with condition C1, r  C2, r  C3, r, our assertion for the existence is only proving C1, r  C2, r  C3, r or A1, r  B1, r  A2, r  B2, r  A3, r  B3, r. Because: A ¼ B C⟺A1, r ¼ B1, r + C1, r , A2,r ¼ B2, r + C2,r , A3, r ¼ B3,r + C3, r To show A1, r  B1, r  A3, r  B3, r it is enough to show B3, r  B1, r  A3, r  A1, r. In the first case suppose that min{A2, r  A1, r, A3, r  A2, r} ¼ A2, r  A1, r > 0. Now we have B3, r  B1, r  A2, r  A1, r  A3, r  A1, r and the proof is completed. In the second case, suppose that min{A2, r  A1, r, A3, r  A2, r} ¼ A3, r  A2, r > 0. Now we have B3, r  B1, r  A3, r  A2, r  A3, r  A1, r and the proof is also completed. To show A1, r  B1, r  A2, r  B2, r it is enough to show B2, r  B1, r  A2, r  A1, r. In the first case, suppose that min{A2, r  A1, r, A3, r  A2, r} ¼ A2, r  A1, r > 0. Now we have B2, r  B1, r  B3, r  B1, r  A2, r  A1, r and the proof is completed. In the second case, suppose that min{A2, r  A1, r, A3, r  A2, r} ¼ A3, r  A2, r > 0. Now we have B2, r  B1, r  B3, r  B1, r  A3, r  A2, r  A2, r  A1, r and the proof is also completed. To show A2, r  B2, r  A3, r  B3, r it is enough to show B3, r  B2, r  A3, r  A2, r. In the first case, suppose that min{A2, r  A1, r, A3, r  A2, r} ¼ A2, r  A1, r > 0. Now we have B3, r  B2, r  B3, r  B1, r  A2, r  A1, r  A3, r  A2, r and the proof is completed. In the second case, suppose that min{A2, r  A1, r, A3, r  A2, r} ¼ A3, r  A2, r > 0. Now we have B3, r  B2, r  B3, r  B1, r  A3, r  A2, r and the proof is also completed.

Uncertain sets

35

2.5.9.4.2 Example As we mentioned, the existence of the difference is conditional. Now in this example, we will see that it does not always exist. Suppose that in A H B: FuzzðBr Þ ¼ B3, r  B1, r > min fA2, r  A1,r , A3,r  A2, r g For instance, A[r] ¼ [1  r, r  1] and B[r] ¼ [2  r, r  2], now: ðA H BÞ½r  ¼ ½Al ðr Þ  Bl ðr Þ, Au ðr Þ  Bu ðr Þ ¼ ½1,  3 As we can see, it is not an interval and for any r. So the difference does not exist.

2.5.9.5 Generalized Hukuhara difference In this case, we can define the difference in another way. Suppose that we want to try B H A ¼ (1)C and the difference C may exist. Considering the level-wise forms of two sides: ðB H AÞ½r  ¼ ½Bl ðr Þ  Al ðr Þ, Bu ðr Þ  Au ðr Þ ¼ ðð1ÞCÞ½r  ¼ ½Cu ðr Þ,  Cl ðr Þ we have: Bl ðr Þ  Al ðr Þ ¼ Cu ðr Þ, Bu ðr Þ  Au ðr Þ ¼ Cl ðr Þ or: Al ðr Þ  Bl ðr Þ ¼ Cu ðr Þ, Au ðr Þ  Bu ðr Þ ¼ Cl ðr Þ In general:

ðB H AÞl ðr Þ, ðB H AÞu ðr Þ ¼ ½Cl ðr Þ, Cu ðr Þ ¼ ½Au ðr Þ  Bu ðr Þ, Al ðr Þ  Bl ðr Þ

:¼ ð1Þ ðA H BÞl ðr Þ; ðA H BÞu ðr Þ Now to define an almost right definition for the difference, we have two cases to consider. 8 A ¼ B C < iÞ or A gH B ¼ C , : iiÞ B ¼ A ð1ÞC The generalized Hukuhara difference is defined in two cases. If the case (i) exists, so there is no need to consider the case (ii). Otherwise we will need the second case. The relation between the two cases can be explained as follows: •

In case (i): A H B ¼ C

36 •

Soft Numerical Computing in Uncertain Dynamic Systems

In case (ii): B H A ¼ ð1ÞC

The relationship is: 

  A gH B i ½r  :¼ ð1Þ A gH B ii ½r 

In the case that both exist, then C ¼ (1)C and it is concluded that both types of the difference are the same and equal (Abbasi and Allahviranloo, 2018). 2.5.9.5.1 The level-wise form of generalized difference As we found, the level-wise form in case (i) is: h i  A gH B l ðr Þ, A gH B u ðr Þ ¼ ½Cl ðr Þ, Cu ðr Þ ¼ ½Al ðr Þ  Bl ðr Þ, Au ðr Þ  Bu ðr Þ and in case (ii) it is as follows: h i  B gH A l ðr Þ, B gH A u ðr Þ ¼ ½Cl ðr Þ, Cu ðr Þ ¼ ½Au ðr Þ  Bu ðr Þ, Al ðr Þ  Bl ðr Þ So to define the endpoints of the difference: Cl ðr Þ ¼ min fAl ðr Þ  Bl ðr Þ, Au ðr Þ  Bu ðr Þg Cu ðr Þ ¼ max fAl ðr Þ  Bl ðr Þ, Au ðr Þ  Bu ðr Þg To show two cases at the same time, we use gH-difference notation and define it in the following form:  A gH B ½r  ¼ ¼ ½ min fAl ðr Þ  Bl ðr Þ, Au ðr Þ  Bu ðr Þg, max fAl ðr Þ  Bl ðr Þ, Au ðr Þ  Bu ðr Þg 2.5.9.5.2 Some properties of gH-difference It should be noted that all of the following properties can be proved in level-wise form easily. 1. 2. 3. 4. 5. 6.

If the gH-difference exists it is unique. A gHA ¼ 0. If A gHB exists in case (i) then B gHA exists in case (ii) and vice versa. In both cases (A B) gHB ¼ A. (It is easy to show in level-wise form.) If A gHB and B gHA exist then 0 gH(A gHB) ¼ B gHA. If A gHB ¼ B gHA ¼ C if and only if C ¼  C and A ¼ B.

Uncertain sets

37

The difference even in the gH-difference case may not exist. It can be, for example, that the gH-difference of two fuzzy numbers is not always a fuzzy number. 2.5.9.5.3 Example This example shows that the generalized Hukuhara difference does not exist for each arbitrary level of difference. Suppose that one of the numbers is triangular and another one is in trapezoidal forms. A ¼ (0, 2, 4) or in parametric form A[r] ¼ [2r, 4  2r] and B ¼ (0, 1, 2, 3) or in parametric form B[r] ¼ [r, 3  r]. In case (i):  A gH B ½r  ¼ ½r, 1  r  If r ¼ 1, the difference is as [1, 0], which is not an interval. In case (ii):  A gH B ½r  ¼ ½1  r, r  If r ¼ 0, the difference is as [1, 0] that is not an interval. So as we see the gHdifference does not exist for all r  [0, 1]. Note. In all methods in this book, we will suppose that the gH-difference always exists. Some other properties of gH-difference are now shown, considering one of the ordering methods of fuzzy numbers. Generally, there are many methods to order or rank the fuzzy numbers. The most useful one is partial ordering, in the case of level-wise form. To define this, consider two fuzzy numbers A and B.

2.5.9.6 Partial ordering For two fuzzy numbers A,BR , we call ≼ a partial order notation and A ≼ B if and only if Al(r)  Bl(r) and Au(r)  Bu(r). We also have the same definition for the strict inequality, A B if and only if Al(r) < Bl(r) and Au(r) < Bu(r). For any r  [0, 1]. 2.5.9.6.1 Some properties of partial ordering 1. If A ≼ B then  B ≼  A. 2. If A ≼ B and B ≼ A then A ¼ B. To prove the properties, we use the level-wise form and they are very clear. For instance, we prove the first property: A ≼ B if and only if Al(r)  Bl(r) and Au(r)  Bu(r)

38

Soft Numerical Computing in Uncertain Dynamic Systems

 B ≼  A if and only if  Bl(r)   Al(r) and  Bu(r)   Au(r) So the proof is completed. The second one is obtained in a similar way. 2.5.9.6.2 Absolute value of a fuzzy number The absolute value of a fuzzy number A is defined as follows:  A, A≽0 j Aj ¼ A, A 0 where 0 fuzzy number is called a singleton fuzzy zero number. 2.5.9.6.3 Some properties of partial ordering in gH-difference 1. If A ≼ B then A gHB ≼ 0. 2. If A ≽ B then A gHB ≽ 0. They are very easy to prove in level-wise form.

2.5.9.7 Approximately generalized Hukuhara difference In the case that the gH-difference does not exist or (A gHB)[r] do not define a fuzzy number for any r  [0, 1], we can use the nested property of the fuzzy numbers and define a proper fuzzy number as a difference. We call this approximately gH-difference, denoted by g, and it is defined in level-wise form as follows: ! [  A g B ½r  ¼ cl A gH B ½β , r½0, 1 βr

If the gH-difference (A gHB)[β] exists or defines a proper fuzzy number for any β  [0, 1], then (A gB)[r] is exactly the same as the gH-difference (A gHB)[r] and it is exactly the same as the (A HB)[r] Hukuhara difference, where: 



"

#

A g B ½r  ¼ inf min fAl ðβÞ  Bl ðβÞ, Au ðβÞ  Bu ðβÞ g, sup max fAl ðβÞ  Bl ðβÞ, Au ðβÞ  Bu ðβÞ g βr

βr

Proposition g.1 For any two fuzzy numbers, A, B  RF, the two of A gB and B gA exist for any r  [0, 1] and A gB ¼  (B gA) where:  A g B ½r  ¼ ½Dl ðr Þ, Du ðr Þ Dl ðr Þ ¼ inf ffAl ðβÞ  Bl ðβÞ j β  r g [ fAu ðβÞ  Bu ðβÞ j β  r gg Du ðr Þ ¼ supffAl ðβÞ  Bl ðβÞ j β  r g [ fAu ðβÞ  Bu ðβÞ j β  r gg

Uncertain sets

39

It should be noted that in the case of finite numbers of level or discretized levels, the abovementioned interval is the same as the gH-difference.  A g B ½r  ¼ ½Dl ðr Þ, Du ðr Þ Dl ðr Þ ¼ min fAl ðr Þ  Bl ðr Þ, Au ðr Þ  Bu ðr Þg Du ðr Þ ¼ max fAl ðr Þ  Bl ðr Þ, Au ðr Þ  Bu ðr Þg Proposition g.2 For any two fuzzy numbers, A, B  RF, A gB always exists or is a fuzzy number, because it satisfies the conditions of fuzzy numbers. inf min fAl ðβÞ  Bl ðβÞ, Au ðβÞ  Bu ðβÞ g  sup max fAl ðβÞ  Bl ðβÞ, Au ðβÞ  Bu ðβÞ g

βr

βr

1. inf βr min fAl ðβÞ  Bl ðβÞ, Au ðβÞ  Bu ðβÞ g is nondecreasing, left continuous, and bounded function for any r  [0, 1]. 2. supβr max fAl ðβÞ  Bl ðβÞ, Au ðβÞ  Bu ðβÞ g is nonincreasing, left continuous, and bounded function for any r  [0, 1]. 2.5.9.7.1 Some properties of g-difference For any two fuzzy numbers A, B  RF: 1. 2. 3. 4. 5.

A gB ¼ A gHB subject to A gHB exists. A gA ¼ 0. (A B) gB ¼ A. 0 g(A gB) ¼ B gA. A gB ¼ B gA ¼ C if and only if C ¼  C, the immediate conclusion is C ¼ 0 and A ¼ B. In conclusion: A g B ¼ B g A , A ¼ B

All the properties can be proved very easily based on the definition of the g-difference in level-wise form. Let us consider some examples when the gH-difference does not exist, while the g-difference exists. 2.5.9.7.2 Example Consider two trapezoidal fuzzy numbers as follows: A ¼ ð0, 2, 2, 4Þ, B ¼ ð0, 1, 2, 3Þ where: Al ðβÞ ¼ 2β, Au ðβÞ ¼ 4  2β, Bl ðβÞ ¼ β, Bu ðβÞ ¼ 3  β

40

Soft Numerical Computing in Uncertain Dynamic Systems

and: inf min fAl ðβÞ  Bl ðβÞ, Au ðβÞ  Bu ðβÞ g ¼ inf min fβ, 1  β g ¼ 0

βr

βr

sup max fAl ðβÞ  Bl ðβÞ, Au ðβÞ  Bu ðβÞ g ¼ sup max fβ, 1  β g ¼ 1 βr

βr

So the g-difference is: 

A g B ½r  ¼ ½0, 1

The corresponding trapezoidal fuzzy number of g-difference is (a, b, c, d) ¼ (0, 0, 1, 1). In the next example, we will see that the gH-difference does not exist. 2.5.9.7.3 Example Consider two trapezoidal fuzzy numbers as follows: A ¼ ð2, 3, 5, 6Þ, B ¼ ð0, 4, 4, 8Þ where: Al ðβÞ ¼ 2 + β, Au ðβÞ ¼ 6  β, Bl ðβÞ ¼ 4β, Bu ðβÞ ¼ 8  4β and: inf min fAl ðβÞ  Bl ðβÞ; Au ðβÞ  Bu ðβÞ g ¼ inf min f2  3β;2 + 3β g

βr

βr

sup max fAl ðβÞ  Bl ðβÞ; Au ðβÞ  Bu ðβÞ g ¼ sup max f2  3β; 2 + 3βg βr

βr

Based on the definition of gH-difference, we cannot claim that: min f2  3r,  2 + 3r g  max f2  3r,  2 + 3r g for any r  [0, 1]. Because: Au ðr Þ  Bu ðr Þ ¼ ðAl ðr Þ  Bl ðr ÞÞ and: If Al(r)  Bl(r) 0 then Au(r)  Bu(r)  Al(r)  Bl(r) then min fAl ðr Þ  Bl ðr Þ, Au ðr Þ  Bu ðr Þ g ¼ Au ðr Þ  Bu ðr Þ max fAl ðr Þ  Bl ðr Þ, Au ðr Þ  Bu ðr Þ g ¼ Al ðr Þ  Bl ðr Þ If Al(r)  Bl(r) < 0 then Au(r)  Bu(r) > Al(r)  Bl(r) then min fAl ðr Þ  Bl ðr Þ, Au ðr Þ  Bu ðr Þ g ¼ Al ðr Þ  Bl ðr Þ max fAl ðr Þ  Bl ðr Þ, Au ðr Þ  Bu ðr Þ g ¼ Au ðr Þ  Bu ðr Þ

Uncertain sets

41

Now here for 0  r  23 : min fAl ðr Þ  Bl ðr Þ, Au ðr Þ  Bu ðr Þ g ¼ 2 + 3r max fAl ðr Þ  Bl ðr Þ, Au ðr Þ  Bu ðr Þ g ¼ 2  3r and otherwise for 23 < r  1 : min fAl ðr Þ  Bl ðr Þ, Au ðr Þ  Bu ðr Þ g ¼ 2  3r max fAl ðr Þ  Bl ðr Þ, Au ðr Þ  Bu ðr Þ g ¼ 2 + 3r Then the gH-difference does not exist. However, the g-difference is as for 0  r  0.32:    100 100 A g B ½r ¼ r  2, r+2 32 32 and for 0.32 < r  1:

 A g B ½r  ¼ ½1, 1

2.5.9.8 Generalized division Now like generalized difference, for any two arbitrary fuzzy numbers, A and B, the generalized division can be defined as follows: 8 A ¼ B⊙C < iÞ A g B ¼ C , or : iiÞ B ¼ A⊙C1 It is clear that if both cases are true then A ¼ B ⊙ C ¼ A ⊙ C1 ⊙ C and in this case the only option is C1 ⊙ C ¼ C ⊙ C1 ¼ {1}, where 1 is a singleton fuzzy number or a real scalar, and C is also a nonzero real scalar. The generalized division in the level-wise or interval form can be defined as follows: 8 A½r  ¼ B½r   C½r  < iÞ A½r =B½r  ¼ C½r  , or : iiÞ B½r  ¼ A½r   C1 ½r  h i where C1 ½r ≔ Cu1ðrÞ , Cl1ðrÞ and does not contain zero for all r  [0, 1]. In case (i): ½Al ðr Þ, Au ðr Þ ¼ ½Bl ðr Þ, Bu ðr Þ  ½Cl ðr Þ, Cu ðr Þ and indeed: ½Cl ðr Þ, Cu ðr Þ ¼ ½Al ðr Þ, Au ðr Þ=½Bl ðr Þ, Bu ðr Þ

42

Soft Numerical Computing in Uncertain Dynamic Systems

then: 

 Al ðr Þ Al ðr Þ Au ðr Þ Au ðr Þ , , , , Cl ðr Þ ¼ min Bl ð r Þ B u ð r Þ B l ð r Þ B u ð r Þ   A l ð r Þ Al ð r Þ Au ð r Þ Au ð r Þ , , , Cu ðr Þ ¼ max B l ð r Þ B u ð r Þ Bl ð r Þ B u ð r Þ Subject to 0 62 [Bl(r), Bu(r)], it means Bu(r) < 0, or 0 < Bl(r) for all r  [0, 1]. In case (ii):   1 1 , ½Bl ðr Þ, Bu ðr Þ ¼ ½Al ðr Þ, Au ðr Þ  Cu ðr Þ Cl ðr Þ and indeed: 

 1 1 , ¼ ½Bl ðr Þ, Bu ðr Þ=½Al ðr Þ, Au ðr Þ Cu ðr Þ Cl ðr Þ

so:   1 Bl ð r Þ B l ð r Þ Bu ð r Þ Bu ð r Þ ¼ min , , , , Cu ðr Þ Al ðr Þ Au ðr Þ Al ðr Þ Au ðr Þ   1 Bl ð r Þ B l ð r Þ Bu ð r Þ Bu ð r Þ ¼ max , , , C l ðr Þ Al ð r Þ Au ð r Þ Al ð r Þ Au ð r Þ In fact, similar to the two cases of generalized difference, two endpoints changed the roles.   Al ð r Þ A l ð r Þ Au ð r Þ Au ð r Þ , , , , Cu ðr Þ ¼ min Bl ðr Þ Bu ðr Þ Bl ðr Þ Bu ðr Þ   Al ð r Þ A l ð r Þ Au ð r Þ Au ð r Þ , , , Cl ðr Þ ¼ max Bl ð r Þ Bu ð r Þ Bl ð r Þ Bu ð r Þ Remark The following results in both cases of division are easy to investigate. If Al(r) > 0, Bl(r) > 0 then Cl ðr Þ ¼ BAul ððrrÞÞ , Cu ðr Þ ¼ ABul ððrrÞÞ If Au(r) < 0, Bu(r) < 0 then Cl ðr Þ ¼ ABul ððrrÞÞ ,Cl ðr Þ ¼ BAul ððrrÞÞ In general, if: Cl ðr Þ ¼

Aj ð r Þ Ai ð r Þ , Cu ðr Þ ¼ ,i, jfl, ug Bj ð r Þ Bi ð r Þ

then the adequate condition to reach Cl(r)  Cu(r) is: Ai ðr Þ  Bj ðr Þ  Aj ðr Þ  Bi ðr Þ, i, jfl, ug for all r½0, 1

Uncertain sets

43

and Cl(r) and Cu(r) must be left as continuous increasing and decreasing functions, respectively. 2.5.9.8.1 Some properties of division It should be noted that all the following properties can be proved in level-wise form easily. Suppose that A and B are fuzzy numbers and 1 is {1}. 1. 2. 3. 4.

If 0 62 A[r], 8 r  [0, 1] then A gA ¼ 1. If 0 62 B[r], 8 r  [0, 1] then A ⊙ B gB ¼ A. If 0 62 A[r], 8 r  [0, 1] then 1 gA ¼ A1 and 1 gA1 ¼ A. If A B exists, then either B ⊙ (A gB) ¼ A or A ⊙ (A gB)1 ¼ B and both equalities hold if and only if A gB is a real number.

Note. In all methods in this book, we will suppose that the division always exists. 2.5.9.8.2 Examples Here we explain the generalized division based on its level-wise definition. Suppose that A[r] ¼ [1 + 2r, 7  4r] and B[r] ¼ [3 + r,  1  r], now according to case (i): A½r =B½r  ¼ C½r  , A½r  ¼ B½r   C½r 

1 + 2r and is C½r  ¼ 3 + r , 1r . Suppose that A[r] ¼ [7 + 2r  4  r] and B[r] ¼ [12 + 5r,  4  3r], now according to case (ii): 74r

A½r =B½r  ¼ C½r  , B½r  ¼ A½r   C1 ½r 7 + 2r 5 + r

and is C½r  ¼ 12 + 5r , 115r : In some cases the g-division does not exist. For instance, consider: A½r  ¼ ½1 + 0:5r, 5  3:5r  and B½r  ¼ ½4 + 2r,  1  r In this case, as with the g-difference, we have to define an approximate g-division as follows.

2.5.9.9 Approximately generalized division In the case that the g-division does not exist or (A AgB)[r] does not define a fuzzy number for any r  [0, 1], we can use the nested property of the fuzzy numbers and define a proper fuzzy number as a division. We call this approximately g-division, denoted by Ag, and it is defined in level-wise form as follows: ! [  A g B ½β , r½0, 1 A Ag B ½r  ¼ cl βr

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Soft Numerical Computing in Uncertain Dynamic Systems

If the g-division (A gB)[β] exists or defines a proper fuzzy number for any β  [0, 1], then (A AgB)[r] is exactly the same as the gH-difference (A gB)[r] subject to 0 62 B[r], for any r  [0, 1]. Suppose that A AgB ¼ C on a discrete partition 0  r0  r1  ⋯  rn  1 of [0, 1]. A discretize version of A AgB ¼ C is obtained using: A½ri =B½ri  ¼ W ½ri  ¼ ½Wl, i , Wu, i , 0  i  1 and Cl, n ¼ Wl, n, Cu, n ¼ Wu, n also for i ¼ n  1, …, 0 Cl, i ¼ min fCl, i + 1 , Wl, i g Cu, i ¼ max fCu, i + 1 , Wu, i g 2.5.9.9.1 Example Suppose that A[r] ¼ [1 + 0.5r, 5  3.5r] and B[r] ¼ [4 + 2r,  1  r]. The g-division 53:5r 1 + 0:5r A g B ¼ 4 does not define a proper fuzzy number + 2r , 1r 53:5r for any r  [0, 1]. However, using the Ag-division, the result is as A Ag B ¼ 4 + 2r ,  0:75 .

2.5.10

PIECE-WISE

MEMBERSHIP FUNCTION

Sometimes the lower and upper functions are piece-wise. For instance: A1 ðr Þ ¼ x1 + ðx2  x1 ÞL1 1 ðr Þ, r½x1 , x2 , A2 ðr Þ ¼ x2 + ðx3  x2 ÞL2 1 ðr Þ, r½x2 , x3 , A3 ðr Þ ¼ x4  ðx4  x3 ÞR1 1 ðr Þ, r½x3 , x4 , A4 ðr Þ ¼ x5  ðx5  x4 ÞR2 1 ðr Þ, r½x4 , x5 : More information about these computations and those operators are in (Bede and Gal, 2005; Stefanini, 2010) (Fig. 2.18). A(x) 1

A2(r) A3(r)

A1(r) A4(r) 0 x1

Fig. 2.18

x2

x3

Multifunctional membership function.

x4

x5

r

Uncertain sets

2.5.11

SOME

45

PROPERTIES OF ADDITION AND SCALAR PRODUCT ON FUZZY

NUMBERS

In this section, we need to consider and prove the following properties. Here it is supposed that the zero fuzzy number is a singleton fuzzy number. In general, a singleton fuzzy number is defined as follows.

2.5.11.1 Definition—singleton fuzzy number A fuzzy number like a is called a singleton fuzzy number if the membership degree of a is one and the membership degrees for the other members are zero. See the following figure. 1

0

a

Singleton fuzzy number.

Some of the scalar product and addition properties on fuzzy numbers are as follows. For all properties 0, A  FR are singleton zero number and any fuzzy number, respectively, of which FR is the set of all fuzzy numbers, those are defined on real numbers. 1. 0 A ¼ A 0. 2. There is no inverse with respect to . It means A (1)A 6¼ 0. 3. For two arbitrary real numbers a, b  R in which a  b 0: ða + bÞA ¼ a⊙A b⊙A 4. For any λ  R and A, B  RF: λ⊙ðA BÞ ¼ λ⊙A λ⊙B 5. For any λ, μ  R and A  RF: λ⊙ðμ⊙AÞ ¼ ðλ  μÞ⊙A All of the abovementioned properties can be proved very easily by using the computations in level-wise form.

2.6

Advanced uncertainties and their properties

In this section, we first introduce a broad version of uncertainty called “pseudooctagonal” uncertain sets. Special cases are then discussed as “pseudo-triangular” and “trapezoidal.”

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Soft Numerical Computing in Uncertain Dynamic Systems

The motivation is that, in most of the social sciences, the use of uncertainty in triangular and trapezoidal forms may not usually be used sufficiently to measure the characteristics of personal beliefs and beliefs that lead to the fragment. It is common information that can be further expressed from four different points of view in the real line. Thus, even unspecified trapezoidal collections cannot be sufficiently plausible to show such cases from social science measurements. Therefore, in order to fill this gap in the uncertainty literature, the concept of a pseudo-octagonal uncertain set is introduced. The second part of this section is about introducing other uncertain advanced collections and combining them together. The idea was put forward by Professor Lotfi A. Zadeh as “Z-numbers.” Here we will also talk about Z-process improvements. All this information is also explained in Abbasi et al. (2018), Abbasi and Allahviranloo (2018), Alive et al. (2016), Allahviranloo and Ezadi (2019), and Zadeh (2011).

2.6.1

PSEUDO-OCTAGONAL

SETS

An uncertain set, A, is called a pseudo-octagonal uncertain set if its membership function A(x) is as follows (Abbasi and Allahviranloo, 2018): 8 l1, A ðxÞ, a1  x  a2 > > > 1 > > , a 2  x  a3 > > > 2 > > > l ðxÞ, a3  x  _a > < 2, A 1, a xa _ Að x Þ ¼ ð x Þ, a  x  a4 r > 2, A > > 1 > > > , a 4  x  a5 > > 2 > > > > : r1,A ðxÞ, a5  x  a6 0, otherwise This uncertain set is denoted by:   A ¼ a1 , a2 , a3 , a_ , a, a4 , a5 , a6 , ðl1, A ðxÞ, l2, A ðxÞÞ, ðr2, A ðxÞ, r1, A ðxÞÞ where the pair of functions (l1, A(x), l2, A(x)) contains two nondecreasing functions and (r2, A(x), r1, A(x)) contains two nonincreasing functions. This presentation of an uncertain set is more general and without 0.5-level it is changed to the normal pseudo-trapezoidal and trapezoidal fuzzy numbers. In the case of _a ¼ a, it is an exactly triangular fuzzy number. The level-wise form or r-level of a pseudo-octagonal uncertain set is defined as follows: 0 1 0 1 [ [ [ [ [

C

C B B 1 1 r l1 r l1 A½r  ¼ @ ½a2 , a3  ½a4 , a5  @ 1, A ðr Þ, r1, A ðr Þ A 2, A ðr Þ, r2, A ðr Þ A rð0, 12 r½12, 1

Uncertain sets

47

r 1 r1,–1A(r)

–1 l2,A (r)

1 2 r2,–1A(r)

–1 l1,A (r)

0

Fig. 2.19

a1

a2

a3

a–

–a

a4

a5

a6

Pseudo-octagonal uncertain set.

1 1 1 1 Note 1. If [l 1, A(r), r1, A(r)] and [l2, A(r), r2, A(r)] are not linear, the set is called pseudo-octagonal (see Fig. 2.19). Note 2. It is clear that if a2 ¼ a3 and a4 ¼ a5, then the set is a pseudo-trapezoidal uncertain set. 1 1 1 Note 3. If [l1 1, A(r), r1, A(r)] and [l2, A(r), r2, A(r)] are linear, the set is called octagonal (see Fig. 2.20). Note 4. Clearly, if a2 ¼ a3 and a4 ¼ a5, then the set is a trapezoidal fuzzy set. The arithmetic on these uncertain sets is not exactly similar to the normal fuzzy numbers, but generally, they have similar rules. To explain, we consider two of these uncertain sets as follows:

A ¼ ða1 , a2 , a3 ; a, _, a, a4 , a5 , a6 , ðl1, A ðxÞ, l2, A ðxÞÞ, ðr2, A ðxÞ, r1, A ðxÞÞÞ  B ¼ b1 , b2 , b3 ; b, _, b, b4 , b5 , b6 , ðl1, B ðxÞ, l2, B ðxÞÞ, ðr2, B ðxÞ, r1, B ðxÞÞ The operations on these sets are now explained in level-wise form of the uncertain sets. Consider:

r 1 r1,–1A(r)

–1 l2,A (r)

1 2 –1 l1,A (r)

0

Fig. 2.20

a1

r2,–1A(r)

a2

Octagonal uncertain set.

a3

a–

–a

a4

a5

a6

48

Soft Numerical Computing in Uncertain Dynamic Systems 0

1 0 1 [ [ [ [ [



B C B C 1 1 r l1 r l1 A½r  ¼ @ ½a2 , a3  ½a4 , a5  @ 1, A ðr Þ, r1, A ðr Þ A 2, A ðr Þ, r2, A ðr Þ A rð0, 12 r½12, 1 0

1 0 1 [ [ [ [ [

C

C B B 1 1 B½r  ¼ @ r l1 r l1 ½b2 , b3  ½b4 , b5  @ 1, B ðr Þ, r1, B ðr Þ A 2, B ðr Þ, r2, B ðr Þ A rð0, 12 r½12, 1

The general form of computations like    { , , ⊙ , } between these uncertain sets can be displayed as the following general form. 0 1 0 1 [ [ [ [ [ B C

B C r ð A1  r ð A2  ðA  BÞ ½ r  ¼ @  B1 Þ ½ r  A _l, l ½_r , r  @  B2 Þ½r A rð0, 12 r½12, 1 where: a¼

_a + a _b + b , b¼ 2 2

For instance, in the case of addition , we have: _l ¼

a + b a2 + b 2 a + b a3 + b3 , l¼ + + 2 2 2 2

_r ¼

a + b a4 + b4 a + b a5 + b 5 , r¼ + + 2 2 2 2

and: 

 1   1  l1, A ðr Þ + l1 r1, A ðr Þ + r1,1B ðr Þ a+b a+b 1,B ðr Þ , + + ðA1 B1 Þ½r  ¼ 2 2 2 2   1    l2, A ðr Þ + l1 r2,1A ðr Þ + r2,1B ðr Þ a+b a+b 2,B ðr Þ , + + ðA2 B2 Þ½r  ¼ 2 2 2 2 In the case of difference :

and:

_l ¼

a  3b a2 + b2 a  3b a3 + b3 , l¼ + + 2 2 2 2

_r ¼

a  3b a4 + b4 a  3b a5 + b5 , r¼ + + 2 2 2 2

Uncertain sets

49

  1   1  1 l ðr Þ + l1 r ðr Þ + r1,B ðr Þ a  3b a  3b 1, B ðr Þ , + 1, A + 1, A 2 2 2 2   1   1  l ðr Þ + l1 r ðr Þ + r2,1B ðr Þ a  3b a  3b 2, B ðr Þ , ðA2 B2 Þ½r ¼ + 2, A + 2,A 2 2 2 2

ðA1 B1 Þðr Þ ¼

In the other cases of multiplication and division, the relations are discussed conditionally and depend on the signs of a and b. In the case of multiplication ⊙: Case 1 If a 0, b 0: b a b a b a b a _l ¼ a2 + b2 , l ¼ a3 + b3 , _r ¼ a4 + b4 , r ¼ a5 + b5 2 2 2 2 2 2 2 2   b a b a 1 ðA1 ⊙B1 Þ½r  ¼ l1 ðr Þ + l1 ðr Þ, r1,1A ðr Þ + r1,B ðr Þ 2 1, A 2 1, B 2 2   b 1 a 1 b 1 a 1 ðA2 ⊙B2 Þ½r  ¼ l2, A ðr Þ + l2, B ðr Þ, r2, A ðr Þ + r2,B ðr Þ 2 2 2 2 Case 2 If a 0, b  0: b a b a b a b a _l ¼ a5 + b2 , l ¼ a4 + b3 , _r ¼ a3 + b4 , r ¼ a2 + b5 2 2 2 2 2 2 2 2   b a b a 1 ðA1 ⊙B1 Þ½r  ¼ r1,1A ðr Þ + l1 ðr Þ, l1 ðr Þ + r1,B ðr Þ 2 2 1, B 2 1, A 2   b 1 a 1 b 1 a 1 ðA2 ⊙B2 Þ½r  ¼ r2, A ðr Þ + l2, B ðr Þ, l2, A ðr Þ + r2,B ðr Þ 2 2 2 2 Case 3 If a  0, b  0: b a b a b a b a _l ¼ a5 + b5 , l ¼ a4 + b4 , _r ¼ a3 + b3 , r ¼ a2 + b2 2 2 2 2 2 2 2 2   b a b a ðA1 ⊙B1 Þ½r  ¼ r1,1A ðr Þ + r1,1B ðr Þ, l1 ðr Þ + l1 ðr Þ 2 2 2 1, A 2 1,B   b 1 a 1 b 1 a 1 ðA2 ⊙B2 Þ½r  ¼ r2, A ðr Þ + r2, B ðr Þ, l2, A ðr Þ + l2,B ðr Þ 2 2 2 2 Case 4 If a  0, b 0: b a b a b a b a _l ¼ a2 + b5 , l ¼ a3 + b4 , _r ¼ a4 + b3 , r ¼ a5 + b2 2 2 2 2 2 2 2 2

50

Soft Numerical Computing in Uncertain Dynamic Systems  b 1 a l ðr Þ + r1,1B ðr Þ, 2 1, A 2  b a ðA2 ⊙B2 Þ½r  ¼ l1 ðr Þ + r2,1B ðr Þ, 2 2, A 2 ðA1 ⊙B1 Þ½r  ¼

b 1 a r ðr Þ + l1 ðr Þ 2 1,A 2 1, B b 1 a r ðr Þ + l1 ðr Þ 2 2,A 2 2, B

 

In the case of multiplication , we consider that AB ¼ A⊙B1 , so first introduce the B1 in level-wise form as follows: 0 1       1 1 1 1 1 B [ C[ 1 r B1 ½r  ¼ @ l ð r Þ, r ð r Þ b , b A 1 1 2 3 b2 1, B b2 1, B b2 b2 rð0, 12 1 0 [   [ [ 1 1 1 1 1 C B r b4 , 2 b5 l ðr Þ, 2 r2,1B1 ðr Þ A @ 2 2,B1 b2 b b b r½12, 1 0 1 0 1 [ [ [ [ [   

B C B C A⊙B1 ½r  ¼ @ r A1  B1 r A2  B1 ½_r , r  @ _l, l 1 ½r A 2 ½r A rð0, 12 r½12, 1 Now, as in the multiplication case, we should consider the previous four cases, because it is indeed a multiplication by an inverse. Here, it should be mentioned that b is not zero, b 6¼ 0. Case 1 If a 0, b > 0: _l ¼

1 a 1 a 1 a 1 a a2 + 2 b2 , l ¼ a3 + 2 b3 , _r ¼ a4 + 2 b4 , r ¼ a5 + 2 b5 2b 2b 2b 2b 2b 2b 2b 2b    1 1 a 1 1 1 a 1 l r A1 ⊙B1 ð α Þ + l ð α Þ, ð α Þ + r ð α Þ ½ α  ¼ 1 2b 1, A 2b2 1, B 2b 1, A 2b2 1, B    1 1 a 1 1 1 a 1 l r ð α Þ + l ð α Þ, ð α Þ + r ð α Þ ½ α  ¼ A2 ⊙B1 2 2b 2, A 2b2 2, B 2b 2, A 2b2 2, B

Case 2 If a 0, b < 0: _l ¼

1 a 1 a 1 a 1 a a5 + 2 b2 , l ¼ a4 + 2 b3 , _r ¼ a3 + 2 b4 , r ¼ a2 + 2 b5 2b 2b 2b 2b 2b 2b 2b 2b    1 1 a 1 1 1 a 1 1 A1 ⊙B1 ½α ¼ r ðαÞ + 2 l1,B ðαÞ, l1, A ðαÞ + 2 r1, B ðαÞ 2b 1,A 2b 2b 2b    1 1 a 1 1 1 a 1 ð α Þ + l ð α Þ, ð α Þ + r ð α Þ r l ½ α  ¼ A2 ⊙B1 2 2b 2,A 2b2 2,B 2b 2, A 2b2 2, B

Uncertain sets

51

Case 3 If a  0, b < 0: _l ¼

1 a 1 a 1 a 1 a a5 + 2 b5 , l ¼ a4 + 2 b4 , _r ¼ a3 + 2 b3 , r ¼ a2 + 2 b2 2b 2b 2b 2b 2b 2b 2b 2b    1 1 a 1 1 1 a 1 r l ½ α  ¼ A1 ⊙B1 ð α Þ + r ð α Þ, ð α Þ + l ð α Þ 1 2b 1, A 2b2 1,B 2b 1, A 2b2 1, B    1 1 a 1 1 1 a 1 1 r ðαÞ + 2 r2,B ðαÞ, l2, A ðαÞ + 2 l2, B ðαÞ A2 ⊙B2 ½α ¼ 2b 2, A 2b 2b 2b

Case 4 If a  0, b > 0: _l ¼

2.6.2

1 a 1 a 1 a 1 a a2 + 2 b5 , l ¼ a3 + 2 b4 , _r ¼ a4 + 2 b3 , r ¼ a5 + 2 b2 2b 2b 2b 2b 2b 2b 2b 2b    1 1 a 1 1 1 a 1 l r A1 ⊙B1 ð α Þ + r ð α Þ, ð α Þ + l ð α Þ ½ α  ¼ 1 2b 1, A 2b2 1, B 2b 1, A 2b2 1, B    1 1 a 1 1 1 a 1 1 l ðαÞ + 2 r2, B ðαÞ, r2, A ðαÞ + 2 l2, B ðαÞ A2 ⊙B2 ½α ¼ 2b 2, A 2b 2b 2b

Z-PROCESS

At this point, we are interested in expanding the concept of advanced uncertain sets, and we need to explain the benefits and motivations of these sets. In decision science, one who wants to decide on an object must be based on its information. We all know that information about us is linguistic and unclear. This information must be reliable and useful for its usefulness and applicability; it must be mathematically modeled for rational decisions. For example, I may say: “I’ll be at home for about 20 minutes.” Or another example, we can almost predict the economic situation of the country in about 5 years, and any other similarly complex proposition about other matters. It is easy to say that is obviously not easy to deal with, and decision-making has complexity problems. In fact, mathematical decision-making uses a lot of words or language computations. In some cases, the occurrence of some uncertain information is conditional and may depend on another occurrence. Lotfi A. Zadeh was a scientist who introduced and discussed this information as the Z-process. Here we explain these sets and those advanced cases. Basically, in phrases that we use in logical inferences with uncertainty, the ambiguities with reliability do have very strong support in decision-making in comparison with other phrases without any reliability.

52

Soft Numerical Computing in Uncertain Dynamic Systems

2.6.2.1 Definition—Z-process A Z-process introduces another type of uncertainty and is shown by an ordered pair of two linguistic uncertain variables, A and B. It is denoted as Z ¼ (A, B). The first component, A, is the membership function or any uncertain restriction, R(X), on the values of real valued uncertain universe set, X, and the second component, B, is a probability measure of reliability and it mentions certainty of the first component (Zadeh, 2011). Note that the type of uncertain restriction depends on the type of uncertain set, X. For instance, if X is a random variable then the probability distribution plays the role of a probabilistic restriction on X. Mathematically, the relation is defined as follows: ProbabilityðX is AÞ ¼ B Here, the restriction Probability(X is A) is referred to as a possibility restriction or constraint, with A playing the role of the possibility distribution of X. In other words, the restriction R(X) is X is A. Indeed it can be explained as the membership degree of some value of X, which satisfies the restriction and is exactly the possibility measure of the values. RðXÞ : X is A ! Possðx ¼ XÞ ¼ AðxÞ where A(x) is the membership degree of x in A. To complete the discussion, a probabilistic restriction is expressed as: RðXÞ : X is probability px where X is a random variable and here pX is the probability density function of X. So: ð ProbabilityðX is AÞ ¼ B ! AðxÞpX ðxÞdx is B Ð

R

In fact, RμX(x)pX(x)dx is the probability measure. In general, the ordered triple (X, A, B) is referred to as a Z-valuation that is equivalent to an assignment statement, X is (A, B). Basically, uncertain computation is a computation system in which the objects are not the computation of variable values but are constraints on the values of variables. For convenience, the value of X is referred to as X, realizing that if we are talking precisely, A is not an X value but a limit on the values that X can use. The second component, B, is called certainty, and is close to the concepts of sureness, confidence, reliability, strength of belief, probability, possibility, etc. When X is a random variable, certainty may be equated to probability. Informally, B, may be interpreted as a response to the question: “How sure are you that X is A?” Typically, A and B are perception-based and are described in natural language.

Uncertain sets

53

B is very sure

1

A(x) X 50

Fig. 2.21

min

Z ¼ (A, B) ¼ (about 50 minutes, very sure).

2.6.2.2 Example Consider the following Z-process: Z ¼ ðA, BÞ ¼ ðabout 50 minutes, very sureÞ In the example “about 50 min” is a restriction in the role of uncertain set and “very sure” plays the role of reliability of the “about 50 min” as another uncertain set. We can consider the process as “I am very sure that he will arrive about 50 minutes later” (Fig. 2.21). In fact, a Z-number or process may be viewed as a summary of probability distribution that is not known. It is important to note that in day-to-day decisions, most decisions are based on summary information. Viewing a Z-process as a summary is consistent with this fact. In applications for decision analysis, a fundamental problem that arises is the ranking of Z-numbers.

2.6.2.3 Example Is (approximately 80, probably) more than (approximately 80, very likely)? Or (approximately 100, very likely) more than (approximately 90, surely)? Are these meaningful questions? It seems that the relations of probability distribution pX and B can be explained immediately. For instance, if Z ¼ (A, B) is a Z-process, its complement can be described as Zc ¼ (Ac, Bc) where Ac is the complement of A and Bc plays the role of complement of B. For example, consider the previous example, if: Z ¼ ðA, BÞ ¼ ðabout 50 minutes, very sureÞ

54

Soft Numerical Computing in Uncertain Dynamic Systems Very sure

Little sure

1

Not A A X 50

Fig. 2.22

min

Zc ¼ (Ac, Bc) ¼ (not about 50 minutes, unsure).

then: Z c ¼ ðAc , Bc Þ ¼ ðNot about 50 minutes, little sureÞ In graphical form, it can be shown as in Fig. 2.22.

2.6.3

COMPUTATIONS

ON

Z-NUMBERS

First, we should determine the meaning of a fuzzy number. In the Z-process like Z ¼ (A, B), the first component, A, is a fuzzy set with the reliability B, and there is no need to be a fuzzy number. But in the case that A is a fuzzy number with the reliability of B, then the Z-process is called a Z-number. Calculating this information is not as easy as we think. Therefore, it is intended to simplify a particular case of this information. To this end, the summarization of the Z-number can be demonstrated as follows: Z ¼ ðA, BÞ ¼ ðAðxÞ, PRX ðxÞÞ or Z ¼ ðA, PRX Þ: where: Possðx ¼ XÞ ¼ AðxÞ ProbabilityðX is AÞ ¼ B And we say probability distribution, pX, is not known and what is known is restriction on pX, which is demonstrated as follows: ð AðxÞpX ðxÞdx is B R

Uncertain sets

55

Now suppose that A(x) and B(y) are two membership functions of two fuzzy numbers and PRX and PRY are two independent probability density functions on two universe uncertain sets on X and Y, respectively. Consider the following two Z-numbers: ZX ¼ ðA1 , PRX Þ, ZY ¼ ðA2 , PRY Þ, such that: ð

ð

PRX ¼

A1 ðxÞpX ðxÞdx, PRY ¼ R

A2 ðyÞpY ðyÞdy R

and: ð

ð pX ðxÞdx ¼ 1, R

pY ðyÞdy ¼ 1 R

For the compatibility of two components of an uncertain set, we should have the following relations: ð ð xμX ðxÞdx xpX ðxÞdx ¼ ðR R μX ðxÞdx ð

ð

R

ypY ðyÞdy ¼ ðR

yμY ðyÞdy

R

μY ðyÞdy

R

  { , , ⊙ , }. The calculations on the Suppose an arbitrary operation, like  two abovementioned Z-numbers can be defined as:

ZX   Z Y ¼ ð A1   A 2 , PR X   PRY Þ ¼ ðA1   A2 , PRX 

RY Þ

 whose operands are memberIt should be mentioned that the binary operation  ship functions is different from the binary operation   whose operands are probability density functions. Actually PRX   PRY is the convulsion of two probability density functions PRX and PRY and it is very difficult to follow. It is exactly dependent on the meaning of extension of uncertainty under the operation   . It means that the uncertainty of ZX   ZY should be increased in comparison of operands uncertainties and increasing the uncertainty is related to not only the first part, but also the second part. So to extend the uncertainty and evaluation of the extension, if-then rules are suggested. Considering the previous concepts, we have:

ZX ¼ ðA1 , PRX Þ, ZY ¼ ðA2 , PRY Þ,

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Soft Numerical Computing in Uncertain Dynamic Systems

then:  ð   ð  ZX ¼ A1 , A1 ðxÞpX ðxÞdx , ZY ¼ A2 , A2 ðyÞpY ðyÞdy , R

R

In general form, based on Zadeh’s extension principle, two of the components do have membership functions and those can be defined as follows: ðA1   A2 ÞðtÞ ¼ sup min fA1 ðr Þ, A2 ðsÞg, t¼r∗s

and the probability distribution of PRX RY is referred to as a convolution of PRX and PRY. ð PRX  RY ðtÞ ¼ ðA1   A2 ÞðtÞpX∗Y ðtÞdt R

2.6.3.1 Summation of two Z-numbers If we suppose that the operation is summation: ZX ZY ¼ ðA1 A2 , PRX RY Þ The first component is the summation of two membership functions of two fuzzy numbers and it can be investigated by using Zadeh’s extension principle. Then we have: ðA1 A2 ÞðtÞ ¼ sup min fA1 ðr Þ, A2 ðsÞg, t¼r + s

and the probability distribution of PRX RY is: ð PRX RY ðtÞ ¼ ðA1 A2 ÞðtÞpX + Y ðtÞdt, R

where: ð pX + Y ð t Þ ¼

pX ðsÞpY ðt  sÞds R

As we can see, the relations are too complicated and the other computations have the same procedure as the summation.

2.6.3.2 Difference of two Z-numbers Now suppose that the operation is the difference between two Z-numbers: ZX H ZY ¼ ðA1 H A2 , PRX RY Þ

Uncertain sets The probability distribution of PRX RY is: ð PRX RY ðtÞ ¼ ðA1 H A2 ÞðtÞpXY ðtÞdt, R

where: ðA1 H A2 ÞðtÞ ¼ sup min fA1 ðr Þ, A2 ðsÞg, t¼rs

and: ð pXY ðtÞ ¼

pX ðsÞpY ðs  tÞds R

The other computations have the same structures.

2.6.3.3 Multiplication of two Z-numbers Now suppose that the operation is multiplication: ZX ⊙ZY ¼ ðA1 ⊙A2 , PRX ⊙RY Þ Again, similar to the previous operations, we have: ð PRX ⊙RY ðtÞ ¼ ðA1 ⊙A2 ÞðtÞpX  Y ðtÞdt R

where: ðA1 ⊙A2 ÞðtÞ ¼ sup min fA1 ðr Þ, A2 ðsÞg, t¼r  s

and: ð pX  Y ðtÞ ¼

pX ðsÞpY R

t 1 ds s |s|

2.6.3.4 Division of two Z-numbers For the division: ZX ZY ¼ ðA1 A2 , PRX RY Þ Again, similar to the previous operations, we have: ð PRX RY ðtÞ ¼ ðA1 A2 ÞðtÞpX=Y ðtÞdt R

57

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Soft Numerical Computing in Uncertain Dynamic Systems

where: ðA1 A2 ÞðtÞ ¼ sup min fA1 ðr Þ, A2 ðsÞg, t¼r=s

and: ð pX  Y ð t Þ ¼

|s|pX ðsÞpY R

 s ds t

2.6.3.5 Level-wise form of a Z-number For a Z-number like Z ¼ (A, B) on X, we suppose the two components are compatible and the second component is a continuous and normal probability density function (central limit theorem) associated with A. For instance: ! 1 ðt  μÞ2 N ðt; μ, σ Þ ¼ exp  , tR 2σ 2 σ√2π Clearly, for μ ¼ 1, σ ¼ 0, we will have the standard normal probability density function. So instead of the second component we can have N(t; μ, σ). To explain the level-wise form of a Z-number, the following examples are provided: I say with high probability that I will be at the airport at about 2 o’clock in the morning. Or I say with low probability that I will be at the airport at about 2 o’clock in the morning. In both phrases, we see high and low probability, and that if the level is close to zero then the probability is low, and in the case of it being close to one, the probability is high. 2.6.3.5.1 High membership degree does have high reliability We are now able to show two components in level-wise form and we have: Z ½r  ¼ ðA, N Þ½r  ¼ ½A½r , N ½r , 0  r  1, where: A½r  ¼ ½Al ðr Þ, Au ðr Þ,N ½r  ¼ ½Nl ðr Þ, Nu ðr Þ, 0  r  1

Uncertain sets

r

59

1 –r

N [r– ]

Fig. 2.23

Z-number in level-wise form of case 1

The relation between r and r ¼ heightðN Þ can be discussed in two cases.  Case 1 0  r  1 ðA½r , N ½r Þ, 0  r  r Z ½r  ¼ ðA½r , N ½r Þ, r  r  1 In this case, if r  r  1, the highest level is considered for the elements of A[r] (Fig. 2.23).  Case 2 r > 1 ðA½r , N ½r Þ, 0  r  1 Z ½r  ¼ ðA½1, N ½r Þ, 1 < r  r This means that the highest level of probabilities are for the elements of A[1] for 1 < r  r (Fig. 2.24). PRx := N

–r r

A

1

r

Nl (r)

Fig. 2.24

Al (r)

Z-number in level-wise form of case 2

Au (r) Nu (r)

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Soft Numerical Computing in Uncertain Dynamic Systems

2.6.3.5.2 Definition—Level-wise form of a standard Z-number The level-wise form of the standard Z-number with standard normal probability density function should have the following conditions for any 0  r  1: • • • •

Al(r)  Au(r) and Nl(r)  Nu(r). Al(r) and Nl(r) are two increasing and left continuous functions on (0, 1] and right continuous at 0 w.r.t. r. Au(r) and Nu(r) are decreasing and left continuous functions on (0, 1] and right continuous at 0 w.r.t. Ð coreðNÞ Ð coreðr. NÞ N ð r Þdr ¼ Nu ðr Þdr ¼ 12. l ∞ ∞

Note that for cases (ii) and (iii), both functions can be bounded. The word standard comes from the standard normal probability density function as a second component. Based on this definition, we suppose that the standard normal probability function is a best approximation to part B being the Z-number.

2.6.3.6 Summation in level-wise form For two standard Z-numbers like Z1 ¼ (A1, N1), Z2 ¼ (A2, N2): ðZ1 + Z2 Þ½r  ¼ Z1 ½r  + Z2 ½r  ¼ ½½A1 ½r , N1 ½r  + ½A2 ½r , N2 ½r  ¼ ½A1 ½r  + A2 ½r, N1 ½r  + N2 ½r , ¼ ½ ½A1, l ðr Þ, A1, u ðr Þ + ½A2,l ðr Þ, A2, u ðr Þ, ½N1, l ðr Þ, N1,u ðr Þ + ½N2, l ðr Þ, N2,u ðr Þ  ¼ ½½A1, l ðr Þ + A2,l ðr Þ, A1, u ðr Þ + A2, u ðr Þ, ½N1, l ðr Þ + N2, l ðr Þ, N1,u ðr Þ + N2, u ðr Þ  For example: Z1 ¼ ðabout 2 o0 clock, very sureÞ, Z1 ¼ ðabout 3 o0 clock, a little sureÞ Now the summation is as follows: Z1 + Z2 ¼ ðabout 2 o0 clock + about 3 o0 clock, very sure + a little sureÞ Usually we can say that: Z1 + Z2 ¼ ðabout 5 o0 clock, a little too sureÞ Now: ða little too sureÞ½r  ¼ ðvery sureÞ½r  + ða little sureÞ½r  In (a little too sure)[r], if r ! 1, the highest probability of a little too sure as a reliability occurs. This means the elements in the support of fuzzy number about 5 o0 clock with the membership degree of r do have the degree of r reliability in a little too sure.

Uncertain sets

61

2.6.3.7 Scalar multiplication in level-wise form Suppose that λ  R is a real scalar and Z ¼ (A, N) is a standard normal probability function. We are going to investigate the scalar multiplication in level-wise form.  ð½λAl ðr Þ, λAu ðr Þ, ½λNl ðr Þ, λNu ðr ÞÞ, λ  0 ðλZÞ½r  ¼ ðλA½r , λN ½r Þ ¼ ð½λAu ðr Þ, λAl ðr Þ, ½λNu ðr Þ, λNl ðr ÞÞ, λ < 0 For example: Z ¼ ðabout 1, sureÞ,Z ½r  ¼ ð½1, 2  r, sure½r Þ Now suppose λ ¼  1: ð1ÞZ ¼ ðabout ð1Þ, sureÞ, ðð1ÞZ Þ½r  ¼ ð½r  2,  1, sure½r Þ In the Z-number, (1)Z, the reliability “sure” is for the fuzzy set “about(1)” (see Fig. 2.25).

2.6.3.8 Hukuhara difference in level-wise form As mentioned earlier, the H-difference and gH-difference between two fuzzy numbers are different from the negative scalar multiplication and here we will have the similar discussions with some generalizations (Allahviranloo and Ezadi, 2019). Again, suppose two Z-numbers as follows: Z1 ¼ ðA1 , N1 Þ, Z2 ¼ ðA2 , N2 Þ The H-difference of these two Z-numbers Z1 HZ2 exists and is equal to Z3 if and only if: 9Z3 , Z1 H Z2 ¼ Z3 , Z1 ¼ Z2 Z3 where Z3 is another Z-number. This means the existence of the H-difference depends on Z3 and it should be a Z-number as well. Now we can have the following, as mentioned previously: Z1 ½r  ¼ ðZ2 Z3 Þ½r  ¼ Z2 ½r  + Z3 ½r 

(–1)Au (r) = r – 2

Au (r) = 2 – r Sure

–2

Fig. 2.25

–1

Z and (1)Z in level-wise form.

Sure

1

2

62

Soft Numerical Computing in Uncertain Dynamic Systems ¼ ½ ½A2, l ðr Þ, A2, u ðr Þ + ½A3,l ðr Þ, A3, u ðr Þ, ½N2, l ðr Þ, N2,u ðr Þ + ½N3, l ðr Þ, N3,u ðr Þ  ¼ ½½A2, l ðr Þ + A3,l ðr Þ, A2, u ðr Þ + A3, u ðr Þ, ½N2, l ðr Þ + N3, l ðr Þ, N2,u ðr Þ + N3, u ðr Þ 

so: ½A1, l ðr Þ, A1, u ðr Þ ¼ ½A2, l ðr Þ + A3, l ðr Þ, A2, u ðr Þ + A3, u ðr Þ ½N1, l ðr Þ, N1, u ðr Þ ¼ ½N2, l ðr Þ + N3, l ðr Þ, N2, u ðr Þ + N3, u ðr Þ then: A3, l ðr Þ ¼ A1, l ðr Þ  A2,l ðr Þ, A3, u ðr Þ ¼ A1, u ðr Þ  A2, u ðr Þ, N3,l ðr Þ ¼ N1, l ðr Þ  N2,l ðr Þ, N3, u ðr Þ ¼ N1, u ðr Þ  N2, u ðr Þ and: ½½A3,l ðr Þ, A3, u ðr Þ, ½N3, l ðr Þ, N3, u ðr Þ ¼ ¼ ½½A1, l ðr Þ  A2,l ðr Þ, A1, u ðr Þ  A2, u ðr Þ, ½N1, l ðr Þ  N2, l ðr Þ, N1,u ðr Þ  N2, u ðr Þ The conditions for the existence of the Hukuhara difference are as follows: • • • •

A3,l (r)  A3,u (r) and N3,l (r)  N3,u (r). A3,l (r) and N3,l (r) are two increasing and left continuous functions on (0, 1] and right continuous at 0 w.r.t. r. A3, u(r) and N3, u(r) are decreasing and left continuous functions on (0, 1] and right continuous at 0 w.r.t. r. Ð coreðNÞ Ð coreðNÞ N3,l ðr Þdr ¼ ∞ N3, u ðr Þdr ¼ 12. ∞

For any 0  r  1. As we observe that the conditions are very difficult to satisfy, in cases where at least one of the conditions does not work, we are going to consider the H-difference as Z2 HZ1 ¼ (1)Z3. If this one is true then we are going to define the generalized Hukuhara difference.

2.6.3.9 Generalized Hukuhara difference in level-wise form In this case, we can define the difference in another way. Suppose that we want to try Z2 HZ1 ¼ (1)Z3 and let the difference Z3 exist. Considering the level-wise forms of two sides: ðZ2 H Z1 Þ½r ¼ ½½A2, l ðrÞ  A1, l ðrÞ, A2, u ðrÞ  A1, u ðr Þ, ½N2, l ðrÞ  N1, l ðrÞ, N2, u ðr Þ  N1, u ðr Þ ¼ ðð1ÞZ3 Þ½r ¼ ½½A3, u ðr Þ,  A3, l ðrÞ, ½N3, u ðr Þ,  N3, l ðrÞ

Uncertain sets

63

Then we have: ½A2, l ðr Þ  A1, l ðr Þ, A2, u ðr Þ  A1, u ðr Þ ¼ ½A3, u ðr Þ,  A3, l ðr Þ, ½N2,l ðr Þ  N1,l ðr Þ, N2, u ðr Þ  N1, u ðr Þ ¼ ½N3, u ðr Þ,  N3, l ðr Þ Immediately: A2, l ðr Þ  A1, l ðr Þ ¼ A3, u ðr Þ, A2, u ðr Þ  A1, u ðr Þ ¼ A3, l ðr Þ N2, l ðr Þ  N1, l ðr Þ ¼ N3, u ðr Þ, N2, u ðr Þ  N1, u ðr Þ ¼ N3, l ðr Þ and here: ½½A3, l ðr Þ, A3, u ðr Þ, ½N3,l ðr Þ, N3, u ðr Þ ¼ ¼ ½½A1, u ðr Þ  A2, u ðr Þ, A1,l ðr Þ  A2,l ðr Þ, ½N1,u ðr Þ  N2,u ðr Þ, N1, l ðr Þ  N2, l ðr Þ In comparison with previous results, the only difference is interchanging the endpoints of the intervals. So the generalized Hukuhara difference is shown by gH and defined in two cases. •

In case (i): Z1 gH Z2 ¼ Z3 , Z1 ¼ Z2 Z3

and the level-wise form is as follows: ½½A3, l ðr Þ, A3, u ðr Þ, ½N3,l ðr Þ, N3, u ðr Þ ¼ ¼ ½½A1, l ðr Þ  A2, l ðr Þ, A1,u ðr Þ  A2,u ðr Þ, ½N1,l ðr Þ  N2,l ðr Þ, N1, u ðr Þ  N2, u ðr Þ

•

In case (ii): Z2 gH Z1 ¼ ð1ÞZ3 , Z2 ¼ Z1 ð1ÞZ3

and the level-wise form is as follows: ½½A3, l ðr Þ, A3, u ðr Þ, ½N3,l ðr Þ, N3, u ðr Þ ¼ ¼ ½½A1, u ðr Þ  A2, u ðr Þ, A1,l ðr Þ  A2,l ðr Þ, ½N1,u ðr Þ  N2,u ðr Þ, N1, l ðr Þ  N2, l ðr Þ If case (i) exists, there is no need to consider case (ii). Otherwise we will need the second case. The relation between the two cases can be explained as follows:    Z1 gH Z2 i ½r ¼ ð1Þ Z2 gH Z1 ii ½r

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Soft Numerical Computing in Uncertain Dynamic Systems

If both cases exist, then Z3 ¼ (1)Z3 and it is concluded that both types of difference are the same and equal. Considering the two cases, to define the endpoints of the generalized Hukuhara difference: A3,l ðr Þ ¼ min fA1,u ðr Þ  A2, u ðr Þ, A1, l ðr Þ  A2, l ðr Þg A3,u ðr Þ ¼ max fA1, u ðr Þ  A2,u ðr Þ, A1, l ðr Þ  A2, l ðr Þg N3, l ðr Þ ¼ min fN1, u ðr Þ  N2, u ðr Þ, N1,l ðr Þ  N2,l ðr Þg N3, u ðr Þ ¼ max fN1, u ðr Þ  N2, u ðr Þ, N1, l ðr Þ  N2, l ðr Þg





A3 ½r ¼ min A1, u ðrÞ  A2, u ðr Þ, A1, l ðr Þ  A2, l ðrÞ , max A1, u ðr Þ  A2, u ðr Þ, A1, l ðr Þ  A2, l ðrÞ





N3 ½r ¼ min N1, u ðrÞ  N2, u ðr Þ, N1, l ðr Þ  N2, l ðr Þ , max N1, u ðr Þ  N2, u ðr Þ, N1, l ðr Þ  N2, l ðrÞ

2.6.3.10 Some properties of generalized Hukuhara It should be noted that all of the following properties can be proved in level-wise form easily. 1. 2. 3. 4. 5. 6.

If the gH-difference exists, it is unique. Z gHZ ¼ 0. If Z1 gHZ2 exists in case (i), then Z2 gHZ1 exists in case (ii) and vice versa. In both cases (Z1 Z2) gHZ2 ¼ Z1. (It is easy to show in level-wise form.) If Z1 gHZ2 and Z2 gHZ1 exist, then 0 gH(Z1 gHZ2) ¼ Z2 gHZ1. If Z1 gHZ2 ¼ Z2 gHZ1 ¼ Z if and only if Z ¼  Z and Z1 ¼ Z2.

Note. The difference even in the gH-difference case may not exist. It can be said that the gH-difference of two Z-numbers are not always a Z-number. For instance, item 4 in level-wise form can be explained as follows: ðZ1 ½r  Z2 ½r Þ gH Z2 ½r  ¼ ¼ ððA1 ½r, N1 ½r Þ ðA2 ½r , N2 ½r ÞÞ gH ðA2 ½r , N2 ½r Þ ¼ ð½A1, l ðr Þ + A2,l ðr Þ, A1, u ðr Þ + A2, u ðr Þ, ½N1, l ðr Þ + N2, l ðr Þ, N1, u ðr Þ + N2, u ðr ÞÞ ð½A2, l ðr Þ, A2, u ðr Þ, ½N2, l ðr Þ, N2, u ðr ÞÞ ¼ ð½A1,l ðr Þ, A1, u ðr Þ, ½N1, l ðr Þ, N1,u ðr ÞÞ Subtracting component-wise, the right-hand side is achieved very easily.

Uncertain sets

65

References Abbasi, F., Allahviranloo, T., 2018. New operations on pseudo-octagonal fuzzy numbers and its application. Soft Comput. 22, 3077–3095. Abbasi, F., Allahviranloo, T., Abbasbandy, S., 2018. A new attitude coupled with the basic fuzzy thinking to distance between two fuzzy numbers. Iran. J. Fuzzy Syst. 22, 3077–3095. Alive, R.A., Huseynov, O.H., Serdaroglu, R., 2016. Ranking of Z-numbers, and its application in decision making. Int. J. Inform. Technol. Dec. Mak. 15, 1503. Allahviranloo, T., 2020. Uncertain information and linear systems. In: Studies in Systems, Decision and Control. vol. 254. Springer. Allahviranloo, T., Ezadi, S., 2019. Z-advanced number processes. Inform. Sci. 480, 130–143. Bede, B., Gal, S.G., 2005. Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Set Syst. 151, 581–599. Liu, B., 2015. Uncertain Theory. Springer-Verlag, Berlin. Stefanini, L., 2010. A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Set Syst. 161, 1564–1584. Zadeh, L.A., 2011. A note on Z-numbers. Inform. Sci. 181, 2923–2932.

Further reading Allahviranloo, T., Perfilieva, I., Abbasi, F., 2018. A new attitude coupled with fuzzy thinking for solving fuzzy equations. Soft Comput. 22, 3077–3095.

3.1

Chapter 3

Soft computing with uncertain sets Introduction

The aims of this chapter are considering any operators such as differentials, and integral operators on fuzzy numbers. As we know, the distance between two fuzzy numbers is defined as a function and indeed can be known as a type of operator. We will discuss different types of distances of fuzzy numbers, several ranking methods, and expected value and lengths functions. Basically, the operators in this chapter are real and the functions and sets as operands are considered as fuzzy and uncertain sets. In the previous chapter, the main four calculating operators {
,  , ⊙ , } were explained and applied on fuzzy numbers. In this chapter, first, we will discuss the distance between two arbitrary uncertain sets. To explain it, some concepts are considered, such as the expected value in uncertainty.

67 Soft Numerical Computing in Uncertain Dynamic Systems. https://doi.org/10.1016/B978-0-12-822855-5.00003-3 © 2020 Elsevier Inc. All rights reserved.

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Soft Numerical Computing in Uncertain Dynamic Systems

3.2

Expected value

The concept of expected value can be found in a similar way but it depends on the measure of an uncertain set. For a measurable fuzzy set AM with probability measure, we know that Ac M, which is a complement fuzzy set of AM. Suppose that: supx AðxÞ, supx AðxÞ are the supremum membership of an element x and not x in fuzzy set A. The following concept is noted as relation and connection of x and A. 1 Midfx  Ag ¼ ð supx AðxÞ + 1  supx AðxÞÞ 2 Let y be an uncertain variable, then for any relation like y ¼ x, y  x and y x we have:  1
Midfy ¼ xg ¼ supy¼x AðyÞ + 1  supy6¼x AðyÞ : 2  1
Midfy  xg ¼ supyx AðyÞ + 1  supy>x AðyÞ : 2  1
Midfy  xg ¼ supyx AðyÞ + 1  supyx AðyÞ 2 2 2 Since A is a fuzzy number, so if we consider a point like x0 such that A(x0) ¼ 1 then we obviously will have: 8 1 > < supyx AðyÞ, x  x0 Midfy  xg ¼ 2 1 > : 1  supy>x AðyÞ, x > x0 2 and it can be considered in another case: 8 1 > < 1  supyx AðyÞ, Midfy  xg ¼ 1 2 > : supy>x AðyÞ, 2

x  x0 x > x0

Based on the middle functions the expected value is defined as follows: ð ð 1 +∞ 1 x0 Eð y  x Þ ¼ x 0 + supyx AðyÞdx  supyx AðyÞdx

Now we can evaluate the following cases: • •

If y x then supyxA(y) ¼ A(x) for x x0 and supyxA(y) ¼ A(x) for  x0.

Then in each case the expected value is defined as: ð ð 1 +∞ 1 x0 AðxÞdx  AðxÞdx EðAÞ ¼ x0 + 2 x0 2 ∞ RemarkIn accordance with the linearity of the integral the expected value is also linear. This means that for two uncertain sets A and B and two real numbers a and b: EðaA + bBÞ ¼ aEðAÞ + bEðBÞ Because: 1 EðaA + bBÞ ¼ x0 + 2

ð +∞ x0

1 ðaA + bBÞðxÞdx  2

ð x0 ∞

ðaA + bBÞðxÞdx

then:   ð ð 1 +∞ 1 x0 EðaA + bBÞ ¼ a x0 + AðxÞdx  AðxÞdx 2 x0 2 ∞   ð +∞ ð 1 1 x0 BðxÞdx  BðxÞdx ¼ aEðAÞ + bEðBÞ + b x0 + 2 x0 2 ∞ Now let us consider the level-wise form of a fuzzy set A. The expected value in this format can be displayed as: Eð AÞ ¼

¼

1 2

1 2

ð1

ð infA½r + supA½r Þdr

0

ð1

ðAl ðr Þ + Au ðr ÞÞdr

0

In the case of a triangular fuzzy set, A ¼ (A1, A2, A3), which the components are three ordered points in the support of the fuzzy number: Al ðr Þ + Au ðr Þ ¼ A2 + ðA2  A1 Þðr  1Þ + A2 + ðA3  A2 Þð1  r Þ

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and the expected value is: ð 1 1 Eð AÞ ¼ ðA2 + ðA2  A1 Þðr  1Þ + A2 + ðA3  A2 Þð1  r ÞÞdr 2 0 ¼

A1 + 2A2 + A3 4

For any Trapezoidal uncertain set, A ¼ (A1, A2, A3, A4), with: Al ð r Þ + A u ð r Þ ¼ A 2 + ð A 2  A 1 Þ ð r  1Þ + A 3 + ð A 4  A 3 Þ ð 1  r Þ Again, the components are three ordered points in support of the fuzzy number. The expected value is: ð 1 1 Eð AÞ ¼ ðA2 + ðA2  A1 Þðr  1Þ + A3 + ðA4  A3 Þð1  r ÞÞdr 2 0 ¼

A1 + A 2 + A 3 + A 4 4

For more explanation, see Liu (2015) and Allahviranloo (2020).

3.3

Distance of two fuzzy numbers

In this section, we will discuss the several types of distances between two arbitrary fuzzy numbers. Basically, the distance is a type of operator from the space all fuzzy numbers to the set of real numbers. If we suppose that the R is the set of all fuzzy numbers, then: D : R R ! R Clearly, this operator should have some conditions or properties such as following. For any arbitrary fuzzy numbers A, B, C, and D: • • • • •

D(A, B) > 0 D(A, B) ¼ 0 ⟺ A ¼ B D(A
C, B
C) ¼ D(A, C) D(A
B, C
D)  D(A, C) + D(B, D) D(λA, λB) ¼ jλ jD(A, B), λ  R

The first distance is evaluated by the concept of expected value that was discussed in the previous section. For two fuzzy numbers A and B, the absolute value of Hukuhara difference A HB can define a distance. Now the distance is defined using expected value of j AHB j as: DðA, BÞ ¼ EðjA  H BjÞ

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The concept of absolute value means that the members are considered as nonnegative values. Based on the definition of expected value, the distance can be expressed as: DðA, BÞ ¼ EðjA  H BjÞ ¼

1 2

ð +∞
0

 sup|y|x jA  H BjðyÞ + 1  sup|y| 0. This property is very clear to prove, because the expected value is always nonnegative and, in the case of zero, is considered as follows. Ð E(j AHB j) ¼ 0 ⟺ 10(inf j A HB j[r] + sup j A HB j[r])dr ¼ 0 ⟺ inf j A HB j[r] + sup j A HB j[r] ¼ 0 ⟺ inf j A HB j[r] ¼ sup j AHB j[r] ¼ 0 ⟺ A ¼ B E(j (A
C)H(B
C)j ) ¼ E(j A HB j)

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To prove this, suppose that: ðA
CÞ  H ðB
CÞ ¼ M Based on the definition in level-wise form, the left-hand side can be written as: ð 1 1 Eðj ðA
CÞ  H ðB
CÞj Þ ¼ ðj Ml j ðr Þ +j Mu j ðr ÞÞdr 2 0 where: Ml ¼ ðA
CÞl  ðB
CÞl ¼ ðA  H BÞl : Mu ¼ ðA
CÞu  ðB
CÞu ¼ ðA H BÞu Then the proof is completed. •

E(j (A
B)H(C
D)j )  E(j AHC j) + E(j BHD j)

To prove this, again suppose that: ðA
BÞ H ðC
DÞ ¼ M Based on the definition in level-wise form, the left-hand side can be written as: ð 1 1 Eðj ðA
BÞ  H ðC
DÞj Þ ¼ ðj Ml j ðr Þ +j Mu j ðr ÞÞdr 2 0 where: Ml ¼ ðA
BÞl  ðC
DÞl ¼ ðA  H CÞl + ðB  DÞl : Mu ¼ ðA
BÞu  ðC
DÞu ¼ ðA  H CÞu + ðB  DÞu so:

    jMjl ¼ ðA H CÞl + ðB  DÞl   ðA H CÞl  + |ðB  DÞl | :     jMju ¼ ðA H CÞu + ðB  DÞu   ðA H CÞu  + |ðB  DÞu |

The proof is clear based on the triangular inequality property of absolute value. •

E(jλA HλB j) ¼ j λj E(j A HBj),

λR

The proof is clear. For more explanation, see Liu (2015) and Allahviranloo (2020).

3.3.1

P-DISTANCE

For two fuzzy numbers A and B, Dp(A, B)  R0 is defined as follows: ð 1 1 ð1 p p Dp ðA, BÞ ¼ jAl ðr Þ  Bl ðr Þj dr + jAu ðr Þ  Bu ðr Þj dr p , p  1: 0

0

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To prove the properties, the first and second ones are trivial. The third property can be proved: "ð 1

Dp ðA + C, B + CÞ ¼

jAl ðr Þ + Cl ðr Þ  Bl ðr Þ  Cl ðr Þjp dr

0

ð1 +

#1 p

jAu ðr Þ + Cu ðr Þ  Bu ðr Þ  Cu ðr Þjp dr

0

¼

ð 1

p

jAl ðr Þ  Bl ðr Þj dr +

0

ð1

p

jAu ðr Þ  Bu ðr Þj dr

1 p

¼ Dp ðA, BÞ

0

The last property is proved easily: ð 1 1 ð1 Dp ðλA, λBÞ ¼ jλAl ðr Þ  λBl ðr Þjp dr + jλAu ðr Þ  λBu ðr Þjp dr p ¼ |λ|Dp ðA, BÞ 0

0

The other properties, we leave for you as exercises.

3.3.2

HAUSDORFF DISTANCE

Again, for two fuzzy numbers, A and B, DH(A, B)  R0 as Hausdorff distance is defined as follows: DH ðA, BÞ ¼ sup0r1 max fjAl ðr Þ  Bl ðr Þj, jAu ðr Þ  Bu ðr Þjg All of the properties can be verified easily. Here, some of the properties are proved as follows: •

DH(AHB, C HD)  DH(A, C) + DH(B, D), subject that A HB and C HD exist.

To prove it, we have the following relations: jAl ðr Þ  Bl ðr Þj  max fjAl ðr Þ  Bl ðr Þj, jAu ðr Þ  Bu ðr Þjg jAu ðr Þ  Bu ðr Þj  max fjAl ðr Þ  Bl ðr Þj, jAu ðr Þ  Bu ðr Þjg and: jCl ðr Þ  Dl ðr Þj  max fjCl ðr Þ  Dl ðr Þj, jCu ðr Þ  Du ðr Þjg jCu ðr Þ  Du ðr Þj  max fjCl ðr Þ  Dl ðr Þj, jCu ðr Þ  Du ðr Þjg jAl ðr Þ  Bl ðr Þ  Cl ðr Þ + Dl ðr Þj  jAl ðr Þ  Bl ðr Þj + jCl ðr Þ  Dl ðr Þj  max fjAl ðr Þ  Bl ðr Þj, jAu ðr Þ  Bu ðr Þjg + max fjCl ðr Þ  Dl ðr Þj, jCu ðr Þ  Du ðr Þjg jAu ðr Þ  Bu ðr Þ  Cu ðr Þ + Du ðr Þj  jAu ðr Þ  Bu ðr Þj + jCu ðr Þ  Du ðr Þj 

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Soft Numerical Computing in Uncertain Dynamic Systems

max fjAl ðr Þ  Bl ðr Þj, jAu ðr Þ  Bu ðr Þjg + max fjCl ðr Þ  Dl ðr Þj, jCu ðr Þ  Du ðr Þjg so: max fjAl ðr Þ  Bl ðr Þ  Cl ðr Þ + Dl ðr Þj, jAu ðr Þ  Bu ðr Þ  Cu ðr Þ + Du ðr Þjg  max fjAl ðr Þ  Bl ðr Þj, jAu ðr Þ  Bu ðr Þjg + max fjCl ðr Þ  Dl ðr Þj, jCu ðr Þ  Du ðr Þjg then: sup0r1 max fjAl ðr Þ  Bl ðr Þ  Cl ðr Þ + Dl ðr Þj, jAu ðr Þ  Bu ðr Þ  Cu ðr Þ + Du ðr Þjg  sup0r1 max fjAl ðr Þ  Bl ðr Þj, jAu ðr Þ  Bu ðr Þjg + sup0r1 max fjCl ðr Þ  Dl ðr Þj, jCu ðr Þ  Du ðr Þjg The property is now proved and: DH ðA  H B, C  H DÞ  DH ðA, CÞ + DH ðB, DÞ •

DH(A gHB, 0) ¼ DH(A, B)

Based on the definition of Hausdorff distance the proof is easy. Because the left-hand side can be written as:   DH A  gH B, 0 ¼ sup0r1 max fjAl ðr Þ  Bl ðr Þ  0j, jAu ðr Þ  Bu ðr Þ  0jg ¼ sup0r1 max fjAl ðr Þ  Bl ðr Þj, jAu ðr Þ  Bu ðr Þjg ¼ DH ðA, BÞ •

  DH λ ⊙ A  gH μ⊙ A, 0 ¼ jλ  μjDH ðA, 0Þ, AR , λ,μ  0:

To prove the property, in accordance with the property of the distance, the left-hand side can be shown as:   DH λ ⊙ A gH μ ⊙ A, 0 ¼ DH ðλ⊙ A, μ ⊙ AÞ Now, considering the definition of the distance: DH ðλ⊙ A, μ⊙ AÞ ¼ sup0r1 max fjλAl ðr Þ  μAl ðr Þj, jλAu ðr Þ  μAu ðr Þjg ¼ sup0r1 max fjλ  μjAl ðr Þ, jλ  μjAu ðr Þg ¼ jλ  μj sup0r1 max fAl ðr Þ, Au ðr Þg ¼ jλ  μjDH ðA, 0Þ The proof is now completed. For more explanation, see Bede and Gal (2005) and Stefanini and Bede (2009).

3.4

Limit of fuzzy number valued functions

In this section, we are going to display some preliminarily definitions and theorems about fuzzy set number valued functions (Allahviranloo et al., 2015; Gouyandeha et al., 2017).

Soft computing with uncertain sets

3.4.1

DEFINITION—FUZZY

75

SET VALUED FUNCTION

Any function, like x(t), is called a fuzzy set valued function if it is a fuzzy set for any t  R.

3.4.2

DEFINITION—FUZZY

NUMBER VALUED FUNCTION

Any function, like x(t), is called a fuzzy number valued function if it is a fuzzy number for any t  R.

3.4.3

DEFINITION—THE

LIMIT OF

FUZZY

NUMBER VALUED FUNCTION

Suppose that x(t) is a fuzzy number valued function and is defined from any real interval like [a, b] to R . If for any positive E there is a positive δ such that: 8E > 0 9 δ > 0 8 tðjt  t0 j < δ¼)DH ðxðtÞ, LÞ < EÞ where D is the Hausdorff of x(t) and LR . It is equivalent to the following limit: lim xðtÞ ¼ L

t!t0

Note. The limit of a fuzzy number valued function x(t) exists whenever the value L is a fuzzy number not a fuzzy set. Also if we have: lim xðtÞ ¼ xðt0 Þ

t!t0

Then the function x(t) is called continuous at the point t0. It should be noted that the same function is continuous on its real domain [a, b] if it is continuous at all points of the interval. Now we are going to consider some properties on the limit of a fuzzy number valued function.

3.4.4

THEOREM—LIMIT

OF SUMMATION OF FUNCTIONS

Suppose that xðtÞ, yðtÞ : ½a, b ! R are two fuzzy number valued functions. If: lim xðtÞ ¼ L1 , lim yðtÞ ¼ L2

t!t0

t!t0

Such that two values L1 , L2 R . Then: lim ½xðtÞ
yðtÞ ¼ L1
L2

t!t0

Proof. For the function x(t):
E E 8 > 0 9 δ1 > 0 8 t jt  t0 j < δ1 ¼)DH ðxðtÞ, L1 Þ < 2 2

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Soft Numerical Computing in Uncertain Dynamic Systems

Because lim t!t0 xðtÞ ¼ L1 . And also for another one, y(t), we have:
E E 8 > 0 9 δ2 > 0 8 t jt  t0 j < δ2 ¼)DH ðyðtÞ, L2 Þ < 2 2 Now j t  t0 j < δ1 and j t  t0 j < δ2 clearly j t  t0 j < min {δ1, δ2} ¼ δ. Now for any positive 2E there is a positive δ and based on the property of the Hausdorff distance: E E DH ðxðtÞ
yðtÞ, L1
L2 Þ  DH ðxðtÞ, L1 Þ + DH ðyðtÞ, L2 Þ < + ¼ E 2 2 So finally, we prove that: 8E > 0 9 δ > 0 8 tðjt  t0 j < δ¼)DH ðxðtÞ
yðtÞ, L1
L2 Þ < EÞ The proof is completed.

3.4.5

THEOREM—LIMIT

OF DIFFERENCE OF FUNCTIONS

Considering all assumptions of the previous theorem, we can prove that:

lim xðtÞ gH yðtÞ ¼ L1  gH L2 t!t0

Subject to L1 gHL2 existing. Proof. Suppose that x(t)gHy(t) ¼ z(t), so, based on the definition of gHdifference: 8 xðtÞ ¼ yðtÞ
zðtÞ < iÞ xðtÞ gH yðtÞ ¼ zðtÞ , or : iiÞ yðtÞ ¼ xðtÞ
ð1ÞzðtÞ First, case (i). •

In case (i): Since lim t!t0 xðtÞ ¼ L1 then lim t!t0 ðyðtÞ
zðtÞÞ ¼ L1 and based on the previous theorem we have: lim ðyðtÞ
zðtÞÞ ¼ lim yðtÞ
lim zðtÞ ¼ L1

t!t0

t!t0

t!t0

Now it is concluded that: lim zðtÞ ¼ L1  lim yðtÞ ¼ L1  L2

t!t0

•

t!t0

In case (ii): Since lim t!t0 yðtÞ ¼ L2 then lim t!t0 ðxðtÞ
ð1ÞzðtÞÞ ¼ L2 and again based on the previous theorem we have: lim ðxðtÞ
ð1ÞzðtÞÞ ¼ lim xðtÞ
ð1Þ lim zðtÞ ¼ L2

t!t0

t!t0

t!t0

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77

Now it is concluded that: L1
ð1Þ lim zðtÞ ¼ L2 t!t0

and based on the second case of the gH-difference: lim zðtÞ ¼ L1  L2

t!t0

The proof is completed.

3.4.6

THEOREM—LIMIT

OF MULTIPLICATION

Suppose that xðtÞ : ½a, b ! R is a fuzzy number valued function and y(t) is a nonnegative real function. If: lim xðtÞ ¼ L1 , lim yðtÞ ¼ L2

t!t0

t!t0

Such that the two values L1 R ,L2 R + . Then: lim ½xðtÞ ⊙ yðtÞ ¼ L1 ⊙ L2

t!t0

Proof. To prove it, first we investigate the Hausdorff distance of a fuzzy number valued function x(t) and zero.     DH ðxðtÞ0Þ ¼ DH xðtÞ  gH L1
L1 0  DH xðtÞ gH L1 ,0 + DH ðL1 0Þ ¼ DH ðxðxÞL1 Þ + L1 < E + L1 The E is any arbitrary positive number and it can be considered as E ¼ 1, so: DH ðxðtÞ, 0Þ < 1 + L1 For the function x(t) we have the same limit definition as: 8E1 ¼

E > 0 9 δ1 > 0 8 tðjt  t0 j < δ1 ¼)DH ðxðtÞ, L1 Þ < E1 Þ 2L2

The function y(t) is real and a nonnegative function and we have: 8E2 ¼

E > 0 9 δ2 > 0 8 tðjt  t0 j < δ2 ¼)jyðtÞ  L2 j < E2 Þ 2ð1 + L1 Þ

By all assumptions we have:   DH ðxðtÞ⊙ yðtÞ, L1 ⊙ L2 Þ ¼ DH xðtÞ⊙ yðtÞ  gH L1 ⊙ L2 , 0   ¼ DH xðtÞ ⊙ yðtÞ gH xðtÞ⊙ L2
xðtÞ⊙ L2  gH L1 ⊙ L2 , 0 As we know y(t) and L2 are nonnegative and let us consider (y(t)  L2) 0. So the following result is immediately concluded. xðtÞ ⊙ yðtÞ gH xðtÞ⊙ L2 ¼ xðtÞ ⊙ ðyðtÞ  L2 Þ

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Soft Numerical Computing in Uncertain Dynamic Systems

Considering the level cuts of both sides, it is proved easily. Moreover:   xðtÞ⊙ L2  gH L1 ⊙ L2 ¼ xðtÞ gH L1 ⊙ L2 Now the proof is continued as:   DH xðtÞ⊙ yðtÞ  gH xðtÞ⊙ L2
xðtÞ⊙ L2  gH L1 ⊙ L2 , 0 ¼     ¼DH xðtÞ⊙ ðyðtÞ  L2 Þ
xðtÞ gH L1 ⊙ L2 , 0     DH ðxðtÞ⊙ ðyðtÞ  L2 Þ, 0Þ + DH xðtÞ gH L1 ⊙ L2 , 0   ¼jyðtÞ  L2 jDH ðxðtÞ, 0Þ + jL2 jDH xðtÞ  gH L1 , 0 ¼jyðtÞ  L2 jDH ðxðtÞ, 0Þ + jL2 jDH ðxðtÞ, L1 Þ
0 9 δ > 0 DH
Δti ⊙ f ðti Þ, J < E i¼0

Note. The important point is, J should be a fuzzy number not a fuzzy set. In the level-wise form of integral, it can be displayed as: !  ðb n X Δti ⊙ f ðti Þ ½r J ½r ¼ FR f ðtÞdt ½r ≔
a

i¼0

for any 0  r  1. Indeed the definition of the Hausdorff distance needs the level-wise form and it can be brought as: " # n n X X ½Jl ðr Þ, Ju ðr Þ ¼ Δti  fl ðti , r Þ, Δti  fu ðti , r Þ i¼0

i¼0

 ðb  ðb ¼ R fl ðt, r Þdt, R fu ðt, r Þdt a

a

On the one hand: " # n n n X X X Δti  fl ðti , r Þ, Δti  fu ðti , r Þ ¼ Δti ½fl ðti , r Þ, fu ðti , r Þ i¼0

i¼0

i¼0

then:  ðb  ðb ðb R fl ðt, r Þdt, R fu ðt, r Þdt ¼ R ½fl ðt, r Þ, fu ðt, r Þ dt a

a

a

In summary: Jl ð r Þ ¼ R

ðb

fl ðt, r Þdt ¼

n X

a

Ju ðr Þ ¼ R

ðb a

Δti  fl ðti , r Þ :

i¼0

fu ðt, r Þdt ¼

n X i¼0

Δti  fu ðti , r Þ

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Soft Numerical Computing in Uncertain Dynamic Systems

3.5.1 •

SOME

PROPERTIES OF FUZZY

RIEMANN

INTEGRAL

Suppose that the functions f ðtÞ, gðtÞR and are Riemann integrable functions. Then: ðb FR

ðf ðtÞ
gðtÞÞdt ¼


a

n X

Δti ⊙ ðf ðti Þ
gðti ÞÞ

i¼0

This property is easy to prove. •

For the previous function and any c  [a, b]: ðb FR

f ðtÞdt ¼ FR

ðc

a

f ðtÞdt
FR

ðb

a

f ðtÞdt

c

To prove the property, since c  [a, b] the partition for the integral will be as: fa ¼ t0 < t1 < ⋯ < tk ¼ c < ⋯ < tn ¼ bg, 9 k f0, 1, …, ng Now based on the definition: ðc FR

f ðtÞdt ¼


a

ðb FR

k X

Δti ⊙ f ðti Þ

i¼0

f ðtÞdt ¼


c

n X

Δti ⊙ f ðti Þ

i¼k + 1

It is clear that:


k X

Δti ⊙ f ðti Þ


i¼0

•

n X

Δti ⊙ f ðti Þ ¼


i¼k + 1

n X

Δti ⊙ f ðti Þ

i¼0

The property is now proved. Ð Ð (1)FR ba f(t)dt ¼ FR ab f(t)dt

3.6

Differential operator

The main discussion of this chapter is related to the differential operator. Generally, a differential operator is a nonlinear operator on its domain, which is defined by a differential expression. Usually the operator is acting on a space of differentiable vector-valued functions. In this section, we suppose that the operand function is a fuzzy number valued function and we shall consider the effect of a derivative operator on it. Basically, the derivative is defined by the rate of changes and here the changes are

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81

related to the changes of fuzzy set valued function. In this section we will also consider the first and high order of differentiability of a fuzzy number valued function.

3.6.1

DEFINITION—GH-DIFFERENTIABILITY

As before, suppose that the function x(t) is a fuzzy number valued function and t0 is an inner point of its domain as an interval like [a, b]. So we consider t0  (a, b). We also suppose that for any small enough number h ! 0, t0 + h  (a, b). Now the gH-differential of the x(t) at the point t0 is denoted by xgH0 (t0) and defined as: xðt0 + hÞ gH xðt0 Þ h!0 h

x0gH ðt0 Þ ¼ lim

Subject to the gH-difference x(t0 + h)gHx(t0) exists (Bede and Gal, 2005). Note. The point is that the gH-differential exists at the point t0 if xgH0 (t0) is a fuzzy number not a fuzzy set. The level-wise form of this fuzzy number as a derivative can be explained as the following two cases, because the gH-difference in the definition is defined in two cases. So these two cases are considered separately and in general we can claim and prove that the necessary and sufficient conditions for the gH-differentiability a fuzzy number valued function x(t) are: In case (i):


x0igH ðtÞ ½r ¼ x0l ðt0 , r Þ, x0u ðt0 , r Þ In case (ii):






x0iigH ðtÞ ½r ¼ x0u ðt0 , r Þ, x0l ðt0 , r Þ

Subject to the functions xl0 (t0, r) and xu0 (t0, r) being two real valued differentiable functions with respect to t and uniformly with respect to r  [0, 1]. It should be noted that both of the functions are left continuous on r  (0, 1] and right continuous at r ¼ 0. Moreover, the following conditions should be satisfied. • • •

The function xl0 (t0, r) is nondecreasing and the function xu0 (t0, r) is nonincreasing as functions of r and, xl0 (t0, r)  xu0 (t0, r). Or: The function xl0 (t0, r) is nonincreasing and the function xu0 (t0, r) is nondecreasing as functions of r and, xu0 (t0, r)  xl0 (t0, r). (xgH0 (t0))[r] ¼ [min{xl0 (t0, r), xu0 (t0, r)}, max{xl0 (t0, r), xu0 (t0, r)}]

3.6.1.1 Example Let the function x(t) ¼ μ ⊙ t be a fuzzy number function, and μ be a fuzzy number and t a real number (Stefanini and Bede, 2009).

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Soft Numerical Computing in Uncertain Dynamic Systems

Based on the definition of the gH-differentiability we have: xðt + hÞ gH xðtÞ h!0 h

x0igH ðtÞ ¼ lim

Suppose that the gH-difference exists in case (i), it means if we define it as the following: xðt + hÞ  gH xðtÞ ¼ ðμ ⊙ ðt + hÞÞ  gH μ ⊙ t ¼ ½μ ⊙ t
μ ⊙ h  gH μ ⊙t Considering the properties of the gH-difference: xðt + hÞ gH xðtÞ ¼ ðμ ⊙ ðt + hÞÞ gH μ ⊙ t ¼ μ ⊙ h Case (i), If t 0 then h > 0 and x(t + h) ¼ x(t)
μ ⊙ h ¼ μ ⊙ t
μ ⊙ h. This is exactly the definition of gH-difference in case (i). So, the differential also exists in case (i) and: x0igH ðtÞ ¼ lim h↗0

μ⊙h ¼μ h

In the level-wise form:


x0igH ðtÞ ½r ¼ x0l ðt, r Þ, x0u ðt, r Þ ¼ ½μl ðr Þ, μu ðr Þ Case (ii): Based on the definition of gH-difference in case (ii): xðt + hÞ gH xðtÞ ¼ ðμ ⊙ ðt + hÞÞ  gH μ ⊙ t ¼ yðt; hÞ then: μ ⊙ t ¼ ðμ ⊙ ðt + hÞÞ
ð1Þyðt; hÞ ¼ ðμ ⊙ t
μ ⊙ hÞ
ð1Þyðt; hÞ Finally: μ ⊙ t ¼ ðμ ⊙t
μ ⊙ hÞ
ð1Þyðt; hÞ This is also true if y(t; h) ¼  ((1)h) ⊙ μ and it happens only for t < 0 and h < 0: This is investigated by using the level-wise form: ðμ ⊙ tÞl ðr Þ ¼ μu ðr Þt and for the right-hand side: ððμ ⊙ t
μ ⊙ hÞ
ð1Þð  ðð1ÞhÞ ⊙μÞÞl ðr Þ ¼ ððμ ⊙ t
μ ⊙ hÞ ðh ⊙ μÞÞl ðr Þ ¼ μu ðr Þt Finally, the derivative is as: x0igH ðtÞ ¼ lim h↘0

 ðð1ÞhÞ ⊙ μ ¼  ðð1Þ⊙ μÞ h

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In the level-wise form:


x0iigH ðtÞ ½r ¼ x0u ðt, r Þ, x0l ðt, r Þ ¼ ½μu ðr Þ, μl ðr Þ

3.6.1.2 Example Suppose the following function: 8   1 < 2 , a 1 + t cos xðtÞ ¼ t : a,

t 6¼ 0 t¼0

where a[r] ¼ [r  1, 1  r] or a ¼ (1, 0, 1), is a fuzzy number. It is clear that: 8      > < ðr  1Þ 1 + t2 cos 1 , ð1  r Þ 1 + t2 cos 1 , t 6¼ 0 t t xðtÞ½r ¼ > : ½r  1, 1  r , t¼0 It can easily be seen that x(t) is gH-differentiable at the point t ¼ 0 and xgH0 (0) ¼ 0. But the following H-differences do not exist. xðhÞ H xð0Þ because there does not exist a δ > 0 such that the H-difference exists for all h  (0, δ).

3.6.1.3 Definition—gH-differentiability in level-wise form For the fuzzy number valued function x : ½a, b ! R and t0  (a, b) with xl0 (t, r) and xu0 (t, r) both differentiable at t0, the gH-differentiability at the same point is defined in the following cases in level-wise form. •

x is [i  gH]-differentiable at t0 if:



x0igH ðt0 , r Þ ¼ x0l ðt0 , r Þ, x0u ðt0 , r Þ

•

x is [ii  gH]-differentiable at t0 if:

x0iigH ðt0 , r Þ ¼ x0u ðt0 , r Þ, x0l ðt0 , r Þ

3.6.1.4 Definition—Switching points of gH-differentiability We say that a point t0  (a, b) is a switching point for the differentiability of f, if in any neighborhood V of t0 there exist points t1 < t0 < t2 such that: Type (I), at the point t1, xigH0 (t0, r) hold while xiigH0 (t0, r) does not hold and at the point t2, xiigH0 (t0, r) hold while xigH0 (t0, r) does not hold; or Type (II), at the point t1, xiigH0 (t0, r) hold while xigH0 (t0, r) does not hold and at the point t2, xigH0 (t0, r) hold while xiigH0 (t0, r) does not hold.

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In type (II), the left neighborhood of t1is ii  gH differentiability and on the righthand side it is i  gH differentiable. So on the left-hand side:

x0iigH ðt1 , r Þ ¼ x0u ðt1 , r Þ, x0l ðt1 , r Þ , x0u ðt1 , r Þ  x0l ðt1 , r Þ This means: xu ðt1 + hÞ  xu ðt1 Þ  xl ðt1 + hÞ  xl ðt1 Þ then: length½xðt1 + hÞ ¼ xu ðt1 + hÞ  xl ðt1 + hÞ  xu ðt1 Þ  xl ðt1 Þ ¼ length½xðt1 Þ This means that the length operator is a decreasing one. Also on the right-hand side:

x0igH ðt1 , r Þ ¼ x0l ðt2 , r Þ, x0u ðt2 , r Þ , x0l ðt2 , r Þ  x0u ðt2 , r Þ This means: xl ðt2 + hÞ  xl ðt2 Þ  xu ðt2 + hÞ  xu ðt2 Þ then: length½xðt2 Þ ¼ xu ðt2 Þ  xl ðt2 Þ  xu ðt2 + hÞ  xl ðt2 + hÞ ¼ length½xðt2 + hÞ On the right-hand side, the length is increasing (see Fig. 3.1). In type (II), the left neighborhood of t1is i  gH differentiability and on the righthand side it is ii  gH differentiable (see Fig. 3.2).

x′t–gH (t1,r)

x′u–gH (t1,r)

t1 Fig. 3.1

Type II switching point.

t1 + h

t0

t2

t2 + h

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x′u–gH (t1,r)

x′t–gH (t1,r)

t1

t1 + h

t2

t0

t2 + h

Fig. 3.2 Type I switching point.

3.6.1.4.1 Example Let us consider the fuzzy set valued function given level-wise for the interval t  [0, 1].
2  xl ðt, r Þ ¼ tet + r 2 et + t  tet   
2 2 xu ðt, r Þ ¼ et + t + 1  r 2 et  t + et The derivatives of two functions are:
 2 x0l ðt, r Þ ¼ ð1  tÞet + r 2 1 + ðt  1Þet  2tet
 2 2 x0u ðt, r Þ ¼ 4tet + et  r 2 1 + et  2tet It is easy to see it is not gH-differentiable, because if r ¼ 1, both are the same as 2 1  2te t and if r ¼ 0 then: x0l ðt, r Þ ¼ ð1  tÞet x0u ðt, r Þ ¼ 4tet + et 2

and xu0 (t, r)  xl0 (t, r). Fig. 3.3 shows the derivatives in two levels r ¼ 0.0.5.

3.6.1.5 Proposition—Summation in gH-differentiability For two gH-differentiable fuzzy number valued functions like x(t), y(t) it is clearly proved that if: zðtÞ ¼ xðtÞ
yðtÞ

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1

0.8

0.6

0.4

0.2

0

Fig. 3.3

0

0.2

0.4

0.6

0.8

1

The level-wise form of derivatives for r ¼ 0.0.5.

then: z0igH ðtÞ ¼ x0igH ðtÞ
y0igH ðtÞ z0iigH ðtÞ ¼ x0iigH ðtÞ
y0iigH ðtÞ The proof is clear based on the definition of the differential in each type.

3.6.1.6 Proposition—Difference in gH-differentiability Again for two gH-differentiable fuzzy number valued functions like x(t), y(t), it is clearly proved that if: zðtÞ ¼ xðtÞ  gH yðtÞ then: z0igH ðtÞ ¼ x0igH ðtÞ  gH y0igH ðtÞ z0iigH ðtÞ ¼ x0iigH ðtÞ gH y0iigH ðtÞ

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These two equations are proved separately as follows in accordance with the definition of gH-difference:

ði Þ xðtÞ ¼ yðtÞ
zðtÞ or zðtÞ ¼ xðtÞ gH yðtÞ⟺ ðiiÞ zðtÞ ¼ xðtÞ
ð1ÞyðtÞ First consider case (i), and we know that those differentiabilities are the same. x0igH ðtÞ ¼ y0igH ðtÞ
z0igH ðtÞ x0iigH ðtÞ ¼ y0iigH ðtÞ
z0iigH ðtÞ It is concluded that: z0igH ðtÞ ¼ x0igH ðtÞ gH y0igH ðtÞ z0iigH ðtÞ ¼ x0iigH ðtÞ  gH y0iigH ðtÞ Now, in case (ii) we have: z0igH ðtÞ ¼ x0igH ðtÞ
ð1Þy0igH ðtÞ z0iigH ðtÞ ¼ x0iigH ðtÞ
ð1Þy0iigH ðtÞ Because for any real number λ we have:     λ A  igH B ¼ λA  igH λB     λ A  iigH B ¼ λA  iigH λB It can be proved in level-wise form of gH-difference in two cases separately.

3.6.1.7 Proposition—Production in gH-differentiability Suppose that x : ½a, b ! R is a fuzzy number valued function and it is gHdifferentiable on (a, b). Also, let us suppose the function y : [a, b] ! R is a differentiable real function in the same open interval. Then: ðx⊙ yÞ0gH ðtÞ ¼ x0gH ðtÞ ⊙ yðtÞ
xðtÞ⊙ y0 ðtÞ To prove this property, we use the Hausdorff distance. The left-hand side is as: ðx ⊙ yÞ0gH ðtÞ ¼ lim

h!0

xðt + hÞ⊙ yðt + hÞ gH xðtÞ⊙ yðtÞ h

Now considering the distance of two sides of the equation, it is enough to show that the distance intends zero.

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DH  DH

 xðt + hÞ ⊙ yðt + hÞ gH xðtÞ ⊙ yðtÞ 0 , xgH ðtÞ ⊙ yðtÞ
xðtÞ⊙ y0 ðtÞ ¼ h

xðt + hÞ ⊙ yðt + hÞ  gH xðtÞ⊙ yðt + hÞ
xðtÞ⊙ yðt + hÞ gH xðtÞ⊙ yðtÞ 0 , xgH ðtÞ⊙ yðtÞ
xðtÞ⊙ y0 ðtÞ h 0


¼ DH @


 xðt + hÞ  gH xðtÞ ⊙ yðt + hÞ
xðtÞ⊙ yðt + hÞ  gH ⊙ yðtÞ h 0
 DH @

 xðt + hÞ  gH xðtÞ ⊙ yðt + hÞ h

0
DH @ 0
¼ DH @

xðt + hÞ  gH xðtÞ

0

DH @

h

1

1 , xðtÞ ⊙ y0 ðtÞA



1 ⊙ yðt + hÞ, x0gH ðtÞ ⊙ yðtÞA

yðt + hÞ  gH ⊙ yðtÞ h

, x0gH ðtÞ⊙ yðtÞ
xðtÞ⊙ y0 ðtÞA

, x0gH ðtÞ ⊙ yðtÞA


 xðtÞ⊙ yðt + hÞ  gH ⊙ yðtÞ h

1



1 ⊙ xðtÞ, xðtÞ ⊙ y ðtÞA 0

Now, the limits of the two sides when h ! 0 are:  xðt + hÞ⊙ yðt + hÞ  gH xðtÞ⊙ yðtÞ 0 , xgH ðtÞ⊙ yðtÞ
xðtÞ⊙ y0 ðtÞ lim DH h!0 h    xðt + hÞ gH xðtÞ ⊙ yðt + hÞ, x0gH ðtÞ⊙ yðtÞ ¼ lim DH h!0 h   yðt + hÞ  gH ⊙ yðtÞ 0 ⊙ xðtÞ, xðtÞ⊙ y ðtÞ
lim DH h!0 h Based on the properties of the limit and distance (as we expressed before) the proof is completed.  xðt + hÞ gH xðtÞ 0 ⊙ yðtÞ, xgH ðtÞ ⊙ yðtÞ ¼ DH lim h  h!0 yðt + hÞ gH yðtÞ ⊙ xðtÞ, xðtÞ⊙ y0 ðtÞ ¼ 0
DH lim h!0 h

3.6.1.8 Proposition—Composition of gH-differentiability Suppose that x : ½a, b ! R is a fuzzy number valued function and it is gHdifferentiable at y(t) on the interval (a, b) (Allahviranloo et al., 2015). Also, let us



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suppose the function y : [a, b] ! R is a differentiable real function in the same open interval. Then: If y0 (t) 0: ðx o yÞ0igH ðtÞ ¼ y0 ðtÞ⊙ x0igH ðyðtÞÞ : ðx o yÞ0iigH ðtÞ ¼ y0 ðtÞ ⊙ x0iigH ðyðtÞÞ: (if ) y0 (t) < 0, ðx o yÞ0igH ðtÞ ¼ y0 ðtÞ ⊙ x0iigH ðyðtÞÞ, ðx o yÞ0iigH ðtÞ ¼ y0 ðtÞ⊙ x0igH ðyðtÞÞ: The proof for two cases can be displayed in level-wise form. First suppose y0 (t) 0 and (x o y)(t) is differentiable in case (i), then for any r  [0, 1] we determine the level-wise form of composite function as:

ðx o yÞ0igH ðt, r Þ ¼ ðx o yÞ0l ðt, r Þ, ðx o yÞ0u ðt, r Þ Now the level-wise form of the right-hand side is: 0

y ðtÞ  x0l ðyðtÞÞ, y0 ðtÞ  x0u ðyðtÞÞ so: ðx o yÞ0l ðt, r Þ ¼ y0 ðtÞ  x0l ðyðtÞÞ ðx o yÞ0u ðt, r Þ ¼ y0 ðtÞ  x0u ðyðtÞÞ The proof is completed, and for type (ii) differentiability the same holds. Now for the second case we have a similar process, but the coefficient y0 (t) is negative and it will change the endpoints of the intervals in level-wise form:



ðx o yÞ0l ðt, r Þ, ðx o yÞ0u ðt, r Þ ¼ y0 ðtÞ  x0u ðyðtÞÞ, y0 ðtÞ  x0l ðyðtÞÞ then: ðx o yÞ0igH ðtÞ ¼ y0 ðtÞ⊙ x0iigH ðyðtÞÞ The same process is used to prove the type (ii) differentiability of the second case.

3.6.1.9 Proposition—Minimum and maximum For any gH-differentiable fuzzy number function x(t) at the inner point like c  (a, b), if the function has local minimum or maximum then xgH0 (c) ¼ 0.

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To prove the proposition, we use the definition of the minimum and maximum value. The function at the point c has the local maximum value if there is a positive δ and for all t from the neighborhood j t  cj < δ then x(t) ≼ x(c). Then x(t)gHx(c) ≼ 0. On the other hand, for all t  (a, c), t  c < 0. Now it can be concluded that: x0gH ðtÞ ¼ lim

h!0

xðtÞ  gH xðcÞ ≽0 tc

All these properties about the partial ordering have been discussed in Chapter 2. Similarly, the function at the point c has the local minimum value if there is a positive δ and for all t from the neighborhood j t  c j < δ then x(c) ≼ x(t). Finally, we have: x0gH ðtÞ ¼ lim

h!0

xðtÞ  gH xðcÞ ≼0 tc

then: x0gH ðtÞ ≽ 0 and x0gH ðtÞ ≼ 0 then x0gH ðtÞ ¼ 0

3.6.1.10 Definition—Continuous fuzzy number valued function Consider the fuzzy number valued function x : ½a, b ! R . We say the function is continuous at a point like t0  [a, b] if for any E > 0 9 δ > 0 subject to DH(x(t), x(t0)) < E whenever x is an arbitrary value from jx  x0 j < δ. Moreover we say the function is continuous on [a, b] if it is continuous at each point of the interval. Also, another form of definition is level-wise continuity, which means the function as a fuzzy number valued function is continuous if and only if it is continuous in each level. According the ordering method and the definition of the continuity, it is concluded that if the same function x(t) is continuous on the closed and bounded interval [a, b] then it must attain a maximum and minimum each at least once. So: 8t  ½a, b 9 tMin 9 tMax  ½a, b s:t: xðtMin Þ ≼ xðtÞ ≼ xðtMax Þ where max atb xðtÞ ¼ xðtMax Þ and min atb xðtÞ ¼ xðtMin Þ. For more information on this, see Allahviranloo et al. (2015) and Gouyandeha et al. (2017).

3.6.1.11 Proposition Suppose that the fuzzy number valued function x : ½a, b ! R is continuous and gHdifferentiable on (a, b). If x(a) ¼ x(b) then there exists c  (a, b) such that xgH0 (c) ¼ 0.

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To prove the claim, we use x(a) ¼ x(b) and x(tMin) ≼ x(t) ≼ x(tMax). If it is the case that x(tMin) ¼ x(tMax), it is a constant fuzzy number function and immediately we have xgH0 (c) ¼ 0 for all t  [a, b]. Then it is true to say x(tMin) x(tMax) and in this case one of the x(tMin) and x(tMax) do not equal x(a) ¼ x(b). Without loss of the generality, assume that x(tMax) 6¼ x(a) ¼ x(b); the other case is similar and we omit that. Clearly tMax  (a, b) and the function x(t) is gH-differentiable at tMax so from the previous proposition it does have local minimum and maximum and xgH0 (c) ¼ 0.

3.6.1.12 Proposition—Cauchy’s fuzzy mean value theorem Assume that x(t) is a continuous and gH-differentiable fuzzy number valued function on closed and open intervals, respectively, and y(t) is a real valued continuous and differentiable function on the same intervals. Then there is c  (a, b) such that:

xðbÞ gH xðaÞ ⊙ y0 ðcÞ ¼ ½yðbÞ  yðaÞ ⊙ x0gH ðcÞ where the gH-difference exists. Proof. Let us consider a new function as follows:

ϕðtÞ ¼ xðbÞ gH xðaÞ ⊙ yðtÞ  gH ½yðbÞ  yðaÞ ⊙ xðtÞ This function is continuous because: Let us consider x(b)gHx(a) ¼ k and y(b)  y(a) ¼ l so we have: ϕðtÞ ¼ k ⊙ yðtÞ  gH l⊙ xðtÞ Now for any E > 0 9 δ > 0 if for any t in j t  t0 j < δ at an arbitrary point t0 we prove:   DH ðϕðtÞ, ϕðt0 ÞÞ ¼ DH k ⊙ yðtÞ gH l⊙ xðtÞ, k ⊙yðt0 Þ  gH l ⊙ xðt0 Þ Based on the properties of the Hausdorff distance:  DH ðk ⊙ yðtÞ, k ⊙ yðt0 ÞÞ + DH ðl⊙ xðtÞ, l⊙ xðt0 ÞÞ Based on the definition of Hausdorff distance and definition of absolute value of a fuzzy number we can write:  jkjjyðtÞ  yðt0 Þj + jljDH ðxðtÞ, xðt0 ÞÞ and finally this summation is less than E. On the other hand ϕ(a) ¼ ϕ(b), so it is gH-differentiable at the inner point like c and equals zero. Then:

ϕ0gH ðtÞ ¼ xðbÞ gH xðaÞ ⊙ y0 ðtÞ  gH ½yðbÞ  yðaÞ ⊙ x0gH ðtÞ

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and ϕgH0 (c) ¼ 0, then:

xðbÞ gH xðaÞ ⊙ y0 ðcÞ gH ½yðbÞ  yðaÞ ⊙ x0gH ðcÞ ¼ 0 The proof is completed. As a corollary, if we take the real function y(t) ¼ t then we will have the fuzzy mean value theorem.

3.6.1.13 Corollary—Fuzzy mean value theorem Considering all the previous assumptions and y(t) ¼ t we have:

xðbÞ gH xðaÞ ⊙ y0 ðcÞ ¼ ½yðbÞ  yðaÞ ⊙ x0gH ðcÞ¼)

xðbÞ gH xðaÞ ¼ ðb  aÞ ⊙ x0gH ðcÞ then: x0gH ðcÞ ¼

xðbÞ gH xðaÞ ba

3.6.1.14 Proposition—Increasing and decreasing function Let us consider that the fuzzy number valued function x : ½a, b ! R is continuous and gH-differentiable on (a, b). Then: • • •

If xgH0 (t) ≽ 0 for all t  (a, b) then it is increasing on [a, b]. If xgH0 (t) ≼ 0 for all t  (a, b) then it is decreasing on [a, b]. If xgH0 (t) ¼ 0 for all t  (a, b) then it is constant on [a, b].

For the first case, assume that xgH0 (t) ≽ 0 for all t  (a, b). For any t1, t2  (a, b), t1 < t2 there is c  (a, b) subject to: x0gH ðcÞ ¼

xðt2 Þ gH xðt1 Þ t2  t1

It can be concluded that: xðt2 Þ gH xðt1 Þ ¼ x0gH ðcÞ ⊙ ðt2  t1 Þ≽0 Then x(t2) ≽ x(t1) and this means that the function is an increasing one. A similar process is introduced to prove the other cases.

3.6.1.15 Proposition—Integral of gH-differentiability Suppose that the fuzzy number valued function x : ½a, b ! R does not have any switching point in its domain. Then:

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93

x0gH ðtÞdt ¼ xðbÞ  gH xðaÞ

To prove it, we should consider two cases of gH-differentiability of x. Case (i), the integrand function is ii  gH differentiable:  ðb ðb ðb 0

xu ðt, r Þ, x0l ðt, r Þ dt FR x0iigH ðtÞdt ½r ¼ R x0iigH ðt, r Þdt ¼ R a

a

a

 ðb  ðb 0 0 ¼ R xu ðt, r Þdt, R xl ðt, r Þdt ¼ ½xu ðb, r Þ  xu ða, r Þ, xl ðb, r Þ  xl ða, r Þ a

a

¼ ðxðbÞ  xðaÞÞ½r Then: ðb FR a

x0iigH ðtÞdt ¼  ð1ÞðxðbÞ xðaÞÞ

or:  ð1ÞFR

ðb a

x0iigH ðtÞdt ¼ ðxðbÞ xðaÞÞ

The proof for the first case is similar. If we want to consider two types of differentiability for any arbitrary but fixed t, we will have: Case (i), i  gH ðt xðtÞ ¼ xðaÞ
FR x0igH ðsÞds a

Case (ii), ii  gH xðtÞ ¼ xðaÞ  ð1ÞFR

ðt a

x0iigH ðsÞds

3.6.1.16 Proposition—Switching points in integration Let us suppose that the fuzzy number valued function x : ½a, b ! R is gHdifferentiable with n switching points at ci, i ¼ 1, 2, …, n, and a ¼ c0 < c1 < c2 < ⋯ < cn < cn+1 ¼ b. Then:  ð ci ð ci + 1 n  X xðbÞ  gH xðaÞ ¼ FR x0igH ðtÞdt gH ð1ÞFR x0iigH ðtÞdt i¼1

ci1

ci

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and also: ðb FR a

x0gH ðtÞdt ¼


n+1 X

xðci Þ  gH xðci1 Þ

i¼1

3.6.1.17 High order differentiability As before, suppose that the function x(t) is a fuzzy number valued function and t0 is an inner point of its domain as an interval, like [a, b]. So we consider t0  (a, b). Also we suppose that for any enough small number h ! 0, t0 + h  (a, b). Now the n-th order gH-differential of the x(t) at the point t0 is denoted by x(n) gH(t0) and is defined as: ðn1Þ

ðnÞ

xgH ðt0 Þ ¼ lim

ðn1Þ

xgH ðt0 + hÞ gH xgH ðt0 Þ

h!0

h

(n1) x(n1) gH (t0 + h)gHxgH (t0)

Subject to the gH-difference exists for any order of gH-differentiability or any n ¼ 1, 2, …, m. Moreover, we have to assume that all the previous derivatives exist and do not have any switching points. Note. The point is that the n-th order gH-differential exists at the point t0 if x(n) gH(t0) is a fuzzy number not a fuzzy set. The necessary and sufficient conditions for the gH-differentiability as a fuzzy number valued function x(t) are: In case (i): h i ðnÞ ðnÞ xgH ðt0 , r Þ ¼ xl ðt0 , r Þ, xðunÞ ðt0 , r Þ In case (ii): h i ðnÞ ðnÞ xgH ðt0 , r Þ ¼ xðunÞ ðt0 , r Þ, xl ðt0 , r Þ (n) Subject to the functions x(n) l (t0, r) and xu (t0, r) are two real valued differentiable functions with respect to t and uniformly with respect to r  [0, 1]. Note that both of the functions are left continuous on r  (0, 1] and right continuous at r ¼ 0. Moreover, the following conditions should be satisfied for all n.

• •

(n) The function x(n) l (t0, r) is nondecreasing and the function xu (t0, r) is nonincreas(n) (n) ing as functions of r and, xl (t0, r)  xu (t0, r). Or: (n) (n) (n) (n) x(n) gH(t0, r) ¼ [min{xl (t0, r), xu (t0, r)}, max{xl (t0, r), xu (t0, r)}]

3.6.1.18 Extended integral relation Let x(n) gH(t) be a continuous fuzzy number valued function for any s  (a, b), the following integral equations are valid.

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Item 1. Consider x(n) gH(t), n ¼ 1, 2, …, m is (i  gH)-differentiable and the type of differentiability does not change on (a, b).  ðs ðs
 ðnÞ ðnÞ FR xigH ðtÞdt ½r ¼ FR xigH ðtÞ ½r dt ¼ a

a

 ðs  ðs ðnÞ ¼ R xl ðt, r Þdt, R xðunÞ ðt, r Þdt a

h

ðn1Þ

¼ xl

a

ðn1Þ

ðs, r Þ  xl

ða, r Þ, xðun1Þ ðs, r Þ  xðun1Þ ða, r Þ

i

h i h i ðn1Þ ðn1Þ ¼ xl ðs, r Þ, xðun1Þ ðs, r Þ  xl ða, r Þ, xðun1Þ ða, r Þ

 ðn1Þ ðn1Þ ¼ xigH ðsÞ ½r  xigH ðaÞ ½r Then it is concluded: ðs

ðnÞ

ðn1Þ

ðn1Þ

xigH ðtÞdt ¼ xigH ðsÞ xigH ðaÞ

FR a ðn1Þ

ðn1Þ

xigH ðsÞ ¼ xigH ðaÞ
FR

ðs a

ðnÞ

xigH ðtÞdt

Item 2. Consider x(n) gH(t), n ¼ 1, 2, …, m is (ii  gH)-differentiable and the type of differentiability does not change on (a, b).  ðs ðs ðnÞ ðnÞ FR xiigH ðtÞdt ½r ¼ FR xiigH ðt, r Þdt ¼ a

a

 ðs  ðs ðnÞ ðnÞ ¼ R xu ðt, r Þdt, R xl ðt, r Þdt h

a

a

ðn1Þ

¼ xðun1Þ ðs, r Þ  xðun1Þ ða, r Þ, xl

ðn1Þ

ðs, r Þ  xl

ða, r Þ

i

h i h i ðn1Þ ðn1Þ ¼ xðun1Þ ðs, r Þ, xl ðs, r Þ  xðun1Þ ða, r Þ, xl ða, r Þ ðn1Þ

ðn1Þ

¼ xiigH ðs, r Þ  xiigH ða, r Þ Then it is concluded: ðs FR a ðn1Þ

ðnÞ

ðn1Þ

ðn1Þ

xiigH ðtÞdt ¼ xiigH ðsÞ xiigH ðaÞ ðn1Þ

xiigH ðsÞ ¼ xiigH ðaÞ
FR

ðs a

ðnÞ

xiigH ðtÞdt

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Soft Numerical Computing in Uncertain Dynamic Systems

(n1) Item 3. Consider x(n) gH(t) is (i  gH)-differentiable and xgH (t) is (ii  gH)differentiable, then:  ðs ðs ðnÞ ðnÞ  ð1Þ FR xigH ðtÞdt ½r ¼  ð1ÞFR xigH ðt, r Þdt

a

a

 ðs  ðs ðnÞ ¼ R xðunÞ ðt, r Þdt, R xl ðt, r Þdt a

a

h i ðn1Þ ðn1Þ ¼ xðun1Þ ðs, r Þ  xðun1Þ ða, r Þ, xl ðs, r Þ  xl ða, r Þ h i h i ðn1Þ ðn1Þ ¼ xðun1Þ ðs, r Þ, xl ðs, r Þ  xðun1Þ ða, r Þ, xl ða, r Þ ðn1Þ

ðn1Þ

¼ xiigH ðs, r Þ  xiigH ða, r Þ Then it is concluded:  ð1ÞFR

ðs a

ðn1Þ

ðnÞ

ðn1Þ

ðn1Þ

xigH ðtÞdt ¼ xiigH ðsÞ  xiigH ðaÞ ðn1Þ

xiigH ðsÞ ¼ xiigH ðaÞ ð1ÞFR

ðs a

ðnÞ

xigH ðtÞdt

(n1) Item 4. Consider x(n) gH(t) is (ii  gH)-differentiable and xgH (t) is (i  gH)differentiable, then:  ðs ðs ðnÞ ðnÞ  ð1Þ FR xiigH ðtÞdt ½r ¼  ð1ÞFR xiigH ðt, r Þdt

a

a

 ðs  ðs ðnÞ ðnÞ ¼ R xl ðt, r Þdt, R xu ðt, r Þdt h

a

a

ðn1Þ

¼ xl

ðn1Þ

ðs, r Þ  xl

ða, r Þ, xðun1Þ ðs, r Þ  xðun1Þ ða, r Þ

i

h i h i ðn1Þ ðn1Þ ¼ xl ðs, r Þ, xðun1Þ ðs, r Þ  xl ða, r Þ, xðun1Þ ða, r Þ ðn1Þ

ðn1Þ

¼ xigH ðs, r Þ  xigH ða, r Þ Then it is concluded:  ð1ÞFR

ðs a

ðnÞ

ðn1Þ

ðn1Þ

xiigH ðtÞdt ¼ xigH ðsÞ xigH ðaÞ

ðn1Þ ðn1Þ xigH ðsÞ ¼ xigH ðaÞ  ð1ÞFR

ðs a

ðnÞ

xiigH ðtÞdt

Soft computing with uncertain sets

97

3.6.1.19 Part-by-part integration Consider the function x(t) as a gH-differentiable fuzzy number valued function on [a, b] and y(t) as a real valued function on the same interval. Then: ðb ðb 0 xgH ðtÞ ⊙ yðtÞdt ¼ ðxðbÞ ⊙ yðbÞÞ  ðxðaÞ⊙ yðaÞÞ gH xðtÞ ⊙ y0 ðtÞdt a

a

To prove the relation, we use the derivative of the combination of these two functions. ðx⊙ yÞ0gH ðtÞ ¼ x0gH ðtÞ ⊙ yðtÞ
xðtÞ⊙ y0 ðtÞ Integrating both sides with respect to t over the interval [a, b], we will have: ðb
ðb ðb  0 0 ðx⊙ yÞgH ðtÞdt ¼ xgH ðtÞ ⊙ yðtÞ dt
ðxðtÞ⊙ y0 ðtÞÞdt a

a

a

The left-hand side can be obtained immediately as: ðb
ðb  x0gH ðtÞ ⊙ yðtÞ dt
ðxðtÞ⊙ y0 ðtÞÞdt ðxðbÞ ⊙ yðbÞÞ  H ðxðaÞ⊙ yðaÞÞ ¼ a

a

Now, based on the definition of the H-difference, the proof is completed.

3.6.1.20 Taylor expansion Item 1. Let us assume the same continuous fuzzy number valued function and all the derivatives are (i  gH)-differentiable for n ¼ 1, 2, …, m without changing the type of differentiability. Then, based on the previous case, we have the following relation. ðs ð1Þ xðsÞ ¼ xðaÞ
FR xigH ðs1 Þds1 a

and: ð1Þ ð1Þ xigH ðs1 Þ ¼ xigH ðaÞ
FR

ð s1 a

ð2Þ

xigH ðs1 Þds1

Taking the integral of the two sides: ðs ðs ð s1 ðs ð1Þ ð1Þ ð2Þ FR xigH ðs1 Þds1 ¼ FR xigH ðaÞds1
FR FR xigH ðs2 Þds2 ds1 a

a

ð1Þ ¼ xigH ðaÞ⊙ ðs  aÞ
FR

ð s  ð s1 a

On the other hand, the right-hand side is:

a

a

a

ð2Þ xigH ðs2 Þds2

ds1

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Soft Numerical Computing in Uncertain Dynamic Systems

ðs FR a

ð1Þ

xigH ðs1 Þds1 ¼ xðsÞ  H xðaÞ

then: ð1Þ

xðsÞ ¼ xðaÞ
xigH ðaÞ⊙ ðs  aÞ
FR

ð s ð s1 a

a

ð2Þ xigH ðs2 Þds2 ds1

Similarly: ð2Þ

ð2Þ

xigH ðs1 Þ ¼ xigH ðaÞ
FR

ð s1 a

ð3Þ

xigH ðs1 Þds1

Again, applying the integral operator to the two sides: ð s1 ð s1 ð s2 ð2Þ ð2Þ ð3Þ FR xigH ðs2 Þds2 ¼ xigH ðaÞ ⊙ ðs1  aÞ
FR xigH ðs3 Þds3 ds2 a

a

a

Now: ð s ð s1 FR a

a

ðs ð2Þ ð2Þ xigH ðs2 Þds2 ds1 ¼¼ xigH ðaÞ ⊙ FR ðs1  aÞds1 a

ð s  ð s1 ð s2 ð3Þ
FR xigH ðs3 Þds3 ds2 ds1 a

a

a

So by replacement in x(s): ð1Þ

ð2Þ

xðsÞ ¼ xðaÞ
xigH ðaÞ⊙ ðs  aÞ
xigH ðaÞ ðs ð s  ð s1 ð s2 ð3Þ ⊙ FR ðs1  aÞds1
FR FR xigH ðs3 Þds3 ds2 ds1 a

a

a

a

Since: ðs FR

ðs1  aÞds1 ¼

a

ð s  aÞ 2 2!

and finally: ð1Þ

xðsÞ ¼ xðaÞ
xigH ðaÞ⊙ ðs  aÞ ð s  aÞ ð2Þ
xigH ðaÞ⊙ 2!

2


FR

ð s ð s1 ð s2 a

a

a

ð3Þ xigH ðs3 Þds3



ds2 ds1

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99

By continuing in the same way, the general expansion is obtained as follows: ð1Þ

xðsÞ ¼ xðaÞ
xigH ðaÞ ⊙ ðs  aÞ ð2Þ


xigH ðaÞ ⊙

ðs  aÞ2 ðs  aÞm1 ðm1Þ
⋯
xigH ðaÞ ⊙
Rn ða, sÞ 2! ðm  1Þ!

where Rn(a, s) is noted as a reminder term of the expansion and it is: ð s ð s1 ð sn1 ðnÞ Rn ða, sÞ ¼ FR ⋯ xigH ðsn Þdsn dsn1 ⋯ ds1 a

a

a

Item 2. Let us now assume all the derivatives are (ii  gH)-differentiable for n ¼ 1, 2, …, m without changing the type of differentiability. Then, based on the previous case, we have the following relation: ðs ð1Þ xðsÞ ¼ xðaÞ H ð1ÞFR xiigH ðs1 Þds1 a

and: ð1Þ

ð1Þ

xiigH ðs1 Þ ¼ xiigH ðaÞ
FR

ð s1 a

ð2Þ

xiigH ðs1 Þds1

Taking the integral of the two sides: ðs ðs ðs ð s1 ð1Þ ð1Þ ð2Þ FR xiigH ðs1 Þds1 ¼ FR xiigH ðaÞds1
FR FR xiigH ðs2 Þds2 ds1 a

a

ð1Þ

¼ xiigH ðaÞ⊙ ðs  aÞ
FR

ð s  ð s1 a

a

a

a

ð2Þ xiigH ðs2 Þds2 ds1

On the other hand, the right-hand side is: ðs ð1Þ  H ð1ÞFR xiigH ðs1 Þds1 ¼ xðsÞ H xðaÞ a

then: ð1Þ

xðsÞ ¼ xðaÞ  H ð1ÞxiigH ðaÞ ⊙ ðs  aÞ H ð1ÞFR

ð s ð s1 a

a

ð2Þ xiigH ðs2 Þds2 ds1

Similarly: ð2Þ

ð2Þ

xiigH ðs1 Þ ¼ xiigH ðaÞ
FR

ð s1 a

ð3Þ

xiigH ðs1 Þds1

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Soft Numerical Computing in Uncertain Dynamic Systems

Again, applying the integral operator to the two sides: ð s1 ð s1 ð s2 ð2Þ ð2Þ ð3Þ FR xiigH ðs2 Þds2 ¼ xiigH ðaÞ⊙ ðs1  aÞ
FR xiigH ðs3 Þds3 ds2 a

a

a

Now: ð s  ð s1 FR ð2Þ

¼ xiigH ðaÞ ⊙ FR

ðs

a

a

ð2Þ xiigH ðs2 Þds2 ds1 ¼

ðs1  aÞds1


a

ð s  ð s1 ð s2 ð3Þ FR xiigH ðs3 Þds3 ds2 ds1 a

a

a

So by replacement in x(s): ð1Þ

xðsÞ ¼ xðaÞ  H ð1ÞxiigH ðaÞ ⊙ ðs  aÞ ðs ð2Þ  ð1ÞxiigH ðaÞ ⊙ FR ðs1  aÞds1 a

ð s  ð s1 ð s2 ð3Þ  H ð1ÞFR FR xiigH ðs3 Þds3 ds2 ds1 a

a

a

Since: ðs FR

ðs1  aÞds1 ¼

a

ð s  aÞ 2 2!

finally: ð1Þ

ð2Þ

xðsÞ ¼ xðaÞ  H ð1ÞxiigH ðaÞ ⊙ ðs  aÞ H ð1ÞxiigH ðaÞ⊙  ð1ÞFR

ð s ð s1 ð s2 a

a

a

ð3Þ xiigH ðs3 Þds3 ds2 ds1

ð s  aÞ 2 2!

By continuing in the same way, the general expansion is obtained as follows: ð1Þ

ð2Þ

xðsÞ ¼ xðaÞ  H ð1ÞxiigH ðaÞ ⊙ ðs  aÞ H ð1ÞxiigH ðaÞ⊙ ðm1Þ

 H ð1Þ⋯  H ð1ÞxiigH ðaÞ⊙

ð s  aÞ 2 2!

ðs  aÞm1  H ð1ÞRn ðasÞ ðm  1Þ!

where Rn(a, s) is noted as a reminder term of the expansion and it is: ð s ð s1 ð sn1 ðnÞ Rn ða, sÞ ¼ FR ⋯ xiigH ðsn Þdsn dsn1 ⋯ ds1 a

a

a

Soft computing with uncertain sets

101

Item 3. Suppose that the same function is i  gH differentiable for n ¼ 2k  1, k  ℕ and it is ii  gH differentiable for n ¼ 2k, k  ℕ [ {0}. Now x(t) is ii  gH differentiable and: ðs ð1Þ xðsÞ ¼ xðaÞ H ð1ÞFR xiigH ðs1 Þds1 a

xgH0 (t)

is i  gH differentiable and we have: ðs ð1Þ ð1Þ ð2Þ xiigH ðsÞ ¼ xiigH ðaÞ H ð1ÞFR xigH ðtÞdt

According to the hypothesis,

a

Taking the integral of the two sides: ðs ð1Þ FR xiigH ðs1 Þds1 ¼ a

¼ FR

ðs a

ð1Þ xiigH ðaÞds1  H ð1ÞFR

ð1Þ ¼ xiigH ðaÞ ⊙ ðs  aÞ  H ð1ÞFR

ðs

ð s1 FR

a

ð s ð s1 a

a

a

ð2Þ

xiigH ðs2 Þds2 ds1

ð2Þ xiigH ðs2 Þds2

ds1

so: ð1Þ xðsÞ ¼ xðaÞ H ð1ÞxiigH ðaÞ⊙ ðs  aÞ
FR

ð s  ð s1 a

a

ð2Þ xiigH ðs2 Þds2

ds1

Similarly: ð2Þ

ð2Þ

xigH ðs1 Þ ¼ xigH ðaÞ  H ð1ÞFR

ð s1 a

ð3Þ

xiigH ðs1 Þds1

Again, applying the integral operator to the two sides: ð s1 ð2Þ FR xigH ðs2 Þds2 ¼ a

ð2Þ

¼ xigH ðaÞ⊙ ðs1  aÞ  H ð1ÞFR

ð s1 ð s2 a

a

ð3Þ xiigH ðs3 Þds3 ds2

Now: ð s  ð s1 FR a

a

ðs ð2Þ ð2Þ xigH ðs2 Þds2 ds1 ¼¼ xigH ðaÞ⊙ FR ðs1  aÞds1 a

ð s  ð s1 ð s2 ð3Þ  ð1Þ FR xiigH ðs3 Þds3 ds2 ds1 a

a

a

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Soft Numerical Computing in Uncertain Dynamic Systems

So by replacement in x(s): ð1Þ

xðsÞ ¼ xðaÞ ð1ÞxiigH ðaÞ⊙ ðs  aÞ ðs ð2Þ
xigH ðaÞ ⊙ FR ðs1  aÞds1  ð1ÞFR

ðs 

a

ð s1 ð s2

ð3Þ xiigH ðs3 Þds3 ds2 ds1

FR a

a

a

Since: ðs FR

ðs1  aÞds1 ¼

a

ð s  aÞ 2 2!

finally: ð1Þ

xðsÞ ¼ xðaÞ  H ð1ÞxiigH ðaÞ ⊙ ðs  aÞ ð s  aÞ ð2Þ
xigH ðaÞ ⊙

2

 H ð1ÞFR

2!

ð s ð s1 ð s2 a

a

a

ð3Þ xiigH ðs3 Þds3

ds2 ds1

By continuing in the same way, the general expansion is obtained as follows: ð1Þ

xðsÞ ¼ xðaÞ ð1ÞxiigH ðaÞ⊙ ðs  aÞ ð s  aÞ ð2Þ
xigH ðaÞ⊙

2

2!

m1

ð s  aÞ 2
 H ð1Þ⋯  H ð1ÞxiigH ðaÞ⊙  m1 ! 2 ðm1 2 Þ

m

ð s  aÞ 2 xigH ðaÞ⊙
m  H ð1Þ⋯  H ð1ÞRn ðasÞ ! 2 ðm2 Þ

where Rn(a, s) is noted as a reminder term of the expansion and it is: Rn ða, sÞ ¼ FR

ð s ð s1 a

a

⋯

ð sn1 a

ðnÞ xigH ðsn Þdsn



dsn1 ⋯ ds1

Item 4. Suppose that the same function is i  gH differentiable in interval [a, ξ] and ξ is the switching point. Soz: ðξ ð1Þ xðξÞ ¼ xðaÞ
FR xigH ðs1 Þds1 a

Soft computing with uncertain sets

103

and it is ii  gH differentiable in interval [ξ, b] ðs ð1Þ xðsÞ ¼ xðξÞ H ð1ÞFR xiigH ðt1 Þdt1 ξ

By replacement: xðsÞ ¼ xðaÞ
FR

ðξ a

ð1Þ

xigH ðs1 Þds1  H ð1ÞFR

ðs ξ

ð1Þ

xiigH ðt1 Þdt1

Now we are going to find the first integral on the right-hand side. Let us consider ζ 1 as a switching point for the second gH-derivative. And suppose that x(1) igH is ii  gH differentiable on [a, ζ1], then the type of differentiability changes and: ð ζ1 ð1Þ ð1Þ ð2Þ xigH ðζ 1 Þ ¼ xigH ðaÞ H ð1ÞFR xiigH ðs2 Þds2 a

Now for s1  [ζ 1, ξ], is i  gH differentiable on[ζ 1, ξ], then the type of differentiability changes and: ð s1 ð1Þ ð1Þ ð2Þ xigH ðs1 Þ ¼ xigH ðζ 1 Þ
FR xigH ðs3 Þds3 x(1) igH

ζ1

By substituting: ð1Þ ð1Þ xigH ðs1 Þ ¼ xigH ðaÞ  H ð1ÞFR

ð ζ1 a

ð2Þ xiigH ðs2 Þds2
FR

ð s1 ζ1

ð2Þ

xigH ðs3 Þds3

On the other hand: ð2Þ ð2Þ xiigH ðs2 Þ ¼ xiigH ðaÞ
FR

ð s2 ζ1

ð3Þ

xiigH ðs4 Þds4

Using the FR integral on [a, ζ 1] ð ζ1 ð s2 ð ζ1 ð2Þ ð2Þ ð3Þ xiigH ðs4 Þds4 ds2 FR xiigH ðs2 Þds2 ¼ xiigH ðaÞ ⊙ ðζ 1  aÞ
FR a

a

a

and: ð2Þ ð2Þ xigH ðs3 Þ ¼ xigH ðζ 1 Þ
FR

ð s3 ζ1

ð3Þ

xigH ðs5 Þds5

Using the FR integral operator on [ζ 1, s1] ð s1 ð s1 ð s3 ð2Þ ð2Þ ð3Þ FR xigH ðs3 Þds3 ¼ xigH ðζ 1 Þ⊙ ðs1  ζ 1 Þ
FR xigH ðs5 Þds5 ds3 ζ1

ζ1

ζ1

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Soft Numerical Computing in Uncertain Dynamic Systems

(1) To find x(1) igH(s1), we insert two of last equations in xigH(s1).

ð1Þ

xigH ðs1 Þ ¼  ð ζ1 ð s2 ð1Þ ð2Þ ð3Þ ¼ xigH ðaÞ H ð1Þ xiigH ðaÞ ⊙ ðζ 1  aÞ
FR xiigH ðs4 Þds4 ds2 ð2Þ
xigH ðζ 1 Þ⊙ ðs1  ζ 1 Þ
FR ð1Þ

ð s1  ð s3 ζ1

ζ1

ð2Þ

a

a

ð3Þ xigH ðs5 Þds5

ds3

ð2Þ

¼ xigH ðaÞ H xiigH ðaÞ ⊙ ða  ζ 1 Þ
xigH ðζ 1 Þ ⊙ ðs1  ζ 1 Þ  H ð1ÞFR

ð ζ1 ð s2 a

a

ð s1  ð s3 ð3Þ ð3Þ xiigH ðs4 Þds4 ds2
FR xigH ðs5 Þds5 ds3 ζ1

ζ1

Using the FR integral operator on [a, ξ]: ðξ ð1Þ ð1Þ ð2Þ FR xigH ðs1 Þds1 ¼ xigH ðaÞ ⊙ ðξ  aÞ  xiigH ðaÞ⊙ ða  ζ 1 Þ ⊙ ðξ  aÞ a

ð2Þ
xigH ðζ 1 Þ ⊙

 ð1ÞFR

ð ξ ð ζ1 ð s2 a


FR

ðξ  ζ 1 Þ2 ða  ζ 1 Þ2  2 2

a

ð ξ ð s1 ð s3 a

ζ1

ζ1

a

!

ð3Þ xiigH ðs4 Þds4 ds2 ds1

ð3Þ xigH ðs5 Þds5 ds3 ds1

We have this equation: xðsÞ ¼ xðaÞ
FR

ðξ a

ð1Þ xigH ðs1 Þds1  H ð1ÞFR

ðs ξ

ð1Þ

xiigH ðt1 Þdt1

and we obtain the result of the first integral. Now we will express the result of the second integral. (3) Let us suppose that x(2) igH and xigH are ii  gH differentiable on [ξ, b], then the type of differentiability changes and: ð t1 ð1Þ ð1Þ ð2Þ xiigH ðt1 Þ ¼ xiigH ðξÞ
FR xiigH ðt2 Þdt2 ξ

and: ð2Þ

ð2Þ

xiigH ðt2 Þ ¼ xiigH ðξÞ
FR

ð t2 ξ

ð3Þ

xiigH ðt3 Þdt3

Soft computing with uncertain sets

105

Using the FR integral on [ξ, t1] ð t1 ð t2 ð t1 ð2Þ ð2Þ ð3Þ xiigH ðt3 Þdt3 dt2 FR xiigH ðt2 Þdt2 ¼ xiigH ðξÞ ⊙ ðt1  ξÞ
FR ξ

ξ

a

Substituting in x(1) iigH(t1) we have: ð1Þ

ð1Þ

ð2Þ

xiigH ðt1 Þ ¼ xiigH ðξÞ
xiigH ðξÞ ⊙ ðt1  ξÞ
FR

ð t1 ð t2 ξ

a

ð3Þ xiigH ðt3 Þdt3 dt2

Using the FR integral operator on [ξ, s]: ðs ðs  ξÞ2 ð1Þ ð1Þ ð2Þ FR xiigH ðt1 Þdt1 ¼ xiigH ðξÞ ⊙ ðs  ξÞ
xiigH ðξÞ ⊙ 2! ξ ð s ð t ð t2 ð3Þ
FR xiigH ðt3 Þdt3 dt2 dt1 ξ

ξ

a

Now it is the right time to evaluate: ðs ðξ ð1Þ ð1Þ xðsÞ ¼ xðaÞ
FR xigH ðs1 Þds1  ð1ÞFR xiigH ðt1 Þdt1 ξ

a

ð1Þ

ð2Þ

xðsÞ ¼ xðaÞ
xigH ðaÞ⊙ ðξ  aÞ xiigH ðaÞ⊙ ða  ζ 1 Þ ⊙ ðξ  aÞ ! ðξ  ζ 1 Þ2 ða  ζ 1 Þ2 ð2Þ 
xigH ðζ 1 Þ⊙ 2 2  ð1ÞFR

ð ξ ð ζ1 ð s2 a


FR

a

ð ξ ð s1 ð s3 a

ζ1

ζ1

ð1Þ

a

ð3Þ xiigH ðs4 Þds4 ds2 ds1

ð3Þ xigH ðs5 Þds5

ds3 ds1 ð2Þ

 ð1Þ xiigH ðξÞ⊙ ðs  ξÞ
xiigH ðξÞ ⊙
FR

ð s ð t ð t2 ξ

ξ

a

ðs  ξÞ2 2!

ð3Þ xiigH ðt3 Þdt3 dt2 dt1

3.6.1.20.1 Example Let us suppose that xðtÞ ¼ k ⊙ exp ðtÞ,kR and the point that we want to expand the function around is a ¼ 0. First, we introduce the derivatives and the type of differentiability.

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It is clear that x(t) is i  gH differentiable so we can write the derivatives as follows: • • • •

x(t) ¼ k ⊙ exp( t) is i  gH differentiable; x(1) iigH(t) ¼  k ⊙ exp( t) is ii  gH differentiable; x(2) igH(t) ¼ k ⊙ exp( t) is i  gH differentiable; and x(3) iigH(t) ¼  k ⊙ exp( t) is ii  gH differentiable.

It seems that • •

x(2n) igH(t) ¼ k ⊙ exp( t) is i  gH differentiable for n  N [ {0} and x(2n1) igH (t) ¼ k ⊙ exp( t) is ii  gH differentiable for n  N.

Then the Taylor expansion is: ð1Þ

xðtÞ ¼ xð0Þ  ð1Þt⊙ xiigH ðtÞ
 ð1Þ

t2 t3 ð2Þ ð3Þ ⊙ xigH ðtÞ  ð1Þ ⊙ xiigH ðtÞ
⋯ 2! 3!

t2n1 t2n ð2n1Þ ð2nÞ ⊙ xiigH ðtÞ
⊙ xigH ðtÞ
⋯ ð2n  1Þ! ð2nÞ!

ð2n1Þ

ð2nÞ

xð0Þ ¼ k, xiigH ðtÞ ¼ k ⊙ exp ðtÞ, xigH ðtÞ ¼ k ⊙ exp ðtÞ By replacement: xðtÞ ¼ k  t ⊙ k ⊙ exp ðtÞ


t2 t3 ⊙ k ⊙ exp ðtÞ ⊙ k ⊙ exp ðtÞ
⋯ 2! 3!

t2n1 t2n ⊙ k ⊙ exp ðtÞðtÞ
⊙ k ⊙ exp ðtÞ
⋯ ð2n  1Þ! ð2nÞ!

In the next section, we want to express the partial derivative under gH-differential operator. To define it, we need the definition of a two dimensional continuous fuzzy number valued function. Let us assume that  is the area of two dimensional points and:  ¼ fðt, xÞj f ðt, xÞR g This fuzzy number valued function of two variables is called continuous at the point (t0, x0) if for any positive E > 0 there exists a positive δ > 0 such that: k ðt, xÞ  ðt0 , x0 Þ k< δ¼)DH ðf ðt, xÞ, f ðt0 , x0 ÞÞ < E where k  k is a Euclidean norm and DH is the Hausdorff distance. More explanation may be found in Tofigh Allahviranloo and Ahmadi (2010).

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3.6.1.21 gH-partial differentiability The fuzzy number valued function of two variables f ðt, xÞR is called gH-partial differentiable (gH  p) at the point ðt0 , x0 Þ with respect to t and x and denoted by ∂tgHf(t0, x0) and ∂xgHf(t0, x0) if: ∂tgH f ðt0 , x0 Þ ¼ lim

f ðt0 + h, x0 Þ gH f ðt0 , x0 Þ h

∂xgH f ðt0 , x0 Þ ¼ lim

f ðt0 , x0 + kÞ  gH f ðt0 , x0 Þ k

h!0

k!0

Provided that both derivatives ∂tgHf(t0, x0) and ∂xgHf(t0, x0) are fuzzy number valued functions not fuzzy sets. Another way to introduce partial derivatives is using the distance, and based on the properties of the distance we can show this as:  f ðt0 + h, x0 Þ gH f ðt0 , x0 Þ DH lim , ∂tgH f ðt0 , x0 Þ ¼ 0 h!0 h  f ðt0 , x0 + kÞ gH f ðt0 , x0 Þ , ∂xgH f ðt0 , x0 Þ ¼ 0 DH lim k!0 k or: 

f ðt0 + h, x0 Þ gH f ðt0 , x0 Þ , ∂tgH f ðt0 , x0 Þ ! 0 h  f ðt0 , x0 + kÞ  gH f ðt0 , x0 Þ , ∂xgH f ðt0 , x0 Þ ! 0 DH k DH

3.6.1.21.1 Example Consider a real and nonnegative differentiable function like p(t, x) and a scalar fuzzy number uR , then the gH  p derivative for g(t, x) ¼ p(t, x) ⊙ u with respect to t is ∂xgHg(t, x) ¼ ∂xp(t, x) ⊙ u. In accordance with the definition we notice that: gðt, x + kÞ  gH gðt, xÞ ¼ pðt, x + kÞ⊙ u  gH pðt, xÞ ⊙ u ¼ ½pðt, x + kÞ  pðt, xÞ ⊙ u This shows that, considering any sign for [p(t,x +k)  p(t, x)], the gH-difference g(t, x+ k)gHg(t,x) always exists. Dividing both sides by k and using the limit we will have:

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Soft Numerical Computing in Uncertain Dynamic Systems

lim

k!0

gðt, x + kÞ  gH gðt, xÞ ½pðt, x + kÞ  pðt, xÞ ¼ lim ⊙u k!0 k k

Then ∂xgHg(t, x) ¼ ∂xp(t, x) ⊙ u. 3.6.1.21.2 Another simple example Let f ðt, xÞ :  ! R is gh  p differentiable with respect to x and c  R0 be a nonnegative real number. Then ∂xgH(c ⊙ f )(t, x) exists and ∂xgH(c ⊙ f )(t, x) ¼ c ⊙ ∂xgHf(t, x). By using the distance and assumptions:  ðc ⊙ f Þðt, x + kÞ gH ðc ⊙ f Þðt, xÞ DH , c ⊙ ∂xgH f ðt, xÞ k  c⊙ f ðt, x + kÞ  gH c⊙ f ðt, xÞ , c ⊙ ∂xgH f ðt, xÞ ¼ DH k  f ðt, x + kÞ gH f ðt, xÞ , ∂xgH f ðt, xÞ ¼ cDH k Now if k ! 0, then: cDH

 f ðt, x + kÞ gH f ðt, xÞ , ∂xgH f ðt, xÞ ! 0 k

then: ∂xgH ðc ⊙ f Þðt, xÞ ¼ c ⊙ ∂xgH f ðt, xÞ

3.6.1.22 Level-wise form of gH-partial differentiability Suppose that the fuzzy number valued function f ðt, xÞR is gH  p differentiable at the point ðt0 , x0 Þ with respect to t and fl(t, x, r), fu(t, x, r) are real valued functions and partial differentiable with respect to t. We say: •

f(t, x) is (i  gH  p) differentiable w.r.t. t at (t0, x0) if: ∂t, igh f ðt0 , x0 , r Þ ¼ ½∂t fl ðt, x, r Þ, ∂t fu ðt, x, r Þ

•

f(t, x) is (ii  gH  p) differentiable w.r.t. t at (t0, x0) if: ∂t, iigh f ðt0 , x0 , rÞ ¼ ½∂t fu ðt, x, rÞ, ∂t fl ðt, x, r Þ

Note that in each case, the conditions of the definition in level-wise form should be satisfied.

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3.6.1.23 Switching point in gH-partial differentiability For any fixed ξ0 we say the point ðξ0 , xÞ is a switching point for the gH-differentiability of f(t, x) w.r.t. t, if in any neighborhood V of (t0, ξ0) there exist points (t1, x) < (ξ0, x) < (t2, x) for any fixed x such that: Type I at the point (t1, x) is (i  gH  p) differentiable and not (ii  gH  p) differentiable and at the point (t2, x) is (ii  gH  p) differentiable and not (i  gH  p) differentiable. Type II at the point (t1, x) is (ii  gH  p) differentiable and not (i  gH  p) differentiable and at the point (t2, x) is (i  gH  p) differentiable and not (ii  gH  p) differentiable. 3.6.1.23.1 Example Consider the fuzzy number valued function f ðt, xÞ : ½1, 4 ½0, π ! R defined by: f ðt, x, r Þ ¼ ½0:7 + 0:3r, 1:8  0:8r xsin ðtÞ It is clear that: h πi 8 < ð0:7 + 0:3r Þxsin ðtÞ, x½1, 4 , t 0, h 2i fl ðt, x, rÞ ¼ : ð1:8  0:8r Þxsin ðtÞ, x½1, 4 , t π , π 2 h πi 8 < ð1:8  0:8r Þxsin ðtÞ, x½1, 4 , t 0, h 2i fu ðt, x, r Þ ¼ : ð0:7 + 0:3r Þxsin ðtÞ, x½1, 4 , t π , π 2 and the differential in level-wise form is: 8 h πi > < ð0:7 + 0:3r Þx cos ðtÞ, x½1, 4 , t 0, hπ 2 i ∂t fl ðt, x, r Þ ¼ > : ð1:8  0:8r Þx cos ðtÞ, x½1, 4 , t , π 2 8 h πi > < ð1:8  0:8r Þx cos ðtÞ, x½1, 4 , t 0, hπ 2 i ∂t fu ðt, x, rÞ ¼ > : ð0:7 + 0:3r Þx cos ðtÞ, x½1, 4 , t , π 2

π

Clearly π the function f(t, x) is (i  gH  p) differentiable on x½1, 4 , t 0, 2 . At the point 2 , t for allt  [1,  4] the derivative is switched to (ii  gH  p) differentiability. So the points π2 , t for all t  [1, 4] are switching points to the derivative of f(t, x) (Fig. 3.4). We will see the switching points in Fig. 3.5 for the derivatives.

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Soft Numerical Computing in Uncertain Dynamic Systems

6 4 2 0

0

1 1

2 t

2

3

x

4 3 Fig. 3.4

The graph of f(t, x) in level r ¼ 0.

5 0 –5 0

1 1

2 t

2

3

x

4 3 Fig. 3.5

The graph of fgH0 (t, x) in level r ¼ 0.

3.6.1.24 Higher order of gH-partial differentiability Suppose that the fuzzy number valued function ∂tgH f ðt, xÞR is gH  p differentiable at the point ðt0 , x0 Þ with respect to t and there is no switching point. Moreover, suppose that ∂ttfl(t, x, r), ∂ttfu(t, x, r) are real valued functions and partial differentiable with respect to t. We say: •

∂tf(t, x) is (i  gH  p) differentiable w.r.t. t at (t0, x0) if: ∂tt, igh f ðt0 , x0 , r Þ ¼ ½∂tt fl ðt, x, r Þ, ∂tt fu ðt, x, r Þ

Soft computing with uncertain sets •

111

∂tf(t, x) is (ii  gH  p) differentiable w.r.t. t at (t0, x0) if: ∂tt, iigh f ðt0 , x0 , r Þ ¼ ½∂tt fu ðt, x, rÞ, ∂tt fl ðt, x, r Þ

Note. In each case the conditions of the definition in level-wise form should be satisfied and the type of gH-partial differentiability for both functions f(t, x) and ∂tf(t, x) is the same.

3.6.1.25 Integral relation in gH-partial differentiability Suppose that the fuzzy number valued function f ðt, xÞR is continuous and gH  p differentiable with respect to t with no switching point in the interval [a, s], then: ðs ∂xgH f ðt, xÞdx ¼ f ðt, sÞ  gH f ðt, aÞ a

According to the lack of switching point and without loss of generality we assume that the function f(t, x) is (ii  gH  p) differentiable (the proof of (i  gH  p) differentiability is similar). So we have: ðs ðs ∂xgH f ðt, x, r Þdx ¼ ½∂t fu ðt, x, r Þ, ∂t fl ðt, x, r Þ dx a

a

¼ ½fu ðt, s, r Þ  fu ðt, a, r Þ, fl ðt, s, r Þ  fl ðt, a, r Þ ¼ f ðt, sÞ gH f ðt, aÞ

3.6.1.26 Multivariate fuzzy chain rule in gH-partial differentiability Let xi(t) be defined on i ≔½a, b R,i ¼ 1, 2,3 and be strictly increasing and differQ entiable functions. Consider U is an open set of R3 such that 3i¼1 i R: Let us assume that the function f : U ! R is a continuous fuzzy function. Suppose that ∂xi gH f : U ! R , i ¼ 1, 2, 3 the gH  p derivatives of f exist and are fuzzy continuous functions. Call xi ≔ xi(t) and z ≔ z(t) ≔ f(x1, x2, x3). Then ∂tgHz exists and: ∂tgH z ¼ ∂x1 gH f ðx1 , x2 , x3 Þ ⊙ ∂t x1 ðtÞ
∂x2 gH f ðx1 , x2 , x3 Þ ⊙ ∂t x2 ðtÞ
∂x3 gH f ðx1 , x2 , x3 Þ⊙ ∂t x3 ðtÞ where ∂txi(t), i ¼ 1, 2, 3 are derivatives of xi(t) with respect to t. To show the assertion, let t  (a, b) and (x1, x2, x3)  U be fixed and Δxi > 0, i ¼ 1, 2, 3 be small enough. Now set: α1 ¼ f ðx1 + Δx1 , x2 + Δx2 , x3 + Δx3 Þ gH f ðx1 , x2 + Δx2 , x3 + Δx3 ÞR α2 ¼ f ðx1 , x2 + Δx2 , x3 + Δx3 Þ gH f ðx1 , x2 , x3 + Δx3 ÞR α3 ¼ f ðx1 , x2 , x3 + Δx3 Þ gH f ðx1 , x2 , x3 ÞR

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So we have: f ðx1 + Δx1 , x2 + Δx2 , x3 + Δx3 Þ gH f ðx1 , x2 , x3 Þ ¼


3 X

αi R

i¼1

Since the partial gH-derivative ∂xigHf exists, the above gH-differences in αi exist for i ¼ 1, 2, 3 when Δxi ! 0. Here Δxi ¼ xi(t + Δt)  xi(t) ≔ xi + Δxi, i ¼ 1, 2, 3. Now:  f ðx1 + Δx1 , x2 + Δx2 , x3 + Δx3 Þ gH f ðx1 , x2 , x3 Þ lim DH : Δt!0 Δt ! 3 X
∂xi gH f ðx1 , x2 , x3 Þ ⊙∂t xi ðtÞ i¼1

0

1

3 X

αi B
C 3 X B i¼1 C B ¼ lim DH B ,
∂xi gH f ðx1 , x2 , x3 Þ⊙ ∂t xi ðtÞC C Δt!0 Δt @ A i¼1   lim DH Δt!0

f ðx1 + Δx1 , x2 + Δx2 , x3 + Δx3 Þ gH f ðx1 , x2 + Δx2 , x3 + Δx3 Þ Δx1 : ⊙ Δx1 Δt ∂x1 gH f ðx1 , x2 , x3 Þ ⊙ ∂t x1 ðtÞ

 f ðx1 , x2 + Δx2 , x3 + Δx3 Þ gH f ðx1 , x2 , x3 + Δx3 Þ Δx2 : ⊙ + lim DH Δt!0 Δt Δx2 ∂x2 gH f ðx1 , x2 , x3 Þ ⊙ ∂t x2 ðtÞ  + lim DH Δt!0

f ðx1 , x2 , x3 + Δx3 Þ gH f ðx1 , x2 , x3 Þ Δx3 : ⊙ Δx3 Δt ∂x3 gH f ðx1 , x2 , x3 Þ ⊙ ∂t x3 ðtÞ

0ð x B B  lim DH B Δt!0 @

1

+ Δx1

∂x1 gH f ðt, x2 + Δx2 , x3 + Δx3 Þdt

x1

Δx1 ∂x1 gH f ðx1 , x2 , x3 Þ ⊙ ∂t x1 ðtÞ

⊙

Δx1 : Δt

Soft computing with uncertain sets 0ð x B B + lim DH B Δt!0 @

2

+ Δx2

∂x2 gH f ðx1 , t, x3 + Δx3 Þdt

x2

⊙

Δx2

113

Δx2 : Δt

! ∂x2 gH f ðx1 , x2 , x3 Þ ⊙ ∂t x2 ðtÞ   + lim DH ∂x3 gH f ðx1 , x2 , x3 Þ ⊙ x03 ðtÞ, ∂x3 gH f ðx1 , x2 , x3 Þ ⊙ ∂t x3 ðtÞ Δt!0

0 i If the limit operator goes inside the distance then lim Δt!0 Δx Δt ¼ ∂t xi ðtÞ≔x3 ðtÞ. Moreover, in each term the Δxi is a constant with respect to the integral variable. So: ð x1 + Δx1 1  ∂t x1 ðtÞ lim DH ∂x1 gH f ðt, x2 + Δx2 , x3 + Δx3 Þdt, ∂x1 gH f ðx1 , x2 , x3 Þ Δt!0 Δx1 x1 ð x2 + Δx2 1 DH ∂x2 gH f ðx1 , t, x3 + Δx3 Þdt, ∂x2 gH f ðx1 , x2 , x3 Þ + 0 +∂t x2 ðtÞ lim Δt!0 Δx2 x2 ð x1 + Δx1   ∂ t x1 ðt Þ  lim DH ∂x1 gH f ðt, x2 + Δx2 , x3 + Δx3 Þ, ∂x1 gH f ðx1 , x2 , x3 Þ dt Δx1 Δt!0 x1 ð x2 + Δx2   ∂t x2 ðtÞ lim DH ∂x2 gH f ðx1 , t, x3 + Δx3 Þ, ∂x2 gH f ðx1 , x2 , x3 Þ dt + Δx2 Δt!0 x2 !   ∂t x1 ðtÞ  lim sup DH ∂x1 gH f ðτ, x2 + Δx2 , x3 + Δx3 Þ, ∂x1 gH f ðx1 , x2 , x3 Þ Δx1 Δx1 Δt!0 τ½x1 , x1 + Δx1

∂t x2 ðtÞ + lim Δx2 Δt!0

sup τ½x1 , x1 + Δx1



DH ∂x1 gH f ðx1 , t, x3 + Δx3 Þ, ∂x1 gH f ðx1 , x2 , x3 Þ



! Δx2 ! 0

As Δt ! 0 then all Δxi ! 0 and thus τi ! xi for all i ¼ 1, 2. Then by continuity of ∂xigH, two of the terms intend to the zero. The proof is completed.

3.7

The fuzzy Laplace transform operator

In this section we suppose that the Laplace operator acts on a fuzzy number valued function and this is the reason we call it fuzzy Laplace transform (Armand et al., 2019; Salahshour and Allahviranloo, 2013). As before, let us consider the function f is a fuzzy number valued function and s is a real parameter. The fuzzy Laplace transform is defined as follows:

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FðsÞ ¼ Lð f ðtÞÞ ¼

ð∞

est ⊙ f ðtÞdt

0

or: FðsÞ ¼ Lð f ðtÞÞ ¼ lim

ðτ

τ!∞ 0

est ⊙ f ðtÞdt

If we consider the Laplace operator in the level-wise form of f(t) as: Lð f ðt, r ÞÞ ¼ ½lðfl ðt, r ÞÞ, lð fu ðt, r ÞÞ then the level-wise form of the Laplace operator is as follows: ðτ Fðs, r Þ ¼ lim est ⊙ f ðt, r Þdt τ!∞ 0

and:  ½Fl ðs, r Þ, Fu ðs, r Þ ¼ lim

ðτ

τ!∞ 0

e

st

fl ðt, r Þdt, lim

ðτ

τ!∞ 0

st

e

 fu ðt, r Þdt

then: Fl ðs, r Þ ¼ lim

ðτ

τ!∞ 0

Fu ðs, r Þ ¼ lim

ðτ

τ!∞ 0

est fl ðt, r Þdt est fu ðt, r Þdt

To define this operator, the important condition is that the integral must converge to a real number. This means it should be bounded. However, there are some integrals that are not convergent.

3.7.1

EXAMPLE

Suppose the fuzzy number valued function f ðtÞ ¼ c ⊙ et 2 , cR . Then: ðτ 2 Fl ðs, r Þ ¼ lim est cl ðr Þet dt ! 0 τ!∞ 0

Fu ðs, r Þ ¼ lim

ðτ

τ!∞ 0

est cu ðr Þet dt ! 0 2

The integral grows without bound for any s as τ ! ∞. If you remember we defined the absolute value of fuzzy number. Now, in the same way, we can define the absolute value of a fuzzy number valued function.

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115

The absolute value of the same function in level-wise form is defined as: j f ðt, r Þj ¼ ½ min fj fl ðt, r Þj, j fu ðt, r Þjg, max fj fl ðt, r Þj, j fu ðt, r Þjg It can be defined in two cases: Type I. Type 1 absolute value fuzzy number function: j f ðt, r Þj ¼ ½j fl ðt, r Þj, j fu ðt, r Þj In other words, if fl(t, r) 0 for all r then f is a type 1 absolute value fuzzy number function. Type II. Type 2 absolute value function: j f ðt, r Þj ¼ ½j fu ðt, r Þj, j fl ðt, r Þj In other words, if fu(t, r) < 0 for all r then f is a type 2 absolute value fuzzy number function. Moreover, the other conditions of a fuzzy number in level-wise form should be satisfied. Note. The absolute value of a fuzzy number function is always a positive fuzzy number valued function.ExampleConsider the fuzzy number function f(t, r) ¼ c[r]et in the level-wise form where c[r] ¼ [2 + r, 4  r]. As we know: j fl ðt, r Þj ¼ jð2 + r Þet j ¼ ð2 + r Þet , j fu ðt, r Þj ¼ jð4  r Þet j ¼ ð4  r Þet and: j f ðt, r Þj ¼ ½ð2 + r Þet , ð4  r Þet Then f is a type 1 absolute value fuzzy number function.

ExampleConsider the fuzzy number function f(t, r) ¼ c[r]et in the level-wise form

where c[r] ¼ [4 + r,  2  r]. As we know:

j fl ðt, r Þj ¼ jð4 + r Þet j ¼ ð4  r Þet , j fu ðt, r Þj ¼ jð2  r Þet j ¼ ðr + 2Þet and: j f ðt, r Þj ¼ ½ðr + 2Þet , ð4  r Þet Then f is a type 2 absolute value fuzzy number function.

3.7.2

DEFINITION—ABSOLUTELY

CONVERGENCE

The integral operator in the Laplace transformation: ðτ FðsÞ ¼ lim est ⊙ f ðtÞdt τ!∞ 0

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Is said to be absolutely convergent if the following exists: ðτ lim jest ⊙ f ðtÞjdt τ!∞ 0

Considering the level-wise form, both of the following integrals exist. ðτ ðτ st lim e jfl ðt, r Þjdt, lim est jfu ðt, r Þjdt τ!∞ 0

τ!∞ 0

Theoretically, in order to apply the fuzzy Laplace transform to physical problems, it is necessary to involve the inverse transform. If F(s) ¼ L(f(t)) is the Laplace transform, the L1 is known as inverse Laplace transform and we have: L1 ðFðsÞÞ ¼ f ðtÞ, t  0 As with the Laplace transform, the inverse transform is also a linear transform operator. Then, for two fuzzy functions f, g, subject to: Lðf ðtÞÞ ¼ FðsÞ, LðgðtÞÞ ¼ GðsÞ then: L1 ða ⊙ FðsÞ
b ⊙ GðsÞÞ ¼ a ⊙ L1 ðFðsÞÞ
b ⊙ L1 ðGðsÞÞ ¼ a ⊙ f ð t Þ
b ⊙ gð t Þ For any real numbers a, b. One of the important functions occurring in some electrical systems is the delay, which can be displayed as a unit step function like (Fig. 3.6):

1, t  a ua ðtÞ≔uðt  aÞ ¼ 0, t < a For instance, in an electric circuit for a voltage at a particular time t ¼ a. We write such a situation using unit step functions as: V ðtÞ ¼ uðtÞ  uðt  aÞ ua(t) 1

a

Fig. 3.6

ua(t).

t

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117

V(t) 1

a

t

Fig. 3.7 V(t).

It is a shifted unit step. It is clear that u(t) ¼ u(t  a) ¼ 1 and V(t) ¼ 0 for t a 0 and u(t) ¼ 1, u(t  a) ¼ 0 and V(t) ¼ 1 for a > t 0 (Fig. 3.7).

3.7.3

FIRST

TRANSLATION THEOREM

If F(s) ¼ L(f(t)) for s > a then F(s  a) ¼ L(eat ⊙ f(t)) such that a is a real number. The proof is clear from the definition of Laplace transform: ð∞ ð∞ ðsaÞt ⊙ f ðtÞdt ¼ est eat ⊙ f ðtÞdt ¼ Lðeat ⊙ f ðtÞÞ Fðs  aÞ ¼ e 0

3.7.4

SECOND

0

TRANSLATION THEOREM

If F(s) ¼ L(f(t)) for s > a 0 then: eas ⊙ FðsÞ ¼ Lðua ðtÞ ⊙ f ðt  aÞÞ According to the definition: Lðua ðtÞ ⊙ f ðt  aÞÞ ¼

ð∞

est ua ðtÞ ⊙ f ðt  aÞdt

0

Since ua(t) ¼ 0 for 0 < t < a and ua(t) ¼ 1 for t a then: ð∞ Lðua ðtÞ ⊙ f ðt  aÞÞ ¼ est ⊙ f ðt  aÞdt a

Let us suppose that t  a ¼ τ ð∞ ð∞ est ⊙ f ðt  aÞdt ¼ esa ⊙ esτ ⊙ f ðτÞdτ ¼ esa ⊙ FðsÞ a

a

Finally: Lðua ðtÞ ⊙ f ðt  aÞÞ ¼ esa ⊙ FðsÞ

118

3.7.5

Soft Numerical Computing in Uncertain Dynamic Systems

LAPLACE

TRANSFORM ON THE DERIVATIVE

In this section we consider the Laplace operator on the gH-derivative of the fuzzy number valued functions, because one of the key operators for this monograph is the derivative of fuzzy functions. Now we work with two operators, entitled Laplace and derivative, on the fuzzy number valued functions.

3.7.5.1 Derivative theorem Suppose that f and f0 are continuous fuzzy number valued on [0, ∞), then: Lðf 0 ðtÞÞ ¼ s ⊙ Lðf ðtÞÞ H f ð0Þ If f is (i)-differentiable: Lðf 0 ðtÞÞ ¼ ð1Þ ⊙ f ð0Þ  H ðs ⊙ Lðf ðtÞÞÞ If f is (ii)-differentiable for s > 0. Each part can be proved easily using the level-wise form. For instance, we prove (ii)-differentiability, and the first part is proved in a similar way. The level-wise form of (ii)-differentiability is as:

f 0 ðt, r Þ ¼ fu0 ðt, r Þ, fl0 ðt, r Þ then:    

Lðf 0 ðt, r ÞÞ ¼ l fu0 ðt, r Þ , l fl0 ðt, r Þ On the other hand, the level-wise form of the right-hand side is: ½fu ð0, r Þ  ðslðfu ðt, r ÞÞÞ,  fl ð0, r Þ  ðslðfl ðt, r ÞÞÞ ¼ ¼ ½slðfu ðt, r ÞÞ  fu ð0, r Þ, slðfl ðt, r ÞÞ  fl ð0, r Þ so:     l fu0 ðt, r Þ ¼ slðfu ðt, r ÞÞ  fu ð0, r Þ, l fl0 ðt, r Þ ¼ slðfl ðt, r ÞÞ  fl ð0, r Þ The proof is completed.

3.7.5.2 High order derivation theorem Suppose that f, f0 and f00 are continuous fuzzy number valued on [0, ∞) (note that for the second derivative the peace-wise differentiability is enough), then:
00  L f ðtÞ ¼ s2 ⊙ Lðf ðtÞÞ H s ⊙ f ð0Þ H f ð0Þ

Soft computing with uncertain sets

119

If f and f0 are (i)-differentiable:
00    L f ðtÞ ¼ f 0 ð0Þ  H s2 ⊙ Lðf ðtÞÞ
ð1Þs ⊙ f ð0Þ If f is (i)-differentiable and f0 is (ii)-differentiable:
00    L f ðtÞ ¼ s ⊙ f ð0Þ H s2 ⊙ Lðf ðtÞÞ H f 0 ð0Þ If f is (ii)-differentiable and f0 is (i)-differentiable:
00  L f ðtÞ ¼ s2 ⊙ Lðf ðtÞÞ H s ⊙ f ð0Þ
ð1Þf 0 ð0Þ If f and f0 are (ii)-differentiable. The proof processes for all four cases are the same and to show the process we just prove the second case. Let f is (i)-differentiable and f0 is (ii)-differentiable. The levelwise form of the left-hand side is as: 00

00

00

Lðf ðt, r ÞÞ ¼ ½lð fl ðt, r ÞÞ, lðfu ðt, r ÞÞ and the right-hand side: 0



fu ð0, r Þ,  fl0 ð0, r Þ  s2 lðfu ðt, r ÞÞ,  s2 lðfl ðt, r ÞÞ + ½sfu ð0, r Þ,  sfl ð0, r Þ

¼ fu0 ð0, r Þ + s2 lðfu ðt, r ÞÞ  sfu ð0, r Þ,  fl0 ð0, r Þ + s2 lðfl ðt, r ÞÞ  sfl ð0, r Þ then:   l fl00 ðt, r Þ ¼ fu0 ð0, r Þ + s2 lðfu ðt, r ÞÞ  sfu ð0, r Þ   l fu00 ðt, r Þ ¼ fl0 ð0, r Þ + s2 lðfl ðt, r ÞÞ  sfl ð0, r Þ The proof is now completed.

3.8

Fuzzy improper integral

Before the discussion of the Fourier transformation, we need to explain the concept of improper integral as a main operator of the Fourier transformation. As we displayed in the previous section the integral over unbounded region of a fuzzy function is referred to as a fuzzy improper integral. In this subsection, we express the double improper integrals and their relation with the derivative. In the following definition, we define the uniformly convergence using the distance.

120

3.8.1

Soft Numerical Computing in Uncertain Dynamic Systems

DEFINITION—UNIFORM

CONVERGENCE

Let f(x, t) is aÐ continuous fuzzy number valued function on [a, b] [0, ∞). Also, suppose that ∞ c f(x, t)dt convergences (level-wise convergence) for any x  [a, b]. Ð We say F(x) ¼ ∞ c f(x, t)dt converges uniformly on x if for any positive E > 0 there is a number N(E) that depends on the E such that:  ðd DH FðxÞ, f ðx, tÞdt < E c

Whenever d N for all x  [a, b], that is: ð d sup DH f ðx, tÞdt, 0 ! 0 c x½a, b When d ! ∞.

3.8.2

THEOREM—INTERCHANGING

INTEGRALS

Consider that the function f : R R + ! R is a fuzzy number valued function and its level-wise form is: f ðx, t, r Þ ¼ ½fl ðx, t, r Þ, fu ðx, t, r Þ

• •

Moreover, suppose that: Ð∞ f(x, t)dt is convergent for all x  [a, ∞] and Ð c∞ a f(x, t)dt is convergent for all t [c, ∞].

Then we have: ð∞ ð∞ a

f ðx, tÞdtdx ¼

ð∞ ð∞

c

c

f ðx, tÞdxdt 8a8cR

a

This is proved easily using the level-wise form. ð ∞ ð ∞  ð∞ ð∞ ð∞ ð∞ f ðx, t, r Þdtdx ¼ fl ðx, t, rÞdtdx, fu ðx, t, r Þdtdx a

¼

c

ð ∞ ð ∞ c

fl ðx, t, r Þdtdx,

a

The proof is completed.

a

c

c

a

ð∞ ð∞



a

fu ðx, t, rÞdtdx ¼

c

ð∞ ð∞ c

a

f ðx, t, r Þdxdt

Soft computing with uncertain sets

3.8.3

THEOREM—INTEGRAL

121

AND DERIVATIVE

Suppose that both functions f(x, t) and ∂xgH f(x, t) are fuzzy continuous in [a, b]

[c, ∞). Also assume that: Ð • ÐF(x) ¼ ∞ c f(x, t)dt converges for all x  R ∞ • ∂ f(x, t)dt converges uniformly on [a, b] c xgH Then F is gH-differentiable on [a, b] and: ð∞ 0 FgH ðxÞ ¼ ∂xgH f ðx, tÞdt c

To prove it, we use the distance, and are going to show that:  ð Fðx + hÞ  gH FðxÞ ∞ , DH ∂xgH f ðx, tÞdt ! 0,h ! 0 h c To do this, the first part is: ð ∞ ð∞ Fðx + hÞ  gH FðxÞ 1 ¼ ⊙ f ðx + h, tÞdt  gH f ðx, tÞdt h h c c ð∞   1 f ðx + h, tÞ gH f ðx, tÞ dt ¼ ⊙ h c It can be written as the following equation: ðx + h ∂ξgH f ðξ, tÞdξ f ðx + h, tÞ gH f ðx, tÞ ¼ x

Now by replacing we have: ð∞ ðx + h Fðx + hÞ gH FðxÞ 1 ¼ ⊙ ∂ξgH f ðξ, tÞdξdt ¼ h h c x ðk ðx + h 1 ¼ ⊙ lim ∂ξgH f ðξ, tÞdξdt k!∞ c x h With uniformly continuity of ∂xgHf(x, t) in [a, b] [c, ∞):   E 8E > 0 9δ > 0, jx  ξj < δ ¼)DH ∂ξgH f ðξ, tÞ gH ∂xgH f ðx, tÞ, 0 < kc The following equation is true because the integral does not depend on the variable ξ and:

122 ð∞

Soft Numerical Computing in Uncertain Dynamic Systems

∂xgH f ðx, tÞdt ¼

c

1 ⊙ h

ð∞ ðx + h c

∂xgH f ðx, tÞdξdt ¼

x

1 ⊙ lim k!∞ h

ðk ðx + h

∂xgH f ðx, tÞdξdt

c x

Finally: 

ð Fðx + hÞ  gH FðxÞ ∞ DH , ∂xgH f ðx, tÞdt ¼ h c  ðk  ðx + h   1 ¼ DH lim ∂ξgH f ðξ, tÞ  gH ∂xgH f ðx, tÞ dξ dt, 0 ⊙ k!∞ c h x ðk  ðx + h
1 E  E  lim ⊙ dξ dt ¼ lim ðk  cÞ ¼ E k!∞ c h k!∞ kc kc x The proof is completed.

3.9

Fourier transform operator

In this section, we briefly discuss the fuzzy Fourier transform and will show in the next chapters how this transformation can be used to solve the fuzzy partial differential equation (Gouyandeha et al., 2017).

3.9.1

DEFINITION—FUZZY FOURIER

TRANSFORM

Consider the function f : R ! R is the fuzzy valued function. The fuzzy Fourier transform of f(x) denoted by ðF f f ðxÞg : R ! C Þ is given by the following integral: ð 1 ∞ F ff ðxÞg ¼ pffiffiffiffiffi f ðxÞ⊙ eiwx dx ¼ FðwÞ 2π ∞ Here, C is the set of all fuzzy numbers on complex numbers. In a classical approach, it would not be possible to use the Fourier transform for a periodic function that cannot be defined in the space of integrable functions on the interval (∞, ∞). The use of generalized functions, however, frees us of that restriction and makes it possible to look at the Fourier transform of a periodic function. In the following example, it can be shown that the Fourier series coefficients of a periodic function are sampled values of the Fourier transform of one period of the function.

3.9.2

EXAMPLE—FUZZY FOURIER

TRANSFORM

Let us consider the following fuzzy set valued function:

0,  π < x < 0 f ðx Þ ¼ c, 0 0 and all s > 0, the increments Cs+t HCt are identically distributed uncertain variables and Ct0, Ct1 HCt0, Ct2 HCt1, …, Cti+1 HCti are independent uncertain variables as well. 3. Every increment Cs+t HCt is a normal uncertain variable (with normal distributive function) with expected value 0 and variance t2 whose uncertainty distribution is:    πx 1 ΦðxÞ ¼ 1 + exp pffiffiffiffi , xR 3t Remark. It should be noted that here the difference between two uncertain processes is exactly the Hukuhara difference H between these two in the fuzzy case. Because in the above we mention that the difference should have a distributive function, which can be a fuzzy membership function, it means that the result of the difference should be a fuzzy variable or, in this case, a fuzzy number. Example. To complete the concept, for instance, you suppose that C0 ¼ 0 and Ct has stationary and independent increments and every increment Cs+t HCt is a triangular or triangular fuzzy variable, like (at, bt, ct), or exponentially distributed fuzzy variable. Then Ct is an uncertain process and, especially here, is a fuzzy process. Note. Let us consider that for a canonical or standard Ct as a C process, dt is an infinitesimal time interval and dCt is defined as dCt ¼ Ct+dt HCt and it is an uncertain or fuzzy process such that for every t and dCt is a normally distributed uncertain or fuzzy variable.

4.2.2

DEFINITION—LIU

INTEGRAL OF AN UNCERTAIN PROCESS

Let Xt be an uncertain process and Ct a canonical Liu process. For any partition on the interval [a, b] like a ¼ t1 < t2 < ⋯ < tn+1 ¼ b and Δ ¼ max jti + 1  ti j: Then the uncer1in

tain Liu integral of an uncertain process Xt with respect to Ct is defined in the form of a Riemann integral and: ðb n X Xt dCt ¼ lim  Xti ⊙ðCti + 1 H Cti Þ a

Δ!0

i¼1

As we mentioned before, this integral does exist if the limit exists, and this means that the result of the limit is a fuzzy number. Calling this a Liu integral comes from using his canonical process (Liu, 2015).

130

4.2.3

Soft Numerical Computing in Uncertain Dynamic Systems

THEOREM—CHAIN

RULE

Before discussing this subject, the same rule has been considered and established on the derivative of fuzzy functions. Let us consider Ct is a canonical process and the uncertain function Xt ¼ h(t, Ct) is a continuously differentiable function. Then we have the following rule: dXt ¼

∂h ∂h ðt, Ct Þdt ðt, Ct Þ⊙dCt , c≔Ct ∂t ∂c

To prove it, since the function h(t, c) is continuously differentiable, so by using the first order Taylor expansion: ΔXt ¼

∂h ∂h ðt, Ct Þ⊙Δt ðt, Ct Þ⊙ΔCt ∂t ∂c

Taking the integral of the two sides on interval [0, s] and changing: ΔXt ! dXt , Δt ! dt, ΔCt ! dCt then: ðs

ðs

∂h dXt ¼ ðt, Ct Þdt 0 0 ∂t

ðs

∂h ðt, Ct Þ⊙dCt 0 ∂c

The right-hand side is: ðs

dXt ¼ Xs H X0

0

Based on the definition of H-difference, we have: ðs ðs ∂h ∂h Xs ¼ X 0  ðt, Ct Þdt ðt, Ct Þ⊙dCt 0 ∂t 0 ∂c for any s 0. So this is an uncertain function that satisfies: dXt ¼

∂h ∂h ðt, Ct Þdt ðt, Ct Þ⊙dCt ∂t ∂c

For more illustration, the infinitesimal increment dCt may be replaced with the derived C process like: dYt ¼ ut dtvt ⊙dCt where utand vt are absolutely integrable uncertain or fuzzy processes, so the result can be explained as: dhðt, Yt Þ ¼

∂h ∂h ðt, Yt Þdt ðt, Yt Þ⊙dYt ∂t ∂c

Continuous numerical solutions of uncertain differential equations 131 By replacing, we will obtain: ∂h ∂h ðt, Yt Þdt ðt, Yt Þ⊙ðut dtvt d⊙Ct Þ ∂t ∂c ∂h ∂h ∂h dhðt, Yt Þ ¼ ðt, Yt Þdt ðt, Yt Þ⊙ut dt ðt, Yt Þ⊙vt ⊙dCt ∂t ∂c ∂c ∂h ∂h ¼ ðt, Yt Þ⊙ð1ut Þdt ðt, Yt Þ⊙vt ⊙dCt , if ut  0 ∂c ∂c ∂h ∂h ¼ ðt, Yt Þ⊙wt dt ðt, Yt Þ⊙vt ⊙dCt , 1ut ¼ wt ∂c ∂c dhðt, Yt Þ ¼

This is another form for the dXt in chain rule. As with the integration by parts method for fuzzy number valued functions, which was explained in Chapter 3, here, another version is reviewed for uncertain functions. For more information, see Liu (2015).

4.2.4

THEOREM—INTEGRATION

BY PARTS

For any standard canonical process Ct and absolutely continuous function like F(t), the integration can be written as: ðs ðs FðtÞ⊙dCt ¼ FðsÞ⊙Cs H Ct ⊙dFðtÞ 0

0

The proof looks like the previous one in the field of fuzzy sets. It is enough to consider or define h(t, Ct) ¼ F(t) ⊙ Ct and, using the chain rule, in this case we get: dðFðtÞ⊙Ct Þ ¼ Ct ⊙dFðtÞFðtÞ⊙dCt and easily: FðsÞ⊙Cs ¼

ðs

dðFðtÞ⊙Ct Þ ¼

0

ðs

Ct ⊙dFðtÞ

ðs

0

FðtÞ⊙dCt

0

The proof is completed. Now we are going to define the uncertain differential equation as an extension of fuzzy differential equation.

4.2.5

DEFINITION—UNCERTAIN

DIFFERENTIAL EQUATION

We discussed the following equation: dXt ¼

∂h ∂h ðt, Ct Þdt ðt, Ct Þ⊙dCt ∂t ∂c

132

Soft Numerical Computing in Uncertain Dynamic Systems

where Xt ¼ Xt+dt HXt, dCt ¼ Ct+dt HCt are uncertain processes with normally distributed function or a fuzzy process with the same characters for each t, and dt is an infinitesimal time interval. Now consider that: ∂h ∂h ðt, Ct Þ≔f ðt, Xt Þ, ðt, Ct Þ≔gðt, Xt Þ ∂t ∂c such that f and g are some given functions. Then: dXt ¼ f ðt, Xt Þdtgðt, Xt Þ⊙dCt is called an uncertain differential equation or fuzzy differential equation. The solution is the uncertain or fuzzy function Xt that satisfies the main differential equation. In general, we have: 8 dXt ¼ f ðt, Xt Þdtgðt, Xt Þ⊙dCt > > > > < dXt ¼ Xt + dt H Xt does have distributive function or membership function dCt ¼ Ct + dt H Ct does have distributive function or membership function > > ðt, Xt Þ is an uncertain function f > > : gðt, Xt Þ is an uncertain function This uncertain differential equation is equivalent to the uncertain integral equation like: ðs ðs Xs ¼ X0  f ðt, Xt Þdt gðt, Xt Þ⊙dCt 0

0

Clearly, this solution is a Liu process as well. Example. Let consider Ct is a standard canonical process and: dXt ¼ adtb⊙dCt where a is drift coefficient and b is diffusion constant. The solution can be obtained as the following form: Xt ¼ atb⊙dCt because: ðs

dXt ¼

0

ðs

ðs adt

0

b⊙dCt 0

then: Xs ¼ asb⊙dCs , X0 ¼ 0 for any s including t. In other words: Xt + dt ¼ aðt + dtÞb⊙dCt + dt

Continuous numerical solutions of uncertain differential equations 133 and: Xt + dt H Xt ¼ aðt + dtÞb⊙dCt + dt H atH b⊙dCt ¼ adtb⊙dCt This means the solutions satisfies the equation. More information can be found in Yao (2016).

4.2.6

REMARK

Suppose that f(t, Xt) and g(t, Xt) are linear functions as follows: f ðt, Xt Þ ¼ u1t ⊙Xt u2t , gðt, Xt Þ ¼ v1t ⊙Xt v2t where u1t, u2t, v1t, v2t are uncertain functions. Then the uncertain differential equation: dXt ¼ ðu1t ⊙Xt u2t Þdtðv1t ⊙Xt v2t Þ⊙dCt does have the solution as follows: Xt ¼ Ut ⊙Vt where: Ut ¼ exp

ð t

ðt u1s ds

0

V t ¼ X0 

ðt

  u2s g Us ds

0

 v1s ⊙dCs

,

0

ðt



 v2s g Us ⊙dCs

0

We suppose that the generalized divisions exist. To show the assertion, we know that: dUt ¼ u1t ⊙Ut dtv1t ⊙Ut ⊙dCt ,     dVt ¼ u2t g Ut dt v2t g Ut ⊙dCt Now, using the chain rule, we have: Xt ¼ Ut ⊙Vt ) dXt ¼ Ut ⊙dVt Vt ⊙dUt So, by substituting:      dXt ¼ Ut ⊙ u2t g Ut dt v2t g Ut ⊙dCt Vt ⊙ðu1t ⊙Ut dtv1t ⊙Ut ⊙dCt Þ and based on the properties of the generalized division and distributive property, it can be displayed as:

134

Soft Numerical Computing in Uncertain Dynamic Systems dXt ¼ ðu2t dtv2t ⊙dCt Þðu1t ⊙Ut ⊙Vt dtv1t ⊙Ut ⊙Vt ⊙dCt Þ

Subject to u1t ⊙ Ut ⊙ Vt and v1t ⊙ Ut ⊙ Vt are fuzzy numbers. Then: dXt ¼ ðu2t dtv2t ⊙dCt Þðu1t ⊙Xt dtv1t ⊙Xt ⊙dCt Þ ¼ ðu1t ⊙Xt u2t Þdtðv1t ⊙Xt v2t Þ⊙dCt The proof is completed. Example. Consider that f(t, Xt) ¼ ut and g(t, Xt) ¼ vt then the uncertain differential equation is linear and we have: dXt ¼ ut dtvt ⊙dCt then: Ut ¼ exp

ð t 0ds 0

Vt ¼ X 0 



ðt 0⊙dCs

¼1

0

ðt

ðt us ds

0

vs ⊙dCs

0

so the solution is: Xt ¼ X0 

ðt

ðt us ds

0

vs ⊙dCs

0

subject to these two integrals being fuzzy number valued functions. Considering the uncertain differential equation, the left-hand side can be transformed to a derivative of an uncertain function and the left-hand side is another uncertain function. Another type of this equation is a fuzzy differential equations (Yao, 2016).

4.3

Fuzzy differential equations

In this section, we first discuss the existence and uniqueness of the solution of these equations. To this end, we need to introduce the fuzzy differential equation. These equations can be divided into several forms, considering the order and type of differentiability. The general forms of the models can be displayed as a system modeled by differential equations (see Fig. 4.2). This model can be expressed in the following cases: •

One of the components of the input vector is a fuzzy number and the system does not have any fuzzy parameter. Then the output contains at least a fuzzy number component.

Continuous numerical solutions of uncertain differential equations 135

Input vector

A system modeled by the differential equations

Output vector

Fig. 4.2 General form of a system modeled by differential equations.

x (t0) = x0

xc(t) = g(t, x)

x (t)

Fig. 4.3 A first order differential equation.

•

•

The input vector is a real vector without any fuzzy components and the main system does have some fuzzy parameters. Then the output contains at least a fuzzy number component. One of the components of the input vector is a fuzzy number and the main system does have some fuzzy parameters. Then the output contains at least a fuzzy number component.

For the following first order fuzzy differential equation like (Fig. 4.3): x0gH ðtÞ ¼ f ðtxÞ, xðt0 Þ ¼ x0 , t0  t  T This model can be expressed in the following cases: • • •

The initial value x0 is a fuzzy number and the system f(t, x) does not have any fuzzy parameter. Then the output x(t) is a fuzzy number. The initial value x0 is a real number and the system f(t, x) does have some fuzzy parameters. Then the output x(t) is a fuzzy number. The initial value x0 is a fuzzy number and the system f(t, x) does have some fuzzy parameters. Then the output x(t) is a fuzzy number.

Sometimes, the main system is a high order differential equation with several initial values (Fig. 4.4). For this fuzzy differential equation, we also have the same cases. Now let us consider that the initial value of the first order fuzzy differential equation is a fuzzy number:

136

Soft Numerical Computing in Uncertain Dynamic Systems

x(t0) x′(t0)

x(n)(t) = ƒ(t, x,x′,... , x(n–1))

x(t)

x(n–1)(t0)

Fig. 4.4

A high order differential equation.

x0gH ðtÞ ¼ f ðt, xÞ, xðt0 Þ ¼ x0 R The solution set of the equation can be defined as: D ¼ fðt, xÞjt0  t  T,  ∞ < x < ∞, T≔real constantg This is called a fuzzy initial value problem. Considering the fuzzy gH-differential of the left-hand side as: x0gH ðtÞ ¼ lim

h!0

xðt + hÞgH xðtÞ h

subject to the gH-difference x(t + h)gHx(t) existing. As we discussed in Chapter 3, the level-wise form of this fuzzy number as a derivative can be explained as the following two cases. In case (i):  

x0gH ðtÞ ½r  ¼ x0l ðt, r Þ, x0u ðt, r Þ In case (ii): 



x0gH ðtÞ ½r  ¼ x0u ðt, r Þ, x0l ðt, r Þ

subject to the functions xl0 (t0, r) and xu0 (t0, r) being two real valued differentiable functions with respect to t and uniformly with respect to r  [0, 1]. Moreover, they have all the properties of the fuzzy number valued for the fuzzy derivative.

Continuous numerical solutions of uncertain differential equations 137 Now if we consider the level-wise form of the equations: In case (i), xigH(t, r) is known as i  gH-solution and

x0igH ðt, r Þ ¼ x0l ðt, r Þ, x0u ðt, r Þ ¼ ½ fl ðt, r Þ, fu ðt, r Þ ½xl ðt0 , r Þ, xu ðt0 , r Þ ¼ ½xl, 0 ðr Þ, xu,0 ðr Þ In case (ii), xiigH(t, r) is known as ii  gH-solution and:

x0iigH ðt, r Þ ¼ x0u ðt, r Þ, x0l ðt, r Þ ¼ ½ fl ðt, r Þ, fu ðt, r Þ ½xl ðt0 , r Þ, xu ðt0 , r Þ ¼ ½xl, 0 ðr Þ, xu,0 ðr Þ then in case (i), xigH(t, r) is known as the i  gH-solution and:  0 xl ðt, r Þ ¼ fl ðt, r Þ, xl ðt0 , r Þ ¼ xl,0 ðr Þ  0 xu ðt, r Þ ¼ fu ðt, r Þ, xu ðt0 , r Þ ¼ xu,0 ðr Þ for t0  t  T. In case (ii), xiigH(t, r) is known as the ii  gH-solution and:  0 xu ðt, r Þ ¼ fl ðt, r Þ, xl ðt0 , r Þ ¼ xl,0 ðr Þ  0 xl ðt, r Þ ¼ fu ðt, r Þ, xu ðt0 , r Þ ¼ xu,0 ðr Þ for t0  t  T. Indeed, in case (ii) we will have the system of two real first order differential equations, and solving this system is sometimes more difficult. In both cases: fl ðt, r Þ≔fl ðt, xl ðt, r Þ, xu ðt, r ÞÞ ¼ min ff ðt, uÞju½xl ðt, r Þ, xu ðt, r Þg, fu ðt, r Þ≔fl ðt, xl ðt, r Þ, xu ðt, r ÞÞ ¼ max f f ðt, uÞju½xl ðt, r Þ, xu ðt, r Þg, 0  r  1: Essentially, this is an embedding method for transforming the fuzzy differential equation to two real differential equations for any arbitrary but fixed level. This embeds it to the shape of a convex cone. Why a convex cone? Because every nonnegative multiplier of a fuzzy number valued function belongs to the cone. Basically, to ensure that the solutions of two real differential equations are the fuzzy solution of the fuzzy original model for all levels, a characteristic theorem should be considered

138

Fig. 4.5

Soft Numerical Computing in Uncertain Dynamic Systems

The convex cone in embedding method.

and proved (see Fig. 4.5) (Bede and Gal, 2005; Bede et al., 2007; Gouyandeha et al., 2017; Stefanini and Bede, 2009). To express the equivalency of the fuzzy differential equation and two real differential equations in the cone, we need to prove a characteristic theorem. Characteristic theorem If the function f : ½t0 , T  R ! R is a continuous and fuzzy number valued function and gH-differentiable that satisfies the following fuzzy differential equation: x0gH ðtÞ ¼ f ðt, xÞ, xðt0 Þ ¼ x0 R and also suppose the following conditions: • •

f(t, x(t), r) ¼ [ fl(t, xl(t, r), xu(t, r)), fu(t, xl(t, r), xu(t, r))] fl(t, xl(t, r), xu(t, r)) and fu(t, xl(t, r), xu(t, r)) are equicontinuous. it means, for any E > 0 and any (t, u, v)  [t0, T] R2 if k(t, u, v)  (t, u1, v1) k < δ we have the following inequalities for 8 r  [0, 1]: j fl ðt, xl ðt, r Þ, xu ðt, r ÞÞ  fl ðt, xl ðt, r Þ, xu ðt, r ÞÞj < E j fl ðt, xl ðt, r Þ, xu ðt, r ÞÞ  fu ðt, xl ðt, r Þ, xu ðt, r ÞÞj < E j fu ðt, xl ðt, r Þ, xu ðt, r ÞÞ  fl ðt, xl ðt, r Þ, xu ðt, r ÞÞj < E j fu ðt, xl ðt, r Þ, xu ðt, r ÞÞ  fu ðt, xl ðt, r Þ, xu ðt, r ÞÞj < E

Continuous numerical solutions of uncertain differential equations 139 • •

fl(t, xl(t, r), xu(t, r)) and fu(t, xl(t, r), xu(t, r)) are uniformly bounded on any bounded set. Lipschitz property. There exists L > 0 such that: j fl ðt, u1 , v1 , r Þ  fl ðt, u2 , v2 , r Þj < Lmax fju1  u2 j, jv1  v2 jg j fl ðt, u1 , v1 , r Þ  fu ðt, u2 , v2 , r Þj < L max fju1  u2 j, jv1  v2 jg j fu ðt, u1 , v1 , r Þ  fl ðt, u2 , v2 , r Þj < L max fju1  u2 j, jv1  v2 jg j fu ðt, u1 , v1 , r Þ  fu ðt, u2 , v2 , r Þj < Lmax fju1  u2 j, jv1  v2 jg

for any r  [0, 1]. Then the fuzzy differential equation is equivalent to the each of following real differential equations in the cone:  0  0 xl ðt, r Þ ¼ fl ðt, r Þ xu ðt, r Þ ¼ fu ðt, r Þ , xl ðt0 , r Þ ¼ xl, 0 ðr Þ xu ðt0 , r Þ ¼ xu,0 ðr Þ  0  0 xu ðt, r Þ ¼ fl ðt, r Þ xl ðt, r Þ ¼ fu ðt, r Þ, , xl ðt0 , r Þ ¼ xl,0 ðr Þ xu ðt0 , r Þ ¼ xu,0 ðr Þ Proof. The equicontinuity implies the continuity of the function f. From the Lipschitz property, it is ensured that: sup max fj fl ðt, xl ðt, r Þ, xu ðt, r ÞÞ  fl ðt, yl ðt, r Þ, yu ðt, r ÞÞj, j fu ðt, xl ðt, r Þ, xu ðt, r ÞÞ r

fu ðt, yl ðt, r Þ, yu ðt, r ÞÞjg  sup max fjxl ðt, r Þ  yl ðt, r Þj, jxu ðt, r Þ  yu ðt, r Þjg r

This means that: DH ð f ðt, xðtÞÞ, f ðt, yðtÞÞÞ  DH ðxðtÞ, yðtÞÞ By the continuity of f, from the Lipschitz condition and the boundedness condition, it is concluded that the fuzzy differential equation has a unique solution and it is gH-differentiable and so the functions xl(t, r) and xu(t, r) are differentiable and it is concluded that (xl(t, r), xu(t, r)) can be the solution of one of the real equations. Conversely, let us assume that xl(t, r) and xu(t, r) are the solutions of one of the real equations for any fixed r. (They exist and are unique because of the Lipschitz condition.) On the other hand, since x(t) is gH-differentiable, then [xl(t, r), xu(t, r)] is the unique solution of the fuzzy differential equation. The proof is completed. We start our conversation with an easy example.

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Example. Consider: x0gH ðtÞ ¼ λ⊙xðtÞ, xðt0 Þ ¼ x0 R , λ  0 This is a simple fuzzy initial value problem. It means the initial value is a fuzzy number. It is clear that this problem does have an i  gH solution, and to find it we must solve the following initial value problems:  0 xl ðt, r Þ ¼ λxl ðt, r Þ, xl ðt0 , r Þ ¼ xl,0 ðr Þ and the solution is xl(t) ¼ x0,l(r) exp(λt). Also, for the upper solution, the same procedure is used and the solution is xu(t) ¼ x0,u(r) exp(λt) for the following upper differential equation in level-wise form:  0 xu ðt, r Þ ¼ λxu ðt, r Þ, xu ðt0 , r Þ ¼ xu,0 ðr Þ The i  gH-solution is: xigH ðt, r Þ ¼ ½xl,0 ðr Þexp ðλtÞ, xu,0 ðr Þexp ðλtÞ Note. In this example, suppose that problem has the ii  gH solution because wise form can interchange. To find that, differential equations, such as:  0 xl ðt, r Þ ¼ λxu ðt, r Þ, , xl ðt0 , r Þ ¼ xl,0 ðr Þ

the constant λ < 0 is negative, then the the end points of the derivative in levelwe have to solve a system of first order 

x0u ðt, r Þ ¼ λxl ðt, r Þ, xu ðt0 , r Þ ¼ xu,0 ðr Þ

Let us consider the differential operator is noted by: DxðtÞ≔x0 ðtÞ and D2 xðtÞ≔x00 ðtÞ, then: Dxl ðt, r Þ ¼ λxu ðt, r Þ, Dxu ðt, r Þ ¼ λxl ðt, r Þ by the second order operator: D2 xl ðt, r Þ ¼ λDxu ðt, r Þ ¼ λ2 xl ðt, r Þ, D2 xu ðt, r Þ ¼ λDxl ðt, r Þ ¼ λ2 xu ðt, r Þ By using the characteristic polynomial for each equation and finding the roots, the solution of each second order differential equation will be found: r 2  λ2 ¼ 0 ) r ¼ λ, + λ

Continuous numerical solutions of uncertain differential equations 141 and the solutions are: xl ðt, r Þ ¼ A exp ðλtÞ + Bexp ðλtÞ xu ðt, r Þ ¼ Cexp ðλtÞ + D exp ðλtÞ subject to: xl ðt0 , r Þ ¼ xl,0 ðr Þ, xu ðt0 , r Þ ¼ xu,0 ðr Þ We have two equations with four unknowns, and need two other equations. To this end, we can use the derivative of these equations as other initial values of the second order differential equations: Dxl ðt, r Þ ¼ ðλÞA exp ðλtÞ + λBexp ðλtÞ, Dxl ðt0 , r Þ ¼ λxu,0 ðr Þ Dxu ðt, r Þ ¼ ðλÞC exp ðλtÞ + λD exp ðλtÞ, Dxu ðt0 , r Þ ¼ λxl, 0 ðr Þ Now, considering the initial values xl(t0, r) ¼ xl,0(r), xu(t0, r) ¼ xu,0(r) in the solutions: A exp ðλt0 Þ + Bexp ðλt0 Þ ¼ xl,0 ðr Þ, C exp ðλtÞ + D exp ðλtÞ ¼ xu,0 ðr Þ On the other hand: Aexp ðλt0 Þ + B exp ðλt0 Þ ¼ xu,0 ðr Þ,  C exp ðλt0 Þ + D exp ðλt0 Þ ¼ xl,0 ðr Þ By solving these four equations, the unknown constants A, B, C, D will be found as follows:     xl, 0 ðr Þ + xu,0 ðr Þ xl,0 ðr Þ + xu,0 ðr Þ A ¼ exp ðλt0 Þxl,0 ðr Þ  , B ¼ exp ðλt0 Þ 2 2     xl,0 ðr Þ + xu,0 ðr Þ xl,0 ðr Þ + xu,0 ðr Þ  exp ðλt0 Þxl,0 ðr Þ, D ¼ exp ðλt0 Þ C¼ 2 2 Indeed A ¼  C, B ¼ D. Finally, the solutions are: xl ðt, r Þ ¼ A exp ðλtÞ + Bexp ðλtÞ xu ðt, r Þ ¼ Cexp ðλtÞ + D exp ðλtÞ where:     xl, 0 ðr Þ + xu,0 ðr Þ xl,0 ðr Þ + xu,0 ðr Þ , B ¼ exp ðλt0 Þ 2 2     xl,0 ðr Þ + xu,0 ðr Þ xl,0 ðr Þ + xu,0 ðr Þ  exp ðλt0 Þxl,0 ðr Þ, D ¼ exp ðλt0 Þ C¼ 2 2 A ¼ exp ðλt0 Þxl,0 ðr Þ 

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2

1

–4

–3

–2

–1

0

0 1

2

3

4

–1

–2

Fig. 4.6

ii  gH solution for λ ¼  1.

It is clear that for a larger value of λ, the first term of the solutions vanishes and the second term intends infinity. To simplify, let us suppose that t0 ¼ 0. In this case: A¼

xl,0 ðr Þ  xu,0 ðr Þ xl,0 ðr Þ + xu,0 ðr Þ ¼ C, B ¼ D ¼ 2 2

For more illustration, consider [xl, 0(r), xu, 0(r)] ¼ [r  1, 1  r] and for λ ¼  1, we have the solutions as: xl ðt, r Þ ¼ ð1  r Þ exp ðtÞ, xu ðt, r Þ ¼ ðr  1Þexp ðtÞ The ii  gH solution is (Fig. 4.6): xiigH ðt, r Þ ¼ ½ð1  r Þ exp ðtÞ, ðr  1Þ exp ðtÞ The i  gH-solution for λ ¼ 1 is (Fig. 4.7): xigH ðt, r Þ ¼ ½ðr  1Þexp ðtÞ, ð1  r Þexp ðtÞ Having analyzed the fuzzy initial value problem above, it is now time to mention the existence and uniqueness of the solutions.

4.3.1

THEOREM—EXISTENCE

AND UNIQUENESS

Let us assume that the following conditions hold for fuzzy differential equations: x0gH ðtÞ ¼ f ðt, xÞ, xðt0 Þ ¼ x0 R 1. The function f : I1 I2 ! R is continuous where I1 is a closed interval, contains the initial value x0, and I2 is any other area such that f is bounded on it.

Continuous numerical solutions of uncertain differential equations 143

2

1

–4

–3

–2

0

–1

0 1

2

3

4

–1

–2

Fig. 4.7 i  gH solution for λ ¼ 1.

2. The function f is bounded. This means 9 M < 0, DH( f(t, x), 0)  M, 8 (t, x)  I1 I2. 3. The real function g : I1 I3 ! R such that g(t, u) ≡ 0 is bounded on I1 I3, 9 M1 > 0, 0  g(t, u)  M1, 8 (t, u)  I1 I3 where I3 is another closed interval and contains u. Moreover, g(t, u) is nondecreasing in u and its corresponding initial value problem: u0 ðtÞ ¼ gðt, uðtÞÞ, uðt0 Þ ¼ 0 has only the solution u(t) ≡ 0 on I1. 4. Also: DH ð f ðt, xÞ, f ðt, yÞÞ  gðt, jx  yjÞ, 8ðt, xÞ8ðt, yÞI1 I2 , x, yI3 Then the fuzzy initial value problem has two (i  gH)-solution xigH(t) and (ii  gH)solution xiigH(t), and the following successive iterations converge to these two solutions: ðt xi, n + 1 ðtÞ ¼ xi,0  f ðt, xi, n ðzÞÞdz, xi,0 ðtÞ ¼ x0 t0

and: xii, n + 1 ðtÞ ¼ xii,0 H ð1Þ⊙

ðt

f ðtxii, n ðzÞÞdz, xii,0 ðtÞ ¼ x0

t0

Proof. We only prove the (ii  gH) solution and other ones are similar: ðt DH ðxii, n + 1 ðtÞ, xii,0 Þ  f ðt, xii, n ðzÞ, 0Þdz t0

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In addition, by the properties of Hausdorff distance, we have:   ðt DH ðxii,1 ðtÞ, xii, 0 ðtÞÞ ¼ DH xii,1 ðtÞ, xii,1 ðtÞð1Þ⊙ f ðt, xii, 0 ðzÞÞdz ¼ DH

ð t

t0

 f ðt, xii, n ðzÞÞdz, 0 

ðt

t0

DH ð f ðt, xii, n ðzÞÞ, 0Þdz

t0

Continuing by properties of the distance again: DH ðxii, n + 1 ðtÞ, xii, n ðtÞÞ ¼ DH ðx0 xii, n + 1 ðtÞ, x0 xii, n ðtÞÞ   ðt ðt ¼ DH ð1Þ⊙ f ðt, xii, n ðzÞÞdz, ð1Þ⊙ f ðt, xii, n1 ðzÞÞdz t0



ðt

t0

DH ð f ðt, xii, n ðzÞÞ, f ðt, xii, n1 ðzÞÞÞdz

t0

For two points like t, t + h around the initial value t0, we observe that: ðt + h f ðtxii, n ðzÞÞdz xii, n + 1 ðtÞH xii, n + 1 ðt + hÞ ¼ ð1Þ⊙ t

because: xii, n + 1 ðt + hÞð1Þ⊙

ðt + h

f ðt, xii, n ðzÞÞdz ¼

t0

¼ x0 H ð1Þ⊙

ðt + h

f ðtxii, n ðzÞÞdzð1Þ⊙

ðt + h

t0

f ðtxii, n ðzÞÞdz

t

¼ x0 H ð1Þ⊙

ðt + h

f ðtxii, n ðzÞÞdz

t0

ð1Þ⊙

ðt + h

f ðtxii, n ðzÞÞdzH ð1Þ⊙

ðt

t0

¼ x0 H ð1Þ⊙

f ðtxii, n ðzÞÞdz

t0

ðt

f ðtxii, n ðzÞÞdz ¼ xii, n + 1 ðtÞ

t0

then: xii, n + 1 ðtÞH xii, n + 1 ðt + hÞ ¼ ð1Þ⊙

ðt + h

f ðtxii, n ðzÞÞdz

t

and passing to limit with h ↘ 0, we have: ðt + h xii, n + 1 ðtÞH xii, n + 1 ðt + hÞ 1 f ðtxii, n ðzÞÞdz ¼ lim ⊙ lim h↘0 h↘0 h h t

Multiplying by

1 h

Continuous numerical solutions of uncertain differential equations 145 Now, the distance between this derivative and the right-hand side of differential equation is:  ðt + h  1 DH ⊙ f ðt, xii, n ðzÞÞdz, f ðt, xii, n ðtÞÞ ¼ h t  ðt + h  ðt + h 1 1 ¼ DH ⊙ f ðt, xii, n ðzÞÞdz, ⊙ f ðt, xii, n ðtÞÞdz h t h t ðt + h 1  ⊙ DH ð f ðt, xii, n ðzÞÞ, f ðt, xii, n ðtÞÞÞdz  sup DH ð f ðt, xii, n ðzÞÞ, f ðt, xii, n ðtÞÞÞ h t jztjh Finally: lim h↘0

xii, n + 1 ðtÞH xii, n + 1 ðt + hÞ ¼ f ðtxii, n ðtÞÞ h

We have the same relation from the left-hand side and: ðt f ðtxii, n ðzÞÞdz xii, n + 1 ðt  hÞH xii, n + 1 ðtÞ ¼ ð1Þ⊙ th

and: xii, n + 1 ðt  hÞH xii, n + 1 ðtÞ ¼ f ðtxii, n ðtÞÞ h↘0 h

lim

Now we claim that the xii,

n+1(t)

is the ii  gH differentiable of:

x0ii, n + 1 ¼ f ðt, xii, n ðtÞÞ for all points in a closed interval around initial value. Now the following relations allow us to prove the existence of ii  gH solution: ðt x0ii, n + 1 ¼ f ðt, xii, n ðtÞÞ, DH ðxii, n + 1 ðtÞ, xii,0 Þ  f ðt, xii, n ðzÞ, 0Þdz t0

DH ðxii, 1 ðtÞ, xii,0 ðtÞÞ 

ðt

DH ðf ðt, xii, n ðzÞÞ, 0Þdz

t0

DH ðxii, n + 1 ðtÞ, xii, n ðtÞÞ 

ðt

DH ð f ðt, xii, n ðzÞÞ, f ðt, xii, n1 ðzÞÞÞdz

t0

4.3.2

FUZZY

DIFFERENTIAL EQUATIONS—VARIATION OF CONSTANTS

First of all, let us analyze a modeling problem as fuzzy differential equations in the following forms: x0 ðtÞ ¼ λ⊙xðtÞ, x0 ðtÞλ⊙xðtÞ ¼ 0

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With fuzzy information, these two fuzzy differential equations are different. Because the zero on the right-hand side is basically a membership function, we can call it a forcing term. So: x0 ðtÞλ⊙xðtÞ ¼ σ ðtÞ, x0 ðtÞ ¼ λ⊙xðtÞσ ðtÞ Now these two different differential equations with the same fuzzy initial value x(t0) ¼ x0 do have different results. To extend our discussion, let us consider the following fuzzy initial value problem:  0 x ðtÞ ¼ aðtÞ⊙xðtÞbðtÞ xð t 0 Þ ¼ x 0 where aðtÞR,x0 R and also bðtÞR 8t: In general, this problem is not equivalent to the following problems:  0 x ðtÞðaðtÞÞ⊙xðtÞ ¼ bðtÞ xðt0 Þ ¼ x0 and: 

x0 ðtÞðbðtÞÞ ¼ aðtÞ⊙xðtÞ xðt0 Þ ¼ x0

Moreover, these last two problems are not equivalent. For instance, suppose that the first and second problems have a common solution, then we will have: aðtÞ⊙xðtÞbðtÞðaðtÞÞ⊙xðtÞ ¼ bðtÞ This causes: aðtÞ⊙xðtÞðaðtÞÞ⊙xðtÞ ¼ 0 and it shows that the solution x(t) is a real valued function. This means that these two problems are not equivalent with a fuzzy (but not real) initial value. Now the question is: which of the above problems should be considered as the fuzzy linear differential equation? Let us remark that in real-world applications, usually a dynamical system under uncertainty is modeled by fuzzification of the crisp (partial) differential equations of the system. So, depending on how we write these three crisp problems and then how we also fuzzify them, we get three different results. This is in contradiction to one of the main requirements of a model, which is that the behavior of the solution should reflect the real behavior of a system and not a particular form of an equation. These three problems are equivalent in the real case but they are inequivalent in the fuzzy case. Then we will provide their solutions at the

Continuous numerical solutions of uncertain differential equations 147 same time and any of the solutions of these problems can be chosen in modeling the real behavior of a dynamic system under uncertainty. In the following theorem, we present the solution of fuzzy differential equations with variation of constants (Allahviranloo and Chehlabi, 2015; Bede and Gal, 2005; Bede et al., 2007; Chehlabi and Allahviranloo, 2018; Stefanini and Bede, 2009).

4.3.3

THEOREM—EXISTENCE

OF THE SOLUTION

Let consider two functions: ð t    ðz   ðt aðzÞdz ⊙ x0  bðzÞ⊙ exp  aðuÞdu dz x1 ðtÞ ¼ exp t0

t0

t0

and: x2 ðtÞ ¼ exp

ð t

   ðz   ðt aðzÞdz ⊙ x0 H ðbðzÞÞ⊙ exp  aðuÞdu dz

t0

t0

t0

provided that the H-difference:  ðz  ðt x0 H ðbðzÞÞ⊙ exp  aðuÞdu dz t0

t0

exists. Now the conditions for the solutions are listed as follows: 1. If a(t) > 0 8 t  (t0, t1) then x1 (i)-differentiable and it is a solution of: 

x0 ðtÞ ¼ aðtÞ⊙xðtÞbðtÞ xðt0 Þ ¼ x0

in the same interval.

Ð Ð 2. If a(t) < 0 and the H-difference x0  tt0 ( b(z)) ⊙ exp( tz0 a(u)du)dz exists for 8 t  (t0, t1) then x1 (ii)-differentiable and it is a solution of the same problem in (1) in the same interval. 3. If a(t) < 0 and x1(t + h)Hx1(t) and x1(t)Hx1(t  h) exist for 8t  (t0, t1) then x1 (i)-differentiable and it is a solution of: 

x0 ðtÞðaðtÞÞ⊙xðtÞ ¼ bðtÞ x ðt 0 Þ ¼ x 0

in the same interval. 4. If a(t) < 0 and x1(t)Hx1(t  h) and x1(t  h)Hx1(t) exist for 8 t  (t0, t1) then x1 (ii)-differentiable and it is a solution of:

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x0 ðtÞðbðtÞÞ ¼ aðtÞ⊙xðtÞ xðt0 Þ ¼ x0

in the same interval.

Ð Ð 5. If a(t) > 0 and the H-difference x0  tt0 ( b(z)) ⊙ exp( tz0 a(u)du)dz exists for 8 t  (t0, t1) and x2(t + h)Hx2(t) and x2(t)Hx2(t  h) exist for 8 t  (t0, t1) then x2 is the solution of: 

x0 ðtÞðbðtÞÞ ¼ aðtÞ⊙xðtÞ xðt0 Þ ¼ x0

in the same interval.

Ð Ð 6. If a(t) > 0 and the H-difference x0  tt0 ( b(z)) ⊙ exp( tz0 a(u)du)dz for 8 t  (t0, t1) and x2(t  h)Hx2(t) and x2(t  h)Hx2(t) exist for 8 t  (t0, t1) then x2 is the solution of: 

x0 ðtÞðaðtÞÞ⊙xðtÞ ¼ bðtÞ xðt0 Þ ¼ x0

in the same interval (Allahviranloo et al., 2015; Gouyandeha et al., 2017). Proof. First suppose that the following term of x1 is (i)-differentiable and its differential is:   ðz  0  ðt  ðt x0  bðzÞ⊙ exp  aðuÞdu dz ¼ bðtÞ⊙ exp  aðuÞdu t0

t0

t0

and also in the second term of x2 if the H-difference:  ðz  ðt x0  ðbðzÞÞ⊙ exp  aðuÞdu dz t0

t0

exists then x2 is (ii)-differentiable and:   ðz  0  ðt  ðt x0  ðbðzÞÞ⊙ exp  aðuÞdu dz ¼ bðtÞ⊙ exp  aðuÞdu t0

t0

Ð

t0

is positive as Case 1. Now case, a(t) > 0 8 t  (t0, t1) then exp( Ð tin the first 0 well. So, (exp( t0 a(z)dz)) ¼ a(t) > 0. Now:  ð t    ðz  0 ðt 0 x1 ðtÞ ¼ exp aðzÞdz ⊙ x0  bðzÞ⊙ exp  aðuÞdu dz t0

t0

t0

t t0 a(z)dz)

Continuous numerical solutions of uncertain differential equations 149  ¼ exp

ð t

0   ðz   ðt aðzÞdz ⊙ x0  bðzÞ⊙ exp  aðuÞdu dz

t0

t0

t0

  ðz  0 ð t  ðt aðzÞdz  x0  bðzÞ⊙ exp  aðuÞdu dz ⊙ exp t0

t0

t0

By substituting: 

x01 ðtÞ ¼ aðtÞ⊙

x0 

ðt

 ðz   bðzÞ⊙ exp  aðuÞdu dz

t0

t0

  ðt  ð t  aðzÞdz ¼ aðtÞ⊙x1 ðtÞbðtÞ  bðtÞ⊙ exp  aðuÞdu ⊙ exp t0

t0

this means that x1 is the solution of:  0 x ðtÞ ¼ aðtÞ⊙xðtÞbðtÞ xðt0 Þ ¼ x0

Ð Case 2. For this section, we have a(t) < 0 8 t  (t0, t1) then exp( tt0 a(z)dz) is again Ðt positive. So (exp( t0 a(z)dz))0 ¼ a(t) < 0. Now x20 is (ii)-differentiable and stated as:  ð t    ðz  0 ðt 0 x2 ðtÞ ¼ exp aðzÞdz ⊙ x0 H ðbðzÞÞ⊙ exp  aðuÞdu dz t0

 ¼ exp

ð t

t0

t0

 exp

ð t

t0

0   ðz   ðt aðzÞdz ⊙ x0 H ðbðzÞÞ⊙ exp  aðuÞdu dz t0

t0

   ðz  0 ðt aðzÞdz ⊙ x0 H ðbðzÞÞ⊙ exp  aðuÞdu dz

t0

t0

t0

By substituting: x02 ðtÞ ¼ aðtÞ⊙  exp

ð t t0

  ðz   ðt x0 H ðbðzÞÞ⊙ exp  aðuÞdu dz t0

t0

   ðt  aðzÞdz ⊙ bðtÞ⊙ exp  aðuÞdu ¼ aðtÞ⊙x1 ðtÞbðtÞ t0

Case 3. We have a(t) < 0 and x1(t + h)Hx1(t) and x1(t)Hx1(t  h) exist for 8 t  (t0, t1) and:

150

Soft Numerical Computing in Uncertain Dynamic Systems  ð t    ðz  0 ðt x01 ðtÞ ¼ exp aðzÞdz ⊙ x0  bðzÞ⊙ exp  aðuÞdu dz t0

¼ exp

ð t

t0

t0

   ðz  0 ðt aðzÞdz ⊙ x0  bðzÞ⊙ exp  aðuÞdu dz

t0

t0

t0

 ð t 0   ðz   ðt H ð1Þ exp aðzÞdz ⊙ x0  bðzÞ⊙ exp  aðuÞdu dz t0

t0

t0

By substituting: x01 ðtÞ ¼ exp

ð t

   ðt  aðzÞdz ⊙ bðtÞ⊙ exp  aðuÞdu

t0

t0

  ðz   ðt H ð1ÞaðtÞ⊙ x0  bðzÞ⊙ exp  aðuÞdu dz t0

t0

¼ bðtÞH ð1ÞaðtÞ⊙x1 ðtÞ this means that x1 is the solution of:  0 x ðtÞðaðtÞÞ⊙xðtÞ ¼ bðtÞ xðt0 Þ ¼ x0 The other cases can be proved similarly. Example. Let us consider an example to investigate the behavior of the (ii)-differentiable solution. Suppose that: x0 ðtÞ ¼ ð1Þ⊙xðtÞt, xð0Þ ¼ ð1, 2, 3Þ The initial value in level-wise form is as x(0, r) ¼ [r + 1, 3  r]. Since the following H-difference exists: ðt x0 H ðzÞ⊙ exp ðzÞdz, 8tð0, ∞Þ 0

the (ii)-solution x2 is:   ðt x2 ðtÞ ¼ exp ðtÞ⊙ ð1, 2, 3ÞH ðzÞ⊙ exp ðzÞdz 0

¼ ðt  1Þ⊙1ð2 exp ðtÞ, 3 exp ðtÞ, 4 exp ðtÞÞ where 1 is a singleton fuzzy number. For more illustration, the level-wise form of (2 exp( t), 3 exp( t), 4 exp( t)) is [r + 2, 4  r] exp( t). The (ii)-differentiable solution is: x2 ðtÞ ¼ ½ðt + r + 1Þexp ðtÞ, ðt + 5  r Þexp ðtÞ For instance, in the levels r ¼ 0, 1, the solutions are plotted in Fig. 4.8.

Continuous numerical solutions of uncertain differential equations 151

5

4

3

2

1

0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Fig. 4.8 The solution in the levels r ¼ 0, 1.

Based on our discussions in Chapter 3 on the diameter of (ii)-solutions, the same conclusion is seen here, and the diameter is decreasing. In the next section, we shall discuss the function entitled length function. After defining this operator, we will use this concept for solving and analyzing fuzzy differential equations with first order.

4.3.4

LENGTH

FUNCTION

First of all, the length, sometimes called the diameter, of the fuzzy function was used for analysis of type 1 and type 2 solutions of fuzzy differential equations. Indeed, in Chapter 3 we pointed out the diameter and here it is explained more with the title of length function.

4.3.4.1 Definition—Length function The length function of a fuzzy number is a function l : R ! S in which S is the space of continuous functions like f : [0, 1] ! R. The length is defined as: lðA, r Þ ¼ Au ðr Þ  Al ðr Þ, 8r½0, 1:

4.3.4.2 Nonlinear property Generally, the length function is not a linear function and this means that if λ, μ  R and A,BR and C ¼ λ ⊙ A  μ ⊙ B, D ¼ λ ⊙ A Hμ ⊙ B subject to the H-difference exists, then: lðC, r Þ ¼ jλjlðA, r Þ + jμjlðB, r Þ lðD, r Þ ¼ jλjlðA, r Þ  jμjlðB, r Þ

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Soft Numerical Computing in Uncertain Dynamic Systems

The proof is easy to check and it is expressed by the signs of the coefficients. Let us here consider that λ 0 and μ < 0. In this case, for all levels r  [0, 1], we have: lðC, r Þ ¼ lðλ⊙Aμ⊙B, r Þ ¼ ðλ⊙Aμ⊙BÞu ðr Þ  ðλ⊙Aμ⊙BÞl ðr Þ ¼ λAu ðr Þ + μBl ðr Þ  λAl ðr Þ  μBu ðr Þ ¼ λðAu ðr Þ  Al ðr ÞÞ + ðμÞðBu ðr Þ  Bl ðr ÞÞ ¼ λlðA, r Þ + ðμÞlðB, r Þ lðD, r Þ ¼ lðλ⊙AH μ⊙B, r Þ ¼ ðλ⊙AH μ⊙BÞu ðr Þ  ðλ⊙AH μ⊙BÞl ðr Þ ¼ λAu ðr Þ  μBl ðr Þ  λAl ðr Þ + μBu ðr Þ ¼ λðAu ðr Þ  Al ðr ÞÞ  ðμÞðBu ðr Þ  Bl ðr ÞÞ ¼ λlðA, r Þ  ðμÞlðB, r Þ The proof for the other cases of the coefficients are similar to this case (Allahviranloo and Chehlabi, 2015; Chehlabi and Allahviranloo, 2018).

4.3.4.3 Theorem—Nonlinear property of fuzzy functions Suppose that f , g : ½a, b ! R are two fuzzy number valued functions. In addition, consider that for all t  [a, b] the function w(t) ¼ λ ⊙ f(t)Hμ ⊙ g(t) exists for some values of λ, μ  R. Moreover, suppose that l(f(t), r) and l(g(t), r) are linearly independent. Then: lðf ðtÞ, r Þ  lðgðtÞ, r Þ ¼ lðλ⊙f ðtÞH μ⊙gðtÞ, r Þ⟺ jλj ¼ jμj ¼ 1 Proof. For the first case, consider w(t) ¼ λ ⊙ f(t)Hμ ⊙ g(t) exists subject to jλ j ¼ jμ j ¼ 1. Now from the previous theorem it is clear that: lðwðtÞ, r Þ ¼ lð f ðtÞ, rÞ  lðgðtÞ, r Þ, 8t½a, b8r½0, 1 Conversely, if l(w(t), r) ¼ l( f(t), r)  l(g(t), r) and from the previous theorem: lðwðtÞ, r Þ ¼ lðλ⊙f ðtÞH μ⊙gðtÞ, r Þ ¼ jλjlðf ðtÞ, r Þ  jμjlðgðtÞ, r Þ so: jλjlð f ðtÞ, rÞ  jμjlðgðtÞ, r Þ ¼ lð f ðtÞ, r Þ  lðgðtÞ, r Þ and: ðjλj  1Þlð f ðtÞ, rÞ  ðjμj  1ÞlðgðtÞ, r Þ ¼ 0 based on the linearly independent property, j λ j ¼ j μ j ¼ 1. The proof is completed.

Continuous numerical solutions of uncertain differential equations 153

4.3.4.4 Remark As a conclusion, let us consider the function: hðtÞ ¼ λ0 ⊙f ðtÞμ0 ⊙gðtÞ, λ0 ,μ0 R, λ0 μ0 6¼ 0: If l(f(t), r) and l(g(t), r) are linear independent with respect to t, r then it is clear that l(h(t), r) and l(g(t), r) are linear independent with respect to t, r as well. If we suppose that λ0 6¼ 0 then we have: 1 1 f ðtÞ ¼ λ⊙hðtÞH μ⊙gðtÞ, λ ¼ 0 , μ ¼ 0 λ μ Therefore, based on the previous theorem, we find: lð f ðtÞ, rÞ ¼ lðhðtÞ, r Þ  lðgðtÞ, r Þ⟺ jλj ¼ jμj ¼ 1 This means: lðhðtÞ, r Þ ¼ lð f ðtÞ, rÞ + lðgðtÞ, r Þ, ⟺ jλ0 j ¼ jμ0 j ¼ 1 for all 8 t  [a, b] 8 r  [0, 1].

4.3.4.5 Remark—Differentiability and length Based on the discussion in Chapter 3, the type of the gH-differentiability can be described by the length operator. As we mentioned before, in the case of type (ii) gH-differentiability the length of the function is decreasing and in type (i) gH-differentiability the length of the function is increasing. Because for a fuzzy number valued function like f(t) the gH-derivative at the point t in the level-wise form is defined as: f ðt + hÞgH f ðtÞ h!0 h

0 fgH ðtÞ ¼ lim

subject to the gH-difference f(t + h)gH f(t) existing, the necessary and sufficient conditions for the gH-differentiability fuzzy number valued function f(t) are: In case (i):  

0 figH ðtÞ ½r ¼ fl0 ðt, r Þ, fu0 ðt, r Þ In case (ii): 



0 ðtÞ ½r  ¼ fu0 ðt, r Þ, fl0 ðt, r Þ fiigH

subject to the functions fl0 (t, r) and fu0 (t, r) being two real valued differentiable functions with respect to t and uniformly with respect to r  [0, 1]. It should be noted that both the functions are left continuous on r  (0, 1] and right continuous at r ¼ 0. Moreover, the following conditions should be satisfied:

154 • •

Soft Numerical Computing in Uncertain Dynamic Systems

The function fl0 (t, r) is nondecreasing and the function fu0 (t, r) is nonincreasing as functions of r and, fl0 (t, r)  fu0 (t, r). Or, The function fl0 (t, r) is nonincreasing and the function fu0 (t, r) is nondecreasing as functions of r and, fu0 (t, r)  fl0 (t, r).

And: 





0 ðtÞ ½r  ¼ min fl0 ðt, r Þ, fu0 ðt, r Þ , max fl0 ðt, r Þ, fu0 ðt, r Þ fgH

Now, considering the definition of the length function and the abovementioned level-wise form of two types of derivative, we have: In case (i):   0 ðtÞ, r ¼ fu0 ðt, r Þ  fl0 ðt, r Þ ¼ l0 ðf ðtÞ, r Þ l figH In case (ii):   0 ðtÞ, r ¼ fl0 ðt, r Þ  fu0 ðt, r Þ ¼ l0 ðf ðtÞ, r Þ l fiigH

4.3.4.6 Theorem—Nonlinear property of fuzzy functions Let us consider f , g : ða, bÞ ! R to be differentiable in the sense of (i) or (ii) on (a, b). If the H-difference λ ⊙ f(t)Hμ ⊙ g(t) exists for any t  (a, b) and some λ, μ  R then the function w(t) ¼ λ ⊙ f(t)Hμ ⊙ g(t) is differentiable and precisely: 1. If f is i  gH differentiable and g is ii  gH differentiable then w is i  gH differentiable. 2. If f is ii  gH differentiable and g is i  gH differentiable then w is ii  gH differentiable. In addition, we have: 0 w0gH ðtÞ ¼ λ⊙fgH ðtÞH μ⊙g0gH ðtÞ, 8tða, bÞ

Proof. For simplicity, we set F(t) ¼ λ ⊙ f(t) and G(t) ¼ μ ⊙ g(t). It is clear that the functions F and G are differentiable in the same differentiability concept of functions f and g, respectively, and also: 0 F0gH ðtÞ ¼ λ⊙ fgH ðtÞ, G0gH ðtÞ ¼ μ⊙g0gH ðtÞ:

Using the length function, we have: lðwðtÞÞ ¼ lðFðtÞH GðtÞÞ ¼ lðFðtÞÞ  lðGðtÞÞ

Continuous numerical solutions of uncertain differential equations 155 Then in case (1), we have:     l F0igH ðtÞ ¼ l0 ðFðtÞÞ, l G0igH ðtÞ ¼ l0 ðGðtÞÞ and:     l0 ðwðtÞÞ ¼ l0 ðFðtÞÞ  l0 ðGðtÞÞ ¼ l F0igH ðtÞ + l G0igH ðtÞ  0 which means w is i  gH differentiable and:

w0 ðt, r Þ ¼ ðFH GÞ0 ðt, r Þ ¼ F0l ðt, r Þ  G0l ðt, r Þ, F0u ðt, r Þ  G0u ðt, r Þ ¼ F0 ðt, r Þ  G0 ðt, r Þ In case (2), we have:     l F0igH ðtÞ ¼ l0 ðFðtÞÞ, l G0igH ðtÞ ¼ l0 ðGðtÞÞ and:     l0 ðwðtÞÞ ¼ l0 ðFðtÞÞ  l0 ðGðtÞÞ ¼  l F0igH ðtÞ + l G0igH ðtÞ  0 which means w is ii  gH differentiable and:

w0 ðt, r Þ ¼ ðFH GÞ0 ðt, r Þ ¼ F0u ðt, r Þ  G0u ðt, r Þ, F0l ðt, r Þ  G0l ðt, r Þ ¼ F0 ðt, r Þ  G0 ðt, r Þ The proof is now completed.

4.3.4.7 Theorem—Derivative of integral equation Let f : ða, bÞ ! R be continuous. In this case: Ð a. The function F(t) ¼ Ðta f(x)dx is i  gH differentiable and F0 (t) ¼ f(t). b. The function G(t) ¼ bt f(x)dx is ii  gH differentiable and G0 (t) ¼ (1) ⊙ f(t). Proof. We directly go through the details of the case (b). Using the length function: ðb ðb lðGðtÞ, r Þ ¼ fu ðx, r Þdx  fl ðx, r Þdx, 8r½0, 1 t

so: Gu ðt, r Þ  Gl ðt, r Þ ¼

t

ðb t

fu ðx, r Þdx 

ðb t

fl ðx, r Þdx

156

Soft Numerical Computing in Uncertain Dynamic Systems

then: Gu ðt, r Þ 

ðb

fu ðx, r Þdx ¼ Gl ðt, r Þ 

ðb

t

fl ðx, r Þdx

t

Taking the differential, we get: G0u ðt, r Þ + fu ðt, r Þ ¼ G0l ðt, r Þ + fl ðt, r Þ Finally: G0u ðt, r Þ  G0l ðt, r Þ ¼ fl ðt, r Þ  fu ðt, r Þ ¼ lðf ðt, r ÞÞ  0 This means that the function G(t) is ii  gH differentiable and: lðG0 ðtÞÞ ¼ lð f ðtÞÞ and about its derivative we can handle it by level-wise form: ð b  ðb ½Gl ðt, r Þ, Gu ðt, r Þ ¼ fl ðx, r Þdx, fu ðx, r Þdx t

t

then:



G0gH ðt, r Þ ¼ min G0l ðt, r Þ, G0u ðt, r Þ , max G0l ðt, r Þ, G0u ðt, r Þ where: G0l ðt, r Þ ¼ fl ðt, r Þ, G0u ðt, r Þ ¼ fu ðt, r Þ Finally: G0gH ðt, r Þ ¼ ½fu ðt, r Þ,  fl ðt, r Þ ¼ ½ fl ðt, r Þ, fu ðt, r Þ ¼ f ðt, r Þ for all 8r  [0, 1]. Then: G0gH ðtÞ ¼ ð1Þ⊙f ðtÞ

4.3.4.8 The length function—Fuzzy differential equations In this section, we consider the first order fuzzy differential equations with a fuzzy initial value. Now let us assume the following fuzzy differential equation: x0gH ðtÞ ¼ f ðtxðtÞÞ, tI ¼ ½0∞, xð0Þ ¼ x0 where f : I R ! R is a continuous fuzzy number valued mapping and x0 R is a fuzzy number. As we mentioned in Chapter 3, if we want to consider two types of differentiability, we will have:

Continuous numerical solutions of uncertain differential equations 157 Case (i): i  gH xðtÞ ¼ x0 FR

ðt a

x0igH ðsÞds ¼ x0 FR

ðt

f ðs, xðsÞÞds

a

Case (ii): ii  gH xðtÞ ¼ x0 H ð1ÞFR

ðt a

x0iigH ðsÞds ¼ x0 H ð1ÞFR

ðt

f ðsxðsÞÞds

a

If the gH-differentiable fuzzy function x : I ! R is i  gH (ii  gH) differentiable and satisfies the fuzzy differential equation, it is called i  solution (ii-solution) of the fuzzy differential equation, respectively. Here again let us consider the following fuzzy initial value problem with variation of constants:  0 xgH ðtÞ ¼ aðtÞ⊙xðtÞbðtÞ xðt0 Þ ¼ x0 where aðtÞR, x0 R and also bðtÞR 8t: Using the length function, we will find all the existing solutions of the above problem. In the first case, we consider a(t) < 0 with ii-differentiability. This means that:

x0iigH ðt, r Þ ¼ x0l ðt, r Þ, x0u ðt, r Þ and lðx0 ðtÞ, rÞ ¼ lðaðtÞxðtÞ, r Þ + lðbðtÞ, r Þ It is concluded: x0l ðt, r Þ  x0u ðt, r Þ ¼ aðtÞxl ðt, r Þ  aðtÞxu ðt, r Þ + lðbðtÞ, r Þ then: x0l ðt, r Þ  aðtÞxl ðt, r Þ ¼ x0u ðt, r Þ  aðtÞxu ðt, r Þ + lðbðtÞ, r Þ Let us consider the right-hand side as: μðtÞ ¼ x0u ðt, r Þ  aðtÞxu ðt, r Þ + lðbðtÞ, r Þ Finally we have: x0l ðt, r Þ  aðtÞxl ðt, r Þ ¼ μðtÞ To find the solution, we use integrating factor method and: ð t    ðu   ðt aðuÞdu xl, 0 ðr Þ + μðuÞ exp  aðvÞdv du xl ðt, r Þ ¼ exp t0

t0

t0

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Soft Numerical Computing in Uncertain Dynamic Systems

Using the value of μ, we can find: ð t   ðt  aðuÞdu xl, 0 ðr Þ + xu ðt, r Þ exp  aðuÞdu  xu,0 ðr Þ xl ðt, r Þ ¼ exp t0

t0

ðt +

 ðu   lðbðuÞ, r Þ exp  aðvÞdv du

t0

t0

The same procedure can be applied for the upper level-wise solution. Then: x0u ðt, r Þ  aðtÞxu ðt, r Þ ¼ x0l ðt, r Þ  aðtÞxl ðt, r Þ  lðbðtÞ, r Þ Let us consider the right-hand side as: μðtÞ ¼ x0l ðt, r Þ  aðtÞxl ðt, r Þ  lðbðtÞ, r Þ Finally, we have: x0u ðt, r Þ  aðtÞxu ðt, r Þ ¼ μðtÞ To find the solution, we use the integrating factor method and: ð t    ðu   ðt aðuÞdu xu,0 ðr Þ + μðuÞ exp  aðvÞdv du xu ðt, r Þ ¼ exp t0

t0

t0

Using the value of μ, we can find: ð t   ðt  aðuÞdu xu,0 ðr Þ + xl ðt, r Þ exp  aðuÞdu  xl,0 ðr Þ xu ðt, r Þ ¼ exp t0



ðt

 ðu   lðbðuÞ, r Þ exp  aðvÞdv du

t0

t0

t0

Now we can obtain the length of x(t) as:

¼ exp

ð t

lðxðtÞ, r Þ ¼ xu ðt, r Þ  xl ðt, r Þ  ð t  ðu   aðuÞdu lðx0 , r Þ  l bðuÞ exp  aðvÞdv du

t0

Ðu

t0

t0

Ð function b(u) exp( t0 a(v)dv)du is continuous and itsÐ integral tt0 b(u) ÐThe Ð exp ( tu0 a(v)dv)du is i  gH differentiable, therefore its length l( tt0 b(u) exp( tu0 a(v) dv)du) is an increasing function of variable t. On the other hand Ð l(x0, r) is not Ð dependent on t, so the solution is described by the initial value x0 and tt0 b(u) exp( tu0 a(v)dv)du as follows: ð t    ðu   ðt aðuÞdu λ⊙x0 H μ⊙ bðuÞ exp  aðvÞdv du xðtÞ ¼ exp t0

t0

where λ, μ  {1, 1} and the H-difference exists.

t0

Continuous numerical solutions of uncertain differential equations 159 Note that all integrations are fuzzy Riemann integrations. Now we can explain the solution with different values of λ and μ such that the different solutions satisfy all the conditions of theorem 4.3.3. For instance, based on initial conditions λ ¼ 1 then for any μ  {1, 1} with a(t) < 0. So considering the second case of theorem 4.3.3, the solution function x(t) is ii  gH differentiable. In general, we can consider the following equation: x0 ðtÞ ¼ aðtÞ⊙xðtÞðμÞ⊙bðtÞ such that for two values of μ  {1, 1} we have different equations with different solutions as follows: ð t    ðu   ðt aðuÞdu x0 μ⊙ bðuÞ exp  aðvÞdv du xðtÞ ¼ exp xðtÞ ¼ exp

ð t

t0

t0

t0

   ðu   ðt aðuÞdu x0 H μ⊙ bðuÞ exp  aðvÞdv du

t0

t0

t0

In case μ ¼  1 ð t    ðu   ðt aðuÞdu x0 H ð1Þ⊙ bðuÞ exp  aðvÞdv du xðtÞ ¼ exp t0

t0

t0

is ii  gH differentiable. Generally, by choosing the different values of μ and two signs of a(t), we will find the same solutions in theorem 4.3.3 (Allahviranloo and Chehlabi, 2015; Chehlabi and Allahviranloo, 2018). Example. Let us assume the linear first order differential equation: x0 ðtÞ ¼ sin t⊙xðtÞγ⊙ sin t, xð0Þ ¼ γ ¼ ð1, 0, 1Þ, t½0, 2π  In this example, a(t) ¼ sin t, which is nonnegative in t  [0, π] and it is nonpositive in the interval t  [π, 2π]. Then the i  solution can be obtained by the following functions with respect to the sign of a(t): ð t   ðt  ðu   xi,1 ðtÞ ¼ exp sin udu ⊙ γ sin u exp  sin vdv ⊙γdu 0

0

0

¼ ð2 exp ð1  cos tÞ  1Þ⊙γ, 0  t  π: Figs. 4.9 and 4.10 are the i-solutions of the problem.and:  ðt   ðt ð u   sin vdv ⊙γdu xi,2 ðtÞ ¼ exp  sin udu ⊙ γ sin u exp π

π

¼ ð2exp ð1 + cos tÞ  1Þ⊙γ, π  t  2π: is the i-solution xi, 2 in π  t  2π.

π

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Soft Numerical Computing in Uncertain Dynamic Systems

10

5

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

–5

–10

Fig. 4.9

The i-solution xi,

1

in 0  t  π.

15 10 5 0 3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6

5.8

6

6.2

–5 –10 –15

Fig. 4.10

the i-solution xi,

2

in π  t  2π.

To find the ii-solutions with assumption of the existence of the following H-differences: ð u  ðt sin vdv ⊙γdu γH ð1Þ⊙ sin u exp 0

0

and: γH ð1Þ⊙

ðt π

 ðu  sin u exp  sin vdv ⊙γdu 0

Continuous numerical solutions of uncertain differential equations 161 we have ii-solution with a(t) 0: ð t   ð u   ðt sin udu ⊙ γH ð1Þ⊙ sin u exp sin vdv ⊙γdu xii,1 ðtÞ ¼ exp 0

0

0

¼ ð2exp ð cos t  1Þ  1Þ⊙γ, 0  t  arccos ð1  ln 2Þ: We have ii-solution with a(t) < 0, ð t    ðu   ðt sin udu ⊙ γH ð1Þ⊙ sin u exp  sin vdv ⊙γdu xii,2 ðtÞ ¼ exp 0

0

0

¼ ð2 exp ð cos t  1Þ  1Þ⊙γ, π  t  2π  arccos ð ln 2  1Þ: These are shown in Figs. 4.11 and 4.12, respectively. Example. Now let us consider other cases of the previous example as the following forms: x0 ðtÞð1Þ sin t⊙xðtÞ ¼ γ⊙ sin t, xð0Þ ¼ γ ¼ ð1, 0, 1Þ,t½0, 2π  x0 ðtÞð1Þγ⊙ sin t ¼ sin t⊙xðtÞ, xð0Þ ¼ γ ¼ ð1, 0, 1Þ,t½0, 2π  Again, using the solution formula, we can find the solution in the form of: ð t    ðu   ðt sin udu γμ⊙ γ⊙ sin u exp  sin vdv du xðtÞ ¼ exp xðtÞ ¼ exp

ð t

0

0

0

   ðu   ðt sin udu γH μ⊙ γ⊙ sinu exp  sin vdv du

0

0

0

such that μ  {1, 1}. 1 0.8 0.6 0.4 0.2 0 0.1

0.2

0.3

0.4

0.5

0.6

–0.2 –0.4 –0.6 –0.8 1

Fig. 4.11

ii-solution xii, 1(t) with a(t) 0.

0.7

0.8

0.9

1

1.1

1.2

162

Soft Numerical Computing in Uncertain Dynamic Systems

1 0.8 0.6 0.4 0.2 0 3.2 –0.2

3.4

3.6

3.8

4

4.2

4.4

–0.4 –0.6 –0.8 1

Fig. 4.12

ii-solution xii, 1(t) with a(t) < 0.

Example. Consider the following equation: x0 ðtÞð1Þ⊙ð1, 2, 3Þt ¼ xðtÞ, xð0Þ ¼ ð2, 3, 4Þ Here a(t) ¼ 1 and: hðtÞ ¼ ð2, 3, 4Þ

ðt

 ðu  bðuÞ exp  dv du

0

¼ ð2, 3, 4Þ ¼ ð2, 3, 4Þ

ðt 0

ðt

0

ðu, 2u, 3uÞ⊙ exp ðuÞdu

0

ðu cosh ðuÞ  3u sinh ðuÞ, 2u cosh ðuÞ  2u sinh ðuÞ, 3u cosh ðuÞ u sinh ðuÞÞdu

and the ii-solution is in (0, ln2), xii,1 ðtÞ ¼ hðtÞ cosh tH ð1ÞhðtÞsinh t xi, 1 ðtÞ ¼ ð5 exp ðtÞ  2 exp ðtÞ  3t  1, 5exp ðtÞ  2t  2, 5 exp ðtÞ  2 exp ðtÞ  t  3Þ

Fig. 4.13 shows the ii-solution of the example.

4.3.5

FUZZY

DIFFERENTIAL EQUATIONS—LAPLACE TRANSFORM

In this section, we are going to use the Laplace transform to solve the fuzzy differential equations. It is clear that these transformations can be used for any order of differentiability of equations. First, we apply them for the first order and then we will use them for the high order of differentiability.

Continuous numerical solutions of uncertain differential equations 163

5

4

3

2

1 0.05 0.1

Fig. 4.13

0.15 0.2 0.25 0.3 0.35 0.4

0.45 0.5

0.55 0.6 0.65

ii-solution of the example.

Now consider the following differential equation with fuzzy initial value problem: x0gH ðtÞ ¼ f ðtxðtÞÞ, xð0Þ ¼ x0 R , t½0T , TR where the function f : R0 R ! R is a continuous fuzzy mapping. The strategy is using the Laplace transform for two sides of the differential equation and initial value. So:   L x0gH ðtÞ ¼ Lðf ðtxðtÞÞÞ In the solving procedure, we should consider the types of differentiability. So the cases are as follows: Case 1. Suppose that the solution function is i-differentiable so the derivative in level-wise form is:

x0igH ðt, r Þ ¼ x0l ðt, r Þ, x0u ðt, r Þ Also suppose that x and x0 are continuous fuzzy number valued on [0, ∞), then:   L x0gH ðtÞ ¼ s⊙LðxðtÞÞH xð0Þ then:

and:

     

l x0igH ðt, r Þ ¼ l x0l ðt, r Þ , l x0u ðt, r Þ   l x0u ðt, r Þ ¼ lð fu ðt, xðtÞ, r ÞÞ ¼ slðxu ðt, r ÞÞ  xu ð0, r Þ,   l x0l ðt, r Þ ¼ lð fl ðt, xðtÞ, r ÞÞ ¼ slðxl ðt, r ÞÞ  xl ð0, r Þ:

164

Soft Numerical Computing in Uncertain Dynamic Systems

By assuming the right-hand sides as new functions and using the inverse Laplace, the solutions can be found as:     l x0l ðt, r Þ ¼ H1 ðs, r Þ, l x0u ðt, r Þ ¼ K1 ðs, r Þ then: x0l ðt, r Þ ¼ l1 ðH1 ðs, r ÞÞ, x0u ðt, r Þ ¼ l1 ðK1 ðs, r ÞÞ Case 2. Suppose that the solution function is ii-differentiable so the derivative in level-wise form is:

x0iigH ðt, r Þ ¼ x0u ðt, r Þ, x0l ðt, r Þ Also suppose that x and x0 are continuous fuzzy number valued on [0, ∞), then: Lðx0 ðtÞÞ ¼ ð1Þ⊙xð0ÞH ðs⊙LðxðtÞÞÞ then:

     

l x0iigH ðt, r Þ ¼ l x0u ðt, r Þ , l x0l ðt, r Þ

and:   l x0u ðt, r Þ ¼ lð fl ðt, xðtÞ, r ÞÞ ¼ xu ð0, r Þ  ðslðxu ðt, r ÞÞÞ,   l x0l ðt, r Þ ¼ lð fu ðt, xðtÞ, r ÞÞ ¼ xl ð0, r Þ  ðslðxl ðt, r ÞÞÞ: By assuming the right-hand sides as new functions and using the inverse Laplace, the solutions can be found as:     l x0l ðt, r Þ ¼ H2 ðs, r Þ, l x0u ðt, r Þ ¼ K2 ðs, r Þ then: x0l ðt, r Þ ¼ l1 ðH2 ðs, r ÞÞ, x0u ðt, r Þ ¼ l1 ðK2 ðs, r ÞÞ Example. Consider the following fuzzy initial value problem: x0gH ðtÞ ¼ xðtÞ, xð0Þ ¼ x0 R , t½0T , TR Now:

  L x0gH ðtÞ ¼ LðxðtÞÞ,   ð∞ L x0gH ðtÞ ¼ est ⊙x0gH ðtÞdt 0

In the case of i-differentiability we have:   L x0gH ðtÞ ¼ s⊙LðxðtÞÞH xð0Þ

Continuous numerical solutions of uncertain differential equations 165 therefore: LðxðtÞÞ ¼ s⊙LðxðtÞÞH xð0Þ since: xðt, r Þ ¼ f ðt, xðtÞ, r Þ ¼ ½xu ðt, r Þ,  xl ðt, r Þ Then in level-wise form: lðxu ðt, r ÞÞ ¼ slðxl ðt, r ÞÞ  xl ð0, r Þ lðxl ðt, r ÞÞ ¼ slðxu ðt, r ÞÞ  xu ð0, r Þ or: lðxu ðt, r ÞÞ ¼ xl ð0, r Þ  slðxl ðt, r ÞÞ lðxl ðt, r ÞÞ ¼ xu ð0, r Þ  slðxu ðt, r ÞÞ Hence the solution can be found as:    s  1 ð 0, r Þ  x u s2  1 s2  1    s  1 lðxu ðt, r ÞÞ ¼ xu ð0, r Þ 2  xl ð0, r Þ 2 s 1 s 1 lðxl ðt, r ÞÞ ¼ xl ð0, r Þ

Thus:    s  1 1 xl ðt, r Þ ¼ xl ð0, r Þl  xu ð0, r Þl s2  1 s2  1    s  1 xu ðt, r Þ ¼ xu ð0, r Þl1 2  xl ð0, r Þl1 2 s 1 s 1 1

Finally: 

   xl ð0, r Þ + xu ð0, r Þ xl ð0, r Þ  xu ð0, r Þ + exp ðtÞ 2 2     xl ð0, r Þ + xu ð0, r Þ xu ð0, r Þ  xl ð0, r Þ + exp ðtÞ xu ðt, r Þ ¼ exp ðtÞ 2 2 xl ðt, r Þ ¼ exp ðtÞ

It should be noted that these solutions are the same as the solutions plotted in Figs. 4.5 and 4.6. In the case of ii-differentiability, we have:   L x0 gH ðtÞ ¼ ð1Þ⊙xð0ÞH ðs⊙LðxðtÞÞÞ therefore: LðxðtÞÞ ¼ ð1Þ⊙xð0ÞH ðs⊙LðxðtÞÞÞ

166

Soft Numerical Computing in Uncertain Dynamic Systems

since: xðt, r Þ ¼ f ðt, xðtÞ, r Þ ¼ ½xu ðt, r Þ,  xl ðt, r Þ Then in level-wise form: lðxl ðt, r ÞÞ ¼ slðxl ðt, r ÞÞ  xl ð0, r Þ lðxu ðt, r ÞÞ ¼ slðxu ðt, r ÞÞ  xu ð0, r Þ or: lðxl ðt, r ÞÞ ¼ xl ð0, r Þ  slðxl ðt, r ÞÞ lðxu ðt, r ÞÞ ¼ xu ð0, r Þ  slðxu ðt, r ÞÞ Hence the solution can be found as: 

 1 lðxl ðt, r ÞÞ ¼ xl ð0, r Þ 1+s   1 lðxu ðt, r ÞÞ ¼ xu ð0, r Þ 1+s Thus:  1 1+s   1 1 xu ðt, r Þ ¼ xu ð0, r Þl 1+s xl ðt, r Þ ¼ xl ð0, r Þl1



Finally: xl ðt, r Þ ¼ exp ðtÞxl ð0, r Þ xu ðt, r Þ ¼ exp ðtÞxu ð0, r Þ Now the solution can be displayed for any initial value (Salahshour and Allahviranloo, 2013; Tofigh Allahviranloo, 2010).

4.3.6

FUZZY

DIFFERENTIAL EQUATIONS—SECOND ORDER

In this section, we will discuss the fuzzy differential equations with order of two differentiability. The higher orders do have a similar procedure to discuss. It is apparent that in these equations we should explain the type of differentiability for any order of differentials separately. Now consider the following general form of second order fuzzy differential equations with fuzzy initial values:

Continuous numerical solutions of uncertain differential equations 167 8   00 > < xgH ðtÞ ¼ f t, xðtÞ, x0gH ðtÞ xðt Þ ¼ x0 R > : gH00 x ðt0 Þ ¼ x1 R where the function f : ½0, T  R R ! R is a fuzzy number valued and continuous function. Here the functions x, x0 , x00 are also fuzzy number valued and continuous ones. It is clear that in accordance with the type of differentiability we will have several differential equations. The function x can be in two forms of differentiability and also x0 , for more illustration, in the level-wise form for 0  r  1. Case I. Both functions x, x0 are i  gH differentiable:



x0igH ðt, r Þ ¼ x0l ðt, r Þ, x0u ðt, r Þ , x00igH ðt, r Þ ¼ x00l ðt, r Þ, x00u ðt, r Þ   8 > < x00igH ðt, r Þ ¼ f t, xðtÞ, x0igH ðtÞ, r xðt0 , r Þ ¼ x0 ðr Þ > : 0 xigH ðt0 , r Þ ¼ x1 ðr Þ and: 8   > < x00l ðt, r Þ ¼ fl t, xðtÞ, x0igH ðtÞ, r x ðt , r Þ ¼ xl,0 > : l0 0 xl ðt0 , r Þ ¼ xl,1

8   > < x00u ðt, r Þ ¼ fu t, xðtÞ, x0igH ðtÞ, r x ðt , r Þ ¼ xu,0 > : u0 0 xu ðt0 , r Þ ¼ xu,1

where:  

fl t, xðtÞ, x0igH ðtÞ, r ¼ min f ðt, u, u0 Þj u½xl ðt, r Þ, xu ðt, r Þ, u0  x0l ðt, r Þ, x0u ðt, r Þ  

fu t, xðtÞ, x0igH ðtÞ, r ¼ max f ðt, u, u0 Þj u½xl ðt, r Þ, xu ðt, r Þ, u0  x0l ðt, r Þ, x0u ðt, r Þ Case II. The function x is i  gH differentiable and x0 is ii  gH differentiable:



x0igH ðt, r Þ ¼ x0l ðt, r Þ, x0u ðt, r Þ , x00iigH ðt, r Þ ¼ x00u ðt, r Þ, x00l ðt, r Þ   8 > < x00iigH ðt, r Þ ¼ f t, xðtÞ, x0igH ðtÞ, r xðt0 , r Þ ¼ x0 ðr Þ > : 0 xigH ðt0 , r Þ ¼ x1 ðr Þ and: 8   > < x00u ðt, r Þ ¼ fl t, xðtÞ, x0igH ðtÞ, r x ðt , r Þ ¼ xl,0 > : l0 0 xl ðt0 , r Þ ¼ xl,1

8   > < x00l ðt, r Þ ¼ fu t, xðtÞ, x0igH ðtÞ, r x ðt , r Þ ¼ xu,0 > : u0 0 xu ðt0 , r Þ ¼ xu,1

168

Soft Numerical Computing in Uncertain Dynamic Systems

Case III. The function x is ii  gH differentiable and x0 is i  gH differentiable:



x0iigH ðt, r Þ ¼ x0u ðt, r Þ, x0l ðt, r Þ , x00igH ðt, r Þ ¼ x00l ðt, r Þ, x00u ðt, r Þ   8 > < x00igH ðt, r Þ ¼ f t, xðtÞ, x0iigH ðtÞ, r xðt0 , r Þ ¼ x0 ðr Þ > : 0 xiigH ðt0 , r Þ ¼ x1 ðr Þ and: 8   > < x00l ðt, r Þ ¼ fl t, xðtÞ, x0iigH ðtÞ, r x ðt , r Þ ¼ xl,0 > : l0 0 xu ðt0 , r Þ ¼ xu,1

8   > < x00u ðt, r Þ ¼ fu t, xðtÞ, x0iigH ðtÞ, r x ðt , r Þ ¼ xu,0 > : u0 0 xl ðt0 , r Þ ¼ xl,1

Case IV. Both functions x, x0 are ii  gH differentiable:



x0iigH ðt, r Þ ¼ x0u ðt, r Þ, x0l ðt, r Þ , x00iigH ðt, r Þ ¼ x00u ðt, r Þ, x00l ðt, r Þ   8 > < x00iigH ðt, r Þ ¼ f t, xðtÞ, x0iigH ðtÞ, r xðt0 , r Þ ¼ x0 ðr Þ > : 0 xiigH ðt0 , r Þ ¼ x1 ðr Þ and: 8   > < x00u ðt, r Þ ¼ fl t, xðtÞ, x0iigH ðtÞ, r x ðt , r Þ ¼ xl,0 > : l0 0 xu ðt0 , r Þ ¼ xl,1

8   > < x00l ðt, r Þ ¼ fu t, xðtÞ, x0iigH ðtÞ, r x ðt , r Þ ¼ xu,0 > : u0 0 xl ðt0 , r Þ ¼ xu,1

Now based on the characteristic theorem, the equations in each of the five cases are equivalent to:   8 00 > < xgH ðtÞ ¼ f t, xðtÞ, x0gH ðtÞ xðt0 Þ ¼ x0 R > : 0 xgH ðt0 Þ ¼ x1 R As we know, the problems in these cases can be solved by any method, and here we are going to use the Laplace transform to solve them. Example. Consider the following second order fuzzy differential equation: 8 00 < xgH ðtrÞ ¼ σ 0 ðr Þ, σ 0 ðr Þ ¼ ½r  1 1  r  xð0r Þ ¼ σ 0 ðr Þ : 0 xgH ð0r Þ ¼ σ 0 ðr Þ Case I. Both functions x, x0 are i  gH differentiable:  00  σ0 L xgH ðtÞ ¼ s2 ⊙LðxðtÞÞH s⊙xð0ÞH xð0Þ ¼ Lðσ 0 Þ ¼ s

Continuous numerical solutions of uncertain differential equations 169 and: s2 ⊙LðxðtÞÞH s⊙σ 0 H σ 0 ¼

σ0 s

in the level-wise form: s2 lðxl ðt, r ÞÞ  sσ l,0 ðr Þ  σ l,0 ðr Þ ¼

σ l,0 ðr Þ s

s2 lðxu ðt, r ÞÞ  sσ u,0 ðr Þ  σ u,0 ðr Þ ¼

σ u,0 ðr Þ s

Finally, after using inverse Laplace: 

 s2 +s+1 2 2  s xu ðt, r Þ ¼ σ u,0 ðr Þ +s+1 2 xl ðt, r Þ ¼ σ l,0 ðr Þ

where σ l, 0(r) ¼ r  1 and σ u, 0(r) ¼ 1  r. Case II. The function x is i  gH differentiable and x0 is ii  gH differentiable. Then by taking the Laplace transform and:     L x00gH ðtÞ ¼ ð1Þ⊙x0gH ð0ÞH s2 ⊙LðxðtÞÞð1Þs⊙xð0Þ     σ0 L x00gH ðtÞ ¼ ð1Þ⊙σ 0 H s2 ⊙LðxðtÞÞð1Þs⊙σ 0 ¼ s in the level-wise form we will have: σ u,0 ðr Þ + s2 lðxu ðt, r ÞÞ  sσ u, 0 ðr Þ ¼ σ l,0 ðr Þ + s2 lðxl ðt, r ÞÞ  sσ l,0 ðr Þ ¼

σ l,0 ðr Þ s

σ u,0 ðr Þ s

Finally, after using inverse Laplace, the solutions are defined on (0, 1) as follows:  2  s xl ðt, r Þ ¼ σ l,0 ðr Þ  + s + 1 2  2  s xu ðt, r Þ ¼ σ u,0 ðr Þ  + s + 1 2 Case III. The function x is ii  gH differentiable and x0 is i  gH differentiable:     L x00gH ðtÞ ¼ s⊙σ 0 H s2 ⊙LðxðtÞÞH σ 0   σ0 s⊙σ 0 H s2 ⊙LðxðtÞÞH σ 0 ¼ s

170

Soft Numerical Computing in Uncertain Dynamic Systems

In the level-wise form we will have: sσ u, 0 ðr Þ + s2 lðxu ðt, r ÞÞ  σ l, 0 ðr Þ ¼

σ l,0 ðr Þ s

sσ l, 0 ðr Þ + s2 lðxl ðt, r ÞÞ  σ u, 0 ðr Þ ¼

σ u,0 ðr Þ s

 pffiffiffi  Finally, after using inverse Laplace, the solutions are defined on 0, 3  1 as follows:  2  s xl ðt, r Þ ¼ σ l,0 ðr Þ   s + 1 2  2  s xu ðt, r Þ ¼ σ u,0 ðr Þ   s + 1 2 Case IV. Both functions x, x0 are ii  gH differentiable:   L x00gH ðtÞ ¼ s2 ⊙ LðxðtÞÞ H s ⊙ xð0Þ  ð1Þ ⊙ x0gH ð0Þ and we have: s2 ⊙ LðxðtÞÞ H s ⊙ σ 0  ð1Þ ⊙ σ 0 ¼

σ0 s

In the level-wise form we will have: s2 lðxl ðt, r ÞÞ  sσ l,0 ðr Þ  σ u,0 ðr Þ ¼

σ l,0 ðr Þ s

s2 lðxu ðt, r ÞÞ  sσ u,0 ðr Þ  σ l,0 ðr Þ ¼

σ u,0 ðr Þ s

Finally, after using inverse Laplace, the solutions are defined on (0, 1) as follows: 2  s s+1 xl ðt, r Þ ¼ σ l,0 ðr Þ 2 2  s s+1 xu ðt, r Þ ¼ σ u,0 ðr Þ 2 Example. Consider the following second order fuzzy differential equation: 8 00 < xgH ðtrÞxðtÞ ¼ σ 0 ðr Þ, σ 0 ðr Þ ¼ ½r2  r  xð0r Þ ¼ ½r  1 1  r  : x0 ð0r Þ ¼ ½r  1 1  r  gH

Continuous numerical solutions of uncertain differential equations 171 Case I. Both functions x, x0 are i  gH differentiable. Applying the Laplace transform, we get: σ0 s2 ⊙LðxðtÞÞH s⊙xð0ÞH x0gH ð0Þ + LðxðtÞÞ ¼ s Similar to the previous example, the solutions in level-wise form are: xl ðt, r Þ ¼ r ð1 + sin tÞ  sin t  cos t xu ðt, r Þ ¼ ð2  r Þð1 + sin tÞ  sin t  cos t The solution x in level-wise form is a fuzzy number valued function and is shown in Fig. 4.14. The solution x in level-wise form is not i  gH differentiable and is shown in Fig. 4.15. This is because: x0l ðt, r Þ ¼ ðr  1Þ cos t + sin t x0u ðt, r Þ ¼ ð1  r Þcos t + sin t and xl0 (t, r) ≰ xu0 (t, r) in all points of the domain and, as we see in the figure, there is a switching point at which the type of differentiability is changed. Based on our previous discussions and definitions, this type of differentiability is called ii  gH differentiability. So in general, there is no solution for this case. Case II. The function x is i  gH differentiable and x0 is ii  gH differentiable. Then by taking the Laplace transform and:   σ0 ð1Þ⊙σ 0 H s2 ⊙LðxðtÞÞð1Þs⊙σ 0 LðxðtÞÞ ¼ s

4 3 2 1

–7

–6

–5

–4

–3

–2

–1

0

0 –1 –2

Fig. 4.14

The i-solution x.

1

2

3

4

172

Soft Numerical Computing in Uncertain Dynamic Systems

1.4 1.2 1 0.8 0.6 0.4 0.2

0

0

Fig. 4.15

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

ii  gH differentiability of the solution.

Similar to the previous case, the solution in level-wise form is: xl ðt, r Þ ¼ r ð1 + sinh tÞ  sinh t  cos t xu ðt, r Þ ¼ ð2  r Þð1 + sinh tÞ  sinh t  cos t In this case, x is i  gH differentiable. Fig. 4.16 shows us the behavior of x0 . It is clear that the lower and upper functions continue regularly without interchanging. In addition, the x0 is ii  gH differentiable because x00 satisfies the definition of type 2 differentiability. Therefore, the solutions are acceptable. Case III. The function x is ii  gH differentiable and x0 is i  gH differentiable. Then by taking the Laplace transform and:   σ0 ð1Þ⊙σ 0 H s2 ⊙LðxðtÞÞH σ 0 LðxðtÞÞ ¼ s

2

1

–3

–2

–1

0

0

1

–1

–2

Fig. 4.16

i  gH differentiability of the solution: x0 .

2

3

4

Continuous numerical solutions of uncertain differential equations 173 similar to the previous case, the solution in level-wise form is: xl ðt, r Þ ¼ r ð1  sinh tÞ + sinh t  cos t xu ðt, r Þ ¼ ð2  r Þð1  sinh tÞ + sinh t  cos t wIfweconsiderthederivativesofthesolution,itisii gHdifferentiableandinlevel-wise formthereisinterchangingoftheendpointsoftheintervalanditlookslikeFig.4.16,butthelower andupperfunctionsarechanged.Inaddition,ithappensforthex00 ,andthismeansx0 isii  gH differentiabletoo.Therefore,thesolutionisnotacceptable. Case IV. Both functions x, x0 are ii  gH differentiable. Then by taking the Laplace transform and: ð1Þ⊙σ 0 s2 ⊙LðxðtÞÞH s⊙σ 0 LðxðtÞÞ ¼

σ0 s

similar to the previous case, the solution in level-wise form is: xl ðt, r Þ ¼ r ð1  sin tÞ + sin t  cos t xu ðt, r Þ ¼ ð2  r Þð1  sin tÞ + sin t  cos t

  If we consider the derivatives of the solution, it is ii  gH differentiable in 0, π2 and in level-wise form there is interchanging of the end points of the interval and it looks like Fig. 4.15, but the lower and upper functions are changed. In addition, it happens for the x00 , and this means x0 is ii  gH differentiable too and looks like Fig. 4.17. Therefore, the solution is not acceptable at any point of the domain (Salahshour and Allahviranloo, 2013).

4.3.7

FUZZY

DIFFERENTIAL EQUATIONS—VARIATIONAL ITERATION METHOD

The aim of this section is introducing the fuzzy variational iteration method to solve the fuzzy differential equations. This method is one of the semianalytical methods and approximates the fuzzy solution functionally with many conditions. Indeed it is

1.5

1

0.5

0 –2

Fig. 4.17

–1.5

1

–0.5

0

ii  gH differentiability of the x0 .

0.5

1

1.5

2

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Soft Numerical Computing in Uncertain Dynamic Systems

conditional and all the derivatives should exist. After discussion on the method, the convergence and error are also investigated. Here a nonlinear fuzzy differential equation is defined as follows in general form (Allahviranloo and Abbasbandy, 2014): LxðtÞNxðtÞgH gðtÞ ¼ 0, t  0 m

where L is a linear operator and is denoted by L ¼ dtd m and N is a nonlinear operator and g is a known fuzzy function. The fuzzy initial values are: xðkÞ ð0Þ ¼ ck R , k ¼ 0,1, …, m  1 subject to all the derivatives existing, i.e., being fuzzy numbers. In addition, the zero number on the right-hand side is a singleton fuzzy number. We know the solution of any fuzzy differential equation is the solution of a corresponding fuzzy integral equation, such as the following form: Case 1. ðt   xn + 1 ðtÞ ¼ xn ðtÞ λðτÞ⊙ LxðτÞNxðτÞgH gðτÞ dτ, n  0 0

if

dm dtm xðtÞ

is i  gH differentiable, and: Case 2. ðt   xn + 1 ðtÞ ¼ xn ðtÞH ð1Þ⊙ λðτÞ⊙ LxðτÞNxðτÞgH gðτÞ dτ, n  0 0

m

is ii  gH differentiable. In both equations the multiplier λ is a general Lagrange multiplier, which can be identified optimally via variational theory and considering limited variations for the function x, it means δx(t) ¼ 0. So after obtaining the optimal multiplier λ, using the initial value x0, the successive approximations xn(t), n 1 will be obtained as a sequence. It is clear that we should show that the sequence does have terms converging to the solution of the fuzzy differential equation: if

d dtm xðtÞ

lim xn ðtÞ ¼ xðtÞ

n!∞

On the other hand, to have a solution with restricted or limited variations we need to denote that the variations of the nonlinear term are also limited. So, we have to consider that δNx(t) ¼ 0. By this condition, the new successive iteration method with limited variations in two cases are: Case 1. ðt   δxn + 1 ðtÞ ¼ δxn ðtÞδ λðτÞ⊙ LxðτÞgH gðτÞ dτ, n  0 0

if

dm dtm xðtÞ

is i  gH differentiable, and:

Continuous numerical solutions of uncertain differential equations 175 Case 2. δxn + 1 ðtÞ ¼ δxn ðtÞH ð1Þ⊙δ

ðt



  λðτÞ⊙ LxðτÞgH gðτÞ dτ, n  0

0 m

if dtd m xðtÞ is ii  gH differentiable. Since the function g is a known one and independent of x, so the variations on it are meaningless. Then, two cases are simplified to the following cases: Case 1. ðt δxn + 1 ðtÞ ¼ δxn ðtÞδ fλðτÞ⊙Lxn ðτÞgdτ, n  0 0

if

dm dtm xðtÞ

is i  gH differentiable, and: Case 2. δxn + 1 ðtÞ ¼ δxn ðtÞH ð1Þ⊙δ

ðt

fλðτÞ⊙Lxn ðτÞgdτ, n  0

0 m

if dtd m xðtÞ is ii  gH differentiable. It should be noted that the multiplier λ depends on τ and even t, and it will be found ultimately based on these variables. As we know, the linear operator is m L ¼ dtd m and by replacing, we will have: Case 1.  ðt  dm δxn + 1 ðtÞ ¼ δxn ðtÞδ λðτÞ⊙ m xn ðτÞ dτ, n  0 dt 0 m

if dtd m xðtÞ is i  gH differentiable, and: Case 2. δxn + 1 ðtÞ ¼ δxn ðtÞH ð1Þ⊙δ

ðt  0

if

 dm λðτÞ⊙ m xn ðτÞ dτ, n  0 dt

dm dtm xðtÞ

is ii  gH differentiable. In two cases, using the part-by-part integration (Chapter 3):   ðt  ðt  dm d m1 dm1 λðτÞ⊙ m xn ðτÞ dτ ¼ λðtÞ⊙ m1 xn ðtÞgH λ0 ðτÞ⊙ m1 xn ðτÞ dτ dt dt dt 0 0

we have: Case 1.    ðt  d m1 dm1 0 δxn + 1 ðtÞ ¼ δxn ðtÞδ λðtÞ⊙ m1 xn ðtÞgH λ ðτÞ⊙ m1 xn ðτÞ dτ dt dt 0 if all derivatives are i  gH differentiable, and:

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Soft Numerical Computing in Uncertain Dynamic Systems

Case 2.    ðt  d m1 dm1 λ0 ðτÞ⊙ m1 xn ðτÞ dτ δxn + 1 ðtÞ ¼ δxn ðtÞH ð1Þδ λðtÞ⊙ m1 xn ðtÞgH dt dt 0 if all derivatives are ii  gH differentiable. Continuing by this integration again:   ðt  ðt  d m1 d m2 dm2 0 0 00 λ ðτÞ⊙ m1 xn ðτÞ dτ ¼ λ ðtÞ⊙ m2 xn ðtÞgH λ ðτÞ⊙ m2 xn ðτÞ dτ dt dt dt 0 0 and finally by considering that: δxn + 1 ðtÞ ¼ 0, and all the gH-differences exist in two cases, it is concluded that the following successive iteration method is found: Case 1.   ðt dm1 d m2 0 ¼ δxn ðtÞδ λ⊙ m1 xn ðtÞλ0 ⊙ m2 xn ðtÞ⋯λðm1Þ xn ðtÞ  λðmÞ ⊙δxn ðτÞdτ dt dt 0 if all derivatives are i  gH differentiable, and: Case 2.   ðt d m1 d m2 0 ¼ δxn ðtÞH ð1Þδ λ⊙ m1 xn ðtÞλ0 ⊙ m2 xn ðtÞ⋯λðm1Þ xn ðtÞ  λðmÞ ⊙δxn ðτÞdτ dt dt 0

if all derivatives are ii  gH differentiable. In two cases, by comparing two sides of the equalities: Case 1. 1 + λðm1Þ ¼ 0, λðmÞ ¼ 0, λ ¼ λ0 ¼ ⋯ ¼ λðm1Þ ¼ 0 Case 2. Note. In this case, the multiplier λ C 1 + λðm1Þ ¼ 0, λðmÞ ¼ 0, λ ¼ λ0 ¼ ⋯ ¼ λðm1Þ ¼ 0 otherwise the abovementioned relations are not true easily. Because in level-wise form of case (2), the relations are:  m1  d dm2 0 ¼ δxn, l ðtÞ + δ λ m1 xn, u ðtÞ + λ0 m2 xn, u ðtÞ + ⋯ + λðm1Þ xn, u ðtÞ dt dt ðt + λðmÞ δxn, u ðτÞdτ 0

Continuous numerical solutions of uncertain differential equations 177  m1  ðt d dm2 0 ¼ δxn, u ðtÞ + δ λ m1 xn, l ðtÞ + λ0 m2 xn, l ðtÞ + ⋯ + λðm1Þ xn, l ðtÞ + λðmÞ δxn, l ðτÞdτ dt dt 0 then there is only one way to have this quality, and it is xn, l(t) ¼ xn, u(t) ¼ 0 and: λ ¼ λ0 ¼ ⋯ ¼ λðm1Þ ¼ λðmÞ ¼ 0 Now we discuss only case (1). In this case, by considering the relations, the multiplier can be found as: λðt, τÞ ¼

ð1Þm ðτ  tÞm1 , 0 < t < τ < T: ðm  1Þ!

So our successive iterates relation is:  ðt    ð1Þm ðτ  tÞm1 ⊙ Lxn ðτÞNxn ðτÞgH gðτÞ dτ xn + 1 ðtÞ ¼ xn ðtÞ 0 ðm  1Þ! for n 0. Now define an operator like A as follows:  ðt   ð1Þm m1  ðτ  tÞ ⊙ LxðτÞNxðτÞgH gðτÞ dτ A½ x  ¼ 0 ðm  1Þ! Then another sequence with components vn is defined: v0 ¼ x0 , v1 ¼ A½x0 , …, vn + 1 ¼ A½v0 v1 ⋯vn  where: v0 ðt Þ ¼ c 0 

m X ck ⊙tk k¼1

vn + 1 ðtÞ ¼

ðt  0

k!



ð1Þm dm ½v0 v1 ⋯vn ðτÞN ½v0 v1 ⋯vn gH gðτÞ ðτ  tÞm1 ⊙ ðm  1Þ! dtm

 dτ

Finally: xðtÞ ¼ lim xn ðtÞ ¼ n!∞

∞ X n¼0

vn , lim DH xn ðtÞ, n!∞

∞ X n¼0

! vn ¼ lim DH ðxn ðtÞ, xðtÞÞ ¼ 0 n!∞

Existence and convergence. Now the convergence of the method should be con∞ P vn ðtÞ such that the sidered. To explain it, we have to show that the series solution n¼0

terms are obtained from: vn + 1 ðtÞ ¼

ðt  0

 m  ð1Þm d ½ v v ⋯v  ð τ ÞN ½ v v ⋯v  g ð τ Þ dτ ðτ  tÞm1 ⊙ n n gH 0 1 0 1 ðm  1Þ! dtm

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Soft Numerical Computing in Uncertain Dynamic Systems

and satisfies the following inequality: DH ðvn + 1 , 0Þ  γDH ðvn , 0Þ, 0 < 9 γ < 1 converging to the exact solution x(t). To show the assertion, first we define the sequence {sn}∞ n¼0 as: s0 ¼ v0 , s1 ¼ v0 v1 ,…, sn ¼ v0 v1 ⋯vn DH ðsn + 1 , sn Þ ¼ DH ðvn + 1 , 0Þ  γDH ðvn , 0Þ  γ 2 DH ðvn1 , 0Þ  ⋯  γ n + 1 DH ðv0 , 0Þ For any j  n  N, we have:     DH sn , sj  DH ðsn , sn1 Þ + DH ðsn1 , sn2 Þ + ⋯ + DH sj + 1 , sj  γ n Dðv0 , 0Þ + γ n1 DH ðv0 , 0Þ + ⋯ + γ j + 1 DH ðv0 , 0Þ ¼ Since 0 < γ < 1, we get:

1  γ nj j + 1 γ DH ðv0 , 0Þ 1γ

  lim DH sn , sj ¼ 0

n, j!∞

Therefore, the sequence {sn}∞ n¼0 is a Cauchy sequence in the normed space. In the same way, we can show that: ! j X 1  γ n1 j + 1 1 j+1 γ D H ð v 0 , 0Þ  DH xðtÞ, γ D H ð v 0 , 0Þ vk  1γ 1γ k¼0 Now the proof is completed. Example. Consider the following fuzzy Duffing’s differential equation: x00 ðtÞ3⊙xðtÞH 2⊙ðxðtÞÞ3 ¼ cos t sin 2t, t  0 xð0, r Þ ¼ ½r  1, 1  r , x0 ð0, r Þ ¼ ½r, 2  r , r½0, 1 Now the initial approximate guesses: v0 ðt, r Þ ¼ ðr  1 + rt, 1  r + ð2  r ÞtÞ The successive iterated relation for m ¼ 2 is formed as: ðt n  o ðτ  tÞ⊙ x00 ðtÞ3⊙xðtÞH 2⊙ðxðtÞÞ3 H cos t sin 2t dτ xn + 1 ðtÞ ¼ xn ðtÞ 0

In order to find it, we have to use the Maclaurin series expansions of sint and cost: cos t ¼ 1 

t2 t24 t6 4t3 4t5 +  , cos t ¼ 2t  + 2 24 720 3 15

Continuous numerical solutions of uncertain differential equations 179 If the H-differences exist, after replacing the Maclaurin series expansions in the abovementioned iterated relation, the solution can be approximated (Allahviranloo and Abbasbandy, 2014).

4.3.8

FUZZY

DIFFERENTIAL EQUATIONS—LEGENDRE DIFFERENTIAL EQUATION

In this section, first the fuzzy generalized power series method, in which the coefficients are fuzzy numbers, is introduced and then the conditions of the uniqueness of the solution and its convergence for the fuzzy differential equation are investigated. Then, using the fuzzy generalized power series method, the fuzzy Legendre differential equation is considered as a case study, and finally, for further illustration, some related examples are solved. For more discussion on the subject, we need to introduce some properties of one of the types of ranking of fuzzy numbers and fuzzy sequences that we are going to use in this section. The ranking that we use is defined as level-wise form. Level-wise ranking. For instance, consider two fuzzy numbers A,BR , if we say A ≼ ()B it means Al(r)  ( : Al ðr ÞBl ðr Þ  Au ðr ÞBu ðr Þ ) Cl ðr Þ ¼ , C u ðr Þ ¼ Bu ð r Þ Bl ðr Þ 2. If 0 < Al(r)  Au(r) and 0 < Bl(r)  Bu(r) then: 8 Al ð r Þ Au ðr Þ > > , Cu ð r Þ ¼ < Al ðr ÞBu ðr Þ  Au ðr ÞBl ðr Þ ) Cl ðr Þ ¼ Bl ð r Þ Bu ðr Þ Au ð r Þ A l ðr Þ > > : Al ðr ÞBu ðr Þ  Au ðr ÞBu ðr Þ ) Cl ðr Þ ¼ , C u ðr Þ ¼ Bu ð r Þ B l ðr Þ 3. If Al(r)  Au(r) < 0 and Bl(r)  Bu(r) < 0 then: 8 Au ð r Þ Al ðr Þ > > , Cu ð r Þ ¼ < Au ðr ÞBl ðr Þ  Al ðr ÞBu ðr Þ ) Cl ðr Þ ¼ Bu ð r Þ Bl ðr Þ A ð r Þ A > l u ðr Þ > Au ðr ÞBl ðr Þ  Al ðr ÞBu ðr Þ ) Cl ðr Þ ¼ : , Cu ð r Þ ¼ Bl ð r Þ Bu ðr Þ 4. If Al(r)  Au(r) < 0 and 0 < Bl(r)  Bu(r) then: 8 Al ð r Þ A u ðr Þ > > , C u ðr Þ ¼ < Al ðr ÞBl ðr Þ  Au ðr ÞBu ðr Þ ) Cl ðr Þ ¼ Bu ð r Þ Bl ðr Þ A ð r Þ Al ðr Þ > u > Al ðr ÞBl ðr Þ  Au ðr ÞBu ðr Þ ) Cl ðr Þ ¼ : , C u ðr Þ ¼ Bl ð r Þ B u ðr Þ If Al(r)  0, Au(r) 0 and Bl(r)  Bu(r) < 0 then: Cl ð r Þ ¼

Au ðr Þ Al ð r Þ , Cu ðr Þ ¼ : B l ðr Þ Bl ð r Þ

If Al(r)  0, Au(r) 0 and 0 < Bl(r)  Bu(r) then: C l ðr Þ ¼

Al ðr Þ Au ð r Þ , C u ðr Þ ¼ : B u ðr Þ Bu ð r Þ

Continuous numerical solutions of uncertain differential equations 181

4.3.8.1 Definition—Power series with fuzzy coefficients The following series with fuzzy coefficients and fuzzy operations is called the fuzzy power series: xðtÞ ¼

∞ X

an ⊙ðt  t0 Þn , 8tR, 9t0 R

n¼0

where the radius of convergence ρ is defined as ρ ¼ lim n!∞ aan +n ð10ð0Þ Þ provided that the limit exists and is a fuzzy number, and also the division exists for two fuzzy numbers an(0), an+1(0). Note. We assume here that ρ > 0 and x(t) and its derivatives are defined in jx  x0 j < ρ. The level-wise form of the series is: xðt, r Þ ¼ ½xl ðt, r Þ, xu ðt, r Þ, where:

Therefore, considering the above level-wise functions, the form of the series with fuzzy operators should be as the following form: X X xðtÞ ¼ an ⊙ðt  t0 Þn H ð1Þ⊙ an ⊙ðt  t0 Þn , t < t0 n:even

n:odd

If we want to consider the gH-derivative of the fuzzy series, we need to define the derivatives of the xl(t, r) and xu(t, r):

182

Soft Numerical Computing in Uncertain Dynamic Systems

and now:





x0gH ðt, r Þ ¼ min x0l ðt, r Þ, x0u ðt, r Þ , max x0l ðt, r Þ, x0u ðt, r Þ

and: x0gH ðtÞ ¼

∞ X

n⊙an ⊙ðt  t0 Þn1

n¼1 ð nÞ

x ð t0 Þ

In the fuzzy series f(x) the coefficient is fined by an ¼ gHn! subject to the nth derivatives existing, which means all of them are fuzzy numbers. Now based on the nature of the gH-differentiability, two types of differentiability appear here again: ∞ X x0igH ðtÞ ¼ n⊙an ⊙ðt  t0 Þn1 n¼1

x0iigH ðtÞ ¼ H ð1Þ

∞ X

n⊙an ⊙ðx  x0 Þn1

n¼1

And the level-wise form is:

x0igH ðt, r Þ ¼ x0l ðt, r Þ, x0u ðt, r Þ

x0iigH ðt, r Þ ¼ x0u ðt, r Þ, x0l ðt, r Þ

4.3.8.2 Some properties of fuzzy series Fuzzy sequence. Let u : Z + ! R (Z+ means set of positive integer numbers) is a fuzzy number valued function. u ≔ {un} is a fuzzy sequence if for all n  N, un R . Bounded sequence. Suppose that kR and k 0: 1. k is a fuzzy lower bound of the fuzzy sequence {un} if for all n  N [ {0}, k ≼ un. 2. k is a fuzzy upper bound of the fuzzy sequence {un} if for all n  N [ {0}, k ≽ un. Convergence. An infinite fuzzy series

∞ P

un with positive terms is convergent if and

n¼1

only if its fuzzy sequence of partial sums has a fuzzy upper bound. ∞ ∞ P P un and vn are two fuzzy series with Fuzzy comparison test. Suppose that n¼1

positive terms such that for all n  N, un ≽ vn then:

n¼1

Continuous numerical solutions of uncertain differential equations 183 P P∞ 1. If the fuzzy series ∞ n¼1 un is convergent, then the fuzzy series n¼1 vn is also convergent. P P∞ 2. If the fuzzy series ∞ n¼1 un is divergent, then the fuzzy series n¼1 vn is also divergent. Two of the properties can be proved easily by the definition of convergence.

4.3.8.3 Fuzzy calculated operations Let us suppose that: x ðt Þ ¼

∞ X

an ⊙ðt  t0 Þn , yðtÞ ¼

n¼0

∞ X

bn ⊙ðt  t0 Þn

n¼0

such that f(x) is a fuzzy series and g(x) is a real one and for both the radius of convergence is positive. We know that: xðtÞ ¼ xðtÞ ¼

X

∞ X

an ⊙ðt  t0 Þn , t  t0

n¼0

an ⊙ðt  t0 Þn H ð1Þ⊙

n:even

X

an ⊙ðt  t0 Þn , t < t0

n:odd

Here, four fuzzy calculated operations of these two series are going to be explained. The sign of coefficients and also x  x0 are considered necessary to obtain a fuzzy number valued series as a result. Summation. •

If t t0: xðtÞyðtÞ ¼

∞ X

ðan bn Þ⊙ðt  t0 Þn

n¼0

•

If t < t0: xðtÞyðtÞ ¼

∞ X

ðan bn Þ⊙ðt  t0 Þn H ð1Þ

n:even

⊙

X

ðan bn Þ⊙ðt  t0 Þn

n:odd

Production. •

If t t0: xðtÞ⊙yðtÞ ¼

∞ X

n X

ðak ⊙bnk Þ⊙ðt  t0 Þn H ð1Þ

n:even k¼0, bnk 0

⊙

∞ X

n X

n:odd k¼0, bnk 0

ðak ⊙bnk Þ⊙ðt  t0 Þn

184 •

Soft Numerical Computing in Uncertain Dynamic Systems

If t < t0: xðtÞ⊙tðtÞ ¼

∞ X

n X

ðak ⊙bnk Þ⊙ðt  t0 Þn H ð1Þ

n:even k¼0, bnk 0

⊙

∞ X

n X

ðak ⊙bnk Þ⊙ðt  t0 Þn

n:odd k¼0, bnk 0

H ð1Þ

n X

∞ X

ðak ⊙bnk Þ⊙ðt  t0 Þn

n:even k¼0, bnk gH gH > < xðt0 Þ ¼ x0 R > > > : 0 xgH ðt0 Þ ¼ x1 R To find the fuzzy power series solution we use the following procedure step by step. 1. The model has the fuzzy power series solution in the form of: xðtÞ ¼

∞ X

an ⊙ðt  t0 Þn

n¼0

2. As we mentioned before, we obtain the x0 (t), x00 (t). 3. Fuzzy arithmetic operations such as production, summation, and difference are calculated according to the previous discussions. 4. The fuzzy series solution and its derivatives as obtained series are replaced in the fuzzy differential equation. 5. We claimed that: ∞ X

an ⊙ðt  t0 Þn ¼ 0 , 8n an ¼ 0

n¼0

and also the fuzzy coefficients are determined. It is confirmed that they are fuzzy numbers. 6. The fuzzy series solutions are now found.

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Soft Numerical Computing in Uncertain Dynamic Systems

Now let us consider the fuzzy Legendre’s equation as follows:   1  t2 ⊙x00gH ðtÞ  2t⊙x0gH ðtÞφðφ + 1Þ⊙xðtÞ ¼ 0 subject to: xð0Þ ¼ x0 R , x0gH ð0Þ ¼ x1 R , φN [ f0g φðφ + 1Þ 2t Since the functions 1t 2 and 1t2 are analytical functions at x0 ¼ 0, then the fuzzy series solution is a unique solution in the form of:

xðtÞ ¼

∞ X

an ⊙ðt  t0 Þn

n¼0

To solve the problem, first we consider that x, x0 are i  gH differentiable. In this case, we have:

and now:



x0gH ðt, r Þ ¼ min x0l ðt, r Þ, x0u ðt, r Þ , max x0l ðt, r Þ, x0u ðt, r Þ and: x0gH ðtÞ ¼ 

∞ X

n⊙an ⊙ðt  t0 Þn1

n¼0

In this case, the coefficients are: a2 ¼  H

φðφ + 1Þ ð φ + 2 Þ ð φ  1Þ ⊙a0 , a3 ¼ H ⊙a1 2 2

and: an + 2 ¼ H

ðφ  nÞðn + φ + 1Þ ⊙a1 , n ¼ 2, 3,… ð n + 2 Þ ð n + 1Þ

Continuous numerical solutions of uncertain differential equations 187 Finally, the solution is:   φðφ + 1Þ 2 ðφ + 2Þφðφ + 1Þðφ + 3Þ 4 xðtÞ ¼ a0 ⊙ 1H x  x H ⋯  2! 4!   ð φ  1 Þ ð φ + 2Þ 3 ð φ  3 Þ ð φ  1 Þ ð φ + 2Þ ð φ + 4 Þ 5 x x H ⋯ a1 ⊙ xH 3! 5! Example. Consider the following fuzzy second order differential equation with the fuzzy series centered about 0: x00gH ðtÞgH 3⊙x0gH ðtÞ2⊙xðtÞ ¼ 0, xð0Þ ¼ ð2, 4, 7Þ,x0 ð0Þ ¼ ð3, 6, 10:5Þ The fuzzy series solution is: ∞ X

xðtÞ ¼

an ⊙tn

n¼0 0

Suppose that x, x are i  gH differentiable. So: x0igH ðtÞ ¼

∞ X

nan ⊙tn1 , x00igH ðtÞ ¼

n¼1

∞ X

nðn  1Þan ⊙ tn2

n¼2

By substituting in the equation, we have: xð0Þ ¼ a0 ¼ ð2, 4, 7Þ, x0igH ð0Þ ¼ a1 ¼ ð3, 6, 10:5Þ ∞ X

nðn  1Þan ⊙tn2 gH 3⊙

n¼2

∞ X

nan ⊙tn1 2⊙

n¼1

∞ X

an ⊙tn ¼ 0

n¼0

and we know that: ∞ X

nðn  1Þan ⊙tn2 ¼

n¼2

∞ X

ðn + 1Þðn + 2Þan + 2 ⊙tn

n¼0 ∞ X n¼1

nan ⊙tn1 ¼

∞ X

ðn + 1Þan + 1 ⊙tn

n¼0

Finally: ∞  X

 ðn + 1Þðn + 2Þan + 2 gH 3⊙ðn + 1Þan + 1 2⊙an ⊙tn ¼ 0

n¼0

Then we obtain: an + 2 ¼

3ðn + 1Þ⊙an + 1 H 2⊙an , n ¼ 0,1, … ð n + 1Þ ð n + 2Þ

188

Soft Numerical Computing in Uncertain Dynamic Systems

The fuzzy series solution is:   14 4 30 5 2 3 xðtÞ ¼ ð2, 4, 7Þ⊙ 1H x H x H x H x H ⋯  24 120   3 15 31 5 x ⋯ ð3, 6, 10:5Þ⊙ x x3  x4  2 24 120 Example. Consider the following fuzzy second order differential equation with the fuzzy series centered about 0: x00gH ðtÞ5⊙x0gH ðtÞ6⊙xðtÞ ¼ 0, xð0Þ ¼ ð1, 3, 4Þ, x0gH ð0Þ ¼ ð10,  7:5,  2:5Þ The fuzzy series solution is: x ðt Þ ¼

∞ X

an ⊙tn

n¼0 0

Suppose that x, x are i  gH differentiable. So: x0iigH ðtÞ ¼ H ð1Þ

∞ X

nan ⊙tn1 ,

n¼1

x00iigH ðtÞ ¼ H ð1Þ

∞ X

nðn  1Þan ⊙tn2

n¼2

Using the same procedure, we have the solution in the form of:   6 30 114 4 xðtÞ ¼ ð1, 3, 4Þ⊙ 1H x2  x3  x H ⋯  2 6 24   5 19 5 ð10,  7:5,  2:5Þ⊙ H ð1ÞxH x2 H ð1Þ x3 H x4 ⋯ 2 6 24

4.3.9

LINEAR

SYSTEMS OF FUZZY DIFFERENTIAL EQUATIONS

In this section, we will discuss the fuzzy linear system of differential equations of the form: x01gH ðtÞ ¼ a11 ðtÞ⊙x1 ðtÞ⋯a1n ðtÞ⊙xn ðtÞf1 ðtÞ x02gH ðtÞ ¼ a21 ðtÞ⊙x1 ðtÞ⋯a2n ðtÞ⊙xn ðtÞf2 ðtÞ ⋮ x0ngH ðtÞ ¼ an1 ðtÞ⊙x1 ðtÞ⋯ann ðtÞ⊙xn ðtÞfn ðtÞ

Continuous numerical solutions of uncertain differential equations 189 such that the functions aij(t), i, j ¼ 1, …, n are real functions and fi(t), i ¼ 1, …, n are fuzzy number valued functions. In the compact form of the system, we have: x0gH ðtÞ ¼ Aðt Þ⊙xðt Þf ðt Þ,xðt 0 Þ ¼ x0 where: 0

x1 ðtÞ

1

0

a11 ðtÞ⋯a1n ðtÞ

1

0

f1 ðtÞ

1

B C B C B C xðt Þ ¼ @ ⋮ A, Aðt Þ ¼ @ ⋮ ⋱ ⋮ A, f ðt Þ ¼ @ ⋮ A xn ðtÞ

an1 ðtÞ⋯ann ðtÞ

fn ðtÞ

such that f(t) is a fuzzy vector number. Indeed, this system is called a nonhomogeneous fuzzy system and in the case that the vector f(t) 5 0, then it is called a homogeneous fuzzy system. In addition, if the coefficient matrix A(t) is a real constant matrix A, then the system is called a fuzzy linear system of differential equations with constant coefficients. It is clear that each equation of this system needs the initial values. The fuzzy initial values are as: x1 ðt0 Þ ¼ A1 , x2 ðt0 Þ ¼ A2 , …, xn ðt0 Þ ¼ An where: A1 ,A2 , …,An R Our discussions will be illustrated in two parts, the first for the homogeneous and the second for nonhomogeneous systems.

4.3.9.1 Homogeneous fuzzy linear differential systems Now consider the following special case of the system with constant coefficient matrix: x0gH ðtÞ ¼ A⊙xðt Þ where: 0

a11 ⋯a1n

1

B C A ¼ @⋮ ⋱ ⋮A an1 ⋯ann with the fuzzy initial values: 0 0 1 0 1 x1 ðtÞ x1,0 x0gH ðtÞ ¼ @ ⋮ A, xð0Þ5@ ⋮ A, xi,0 R , i ¼ 1,…, n x0n ðtÞ xn,0

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It is clear that the solution vector is: xðt Þ5eAt x0 where: 1 1 0 a t a t 10 e 11 ⋯e 1n x1,0 x1 ðt Þ @ ⋮ A ¼ @ ⋮ ⋱ ⋮ A@ ⋮ A ean1 t ⋯eann t xn,0 x n ðt Þ 0

Clearly we can check that this satisfies the system and initial condition. Then it is a solution, and for the uniqueness, suppose that the vector function y(t) is an arbitrary solution of the system. The following vector function can be defined by y(t): Zðt Þ ¼ eAt yðt Þ, Z0gH ðt Þ ¼ eAt y0gH ðt Þð1ÞAeAt yðt Þ ¼ 0 So the vector fuzzy number valued function Z(t) is a constant one and at the initial point is Z(0) 5y(0) 5x0, hence Z(t) 5x0 there for any arbitrary solution y(t) is given by y(t) 5x(t) 5eAt x0. Example—Bimathematic two-compartment model A two-compartment fuzzy model for drug absorption and circulation through the gastrointestinal tract and blood has been formulated in the beginning. In Fig. 4.18, the first compartment corresponds to the gastrointestinal tract and after that, the drug diffuses into the second compartment, blood. Let c1(t) and c2(t) denote the fuzzy valued of concentration of drug in stomach or gastrointestinal tract and blood stream compartments, respectively. If c0 is the fuzzy initial concentration of drug dosage, then the general fuzzy model describing the rate of change in oral drug administration is given as:  0 c1gH ðtÞ ¼ ð1Þk1 ⊙c1 ðtÞ, c 1 ð 0Þ ¼ c 0 c02gH ðtÞ ¼ k1 ⊙c1 ðtÞð1Þc2 ðtÞ, c2 ð0Þ ¼ 0 where c0 R and k1, ke are positive real constants such that they show the rate constant from one compartment to another and the clearance constant. The coefficient

Oral administraction c0 unit of drug

k1 Stomach c1 (t)

Stomach c2 (t)

ke

Fig. 4.18

Drug administration through the stomach and blood.

Continuous numerical solutions of uncertain differential equations 191 matrix is: 

k1 0 A¼ k1 ke



and the compact form of the system is:   0    c1gH k1 0 c 1 ðt Þ ¼ c02gH k1 ke c 2 ðt Þ So the solution can be computed as: cðt Þ ¼ eAt c0    k t   c1 ðt Þ c0 e 1 0 ¼ k1 t ke t 0 c 2 ðt Þ e e then: c1 ðtÞ ¼ c0 ⊙ek1 t , c2 ðtÞ ¼

 c0 ⊙k1  ke t e  ek1 t k1  ke

Example—Biomathematic three-compartment model The blood flow in the cardiovascular system is one directional, so the drug administration through venous blood can be shown by the following pattern (see Fig. 4.19). Consider that the amount of drug flow toward tissue by arterial blood is at the rate of kb and from tissue compartment to the venous blood at the rate ofkt. Moreover, the clearance rate of drug from the blood is ke. Consider c1(t), c2(t), and c3(t) denote the concentration of drug in the arterial blood, tissue, and venous blood compartment, respectively. If c0 is the fuzzy initial value of drug, then the mathematical model of the drug concentration with respect to this compartment is: 8 0 c 1 ð 0Þ ¼ c 0 > < c1gH ðtÞ ¼ ð1Þkb ⊙c1 ðtÞ, c02gH ðtÞ ¼ kb ⊙c1 ðtÞð1Þkt ⊙c2 ðtÞ, c2 ð0Þ ¼ 0 > : c0 ðtÞ ¼ kt ⊙c2 ðtÞð1Þke ⊙c3 ðtÞ, c2 ð0Þ ¼ 0 3gH

Intravenous infusion C0 units of drug

kt

kb Arterial blood C1 (t)

Tissue C2 (t)

Venous blood C3 (t) ke

Elimination

Fig. 4.19

Drug administration through arterial blood and tissue and venous blood.

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Soft Numerical Computing in Uncertain Dynamic Systems

The matrix and the vector solution are: 0 1 0 1 kb 0 c1 ðt Þ 0 A ¼ @ kb kt 0 A, cðt Þ ¼ @ c2 ðtÞ A 0 kt ke c 3 ðt Þ Then, using the same procedure, the solution is: c1 ðtÞ ¼ c0 ⊙ekb t c2 ðtÞ ¼ c 3 ðt Þ ¼ 

 c0 ⊙kb t  kt t e  ekb t kb  kt

c0 ⊙kb kt eðkb + ke + kt Þt ðkb  ke Þðkb  kt Þðke  kt Þ

kb eðkb + ke Þt  kb eðkb + kt Þt  ke eðkb + ke Þt + ke eðke + kt Þt + kt eðkb + ke Þt  kt eðke + kt Þt



4.3.9.2 Nonhomogeneous fuzzy linear differential systems In this section, the following nonhomogeneous fuzzy system of first order linear fuzzy differential equation with real constant coefficients will be solved: x0gH ðtÞ ¼ Aðt Þ⊙xðt Þf ðtÞ, xðt 0 Þ ¼ x0 nR Considering the definition of gH-difference, this equation is equivalent to the following equation: x0gH ðtÞgH Aðt Þ⊙xðt Þ ¼ f ðt Þ Multiplying both sides of the equation by e At 0, we have: eAt ⊙x0gH ðtÞgH eAt Aðt Þ⊙xðt Þ ¼ eAt ⊙f ðt Þ Based on the gH-derivative of composite of two fuzzy functions, the left-hand side can be considered in the form of:  At 0 e ⊙xðt Þ gH ¼ ð1ÞeAt Aðt Þ⊙xðt ÞeAt ⊙x0gH ðtÞ since 0iigHU ¼ (1)U. So:  At 0 e ⊙xðtÞ gH ¼ gH eAt Aðt Þ⊙xðt ÞeAt ⊙x0gH ðtÞ Finally, we have: 

eAt ⊙xðt Þ

0 gH

¼ eAt ⊙f ðt Þ

Continuous numerical solutions of uncertain differential equations 193 As we discussed before, based on two types of differentiability we have two types of solution as well: •

If x(t) is i-differentiable: xðt Þ ¼ eAðtt0 Þ ⊙xðt0 Þ

ðt

eAðtuÞ f ðuÞdu

t0

•

If x(t) is ii-differentiable: xðtÞ ¼ eAðtt0 Þ ⊙xðt0 ÞH ð1Þ

ðt

eAðtuÞ f ðuÞdu

t0

Example—Biomathematic two-compartment model In this model, just two compartments—the tissues and bloodstream—were identified. The mathematical model to describe this is governed by the system of two differential equations such that each equation describes the rate of change of drug concentration with respect to time (see Fig. 4.20). We assume that a drug is ingested at a given rate u such that u is a fuzzy function. Consider that c1(t) and c2(t) denote the concentration of drug in the compartments, the bloodstream and the tissue, respectively:  0 c1gH ðtÞkb ⊙c1 ðtÞ ¼ uðtÞ, c 1 ð 0Þ ¼ 0 0 c2gH ðtÞkt ⊙c2 ðtÞ ¼ kb ⊙c1 ðtÞ, c2 ð0Þ ¼ 0 where kb is the rate constants from the bloodstream. In the following case, we illustrate the use of the fuzzy Laplace transform for solving the metabolism model. The drug lidocaine is used in the treatment of irregular heartbeat. Now assume that u(t) ¼ (1.9,2, 2.2) ⊙ δ(t) mg of lidocaine is injected into the bloodstream and then move into the tissue of the heart, where δ(t) is Kronecker delta function. If c1(t) describes the amount of lidocaine in the bloodstream and c2(t) evaluates the amount of drug in the tissue, then:

u

kb Blood c1 (t)

Tissue c2 (t) kt

Fig. 4.20

Drug administration through blood and tissue.

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Soft Numerical Computing in Uncertain Dynamic Systems 

c01gH ðtÞc1 ðtÞ ¼ ð1:9; 2, 2:2Þ⊙δðtÞ, c02gH ðtÞc2 ðtÞ ¼ c1 ðtÞ,

c1 ð0Þ ¼ 0 c 2 ð 0Þ ¼ 0

As we stated in Chapter 3, taking the Laplace transform of both sides of the equations in the system, we have:  8  < L c01 ðtÞ Lðc1 ðtÞÞ ¼ ð1:9; 2, 2:2Þ⊙LðδðtÞÞ,  gH  : L c0 ðtÞ Lðc2 ðtÞÞ ¼ Lðc1 ðtÞÞ, 2gH Let us consider that: Lðc1 ðtÞÞ ¼ C1 ðsÞ, Lðc2 ðtÞÞ ¼ C2 ðsÞ   L c01gH ðtÞ ¼ s⊙Lðc1 ðtÞÞgH xð0Þ ¼ s⊙C1 ðsÞ   L c02gH ðtÞ ¼ s⊙Lðc2 ðtÞÞgH xð0Þ ¼ s⊙C2 ðsÞ so the systems is as follows:  s⊙C1 ðsÞC1 ðsÞ ¼ ð1:9; 2, 2:2Þ, s⊙C2 ðsÞC2 ðsÞ ¼ C1 ðsÞ, The solutions are: C1 ðsÞ ¼

ð1:9; 2, 2:2Þ ð1:9; 2, 2:2Þ , C 1 ðsÞ ¼ s+1 ð s + 1Þ 2

Fig. 4.21

The solution c1(t).

1.0

1.5

2.0

Now, using the inverse Laplace, the solution of original fuzzy systems is found as follows (Fig. 4.21):

Continuous numerical solutions of uncertain differential equations 195 c1 ðtÞ ¼ L1 c2 ðtÞ ¼ L1



   ð1:9; 2, 2:2Þ 1 ¼ ð1:9; 2, 2:2Þ⊙L1 ¼ ð1:9; 2, 2:2Þ⊙et s+1 s+1 ! ! ð1:9; 2, 2:2Þ 1 1 ¼ ð1:9; 2, 2:2Þ⊙L ¼ ð1:9; 2, 2:2Þ⊙tet ðs + 1Þ2 ð s + 1Þ 2

4.3.9.3 Reduction of a second order fuzzy differential equations to a system of first order equations The second order linear differential equation: x00gH ðtÞa1 ðtÞ⊙x0gH ðtÞa2 ðtÞ⊙xðtÞ ¼ f ðtÞ, with the initial values: xðt0 Þ ¼ A1 , x0gH ðt0 Þ ¼ A2 where the initial values are two fuzzy numbers. To reduce the second order differential equation to the system of first order differential equation, we assume that: xðtÞ ¼ y1 ðtÞ, x0gH ðtÞ ¼ y2 ðtÞ so: y01gH ðtÞ ¼ x0gH ðtÞ ¼ y2 ðtÞ y02gH ðtÞ ¼ x00gH ðtÞ ¼ f ðtÞgH a1 ðtÞ⊙x0gH ðtÞgH a2 ðtÞ⊙xðtÞ By substituting the relations in the last one: y02gH ðtÞ ¼ f ðtÞgH a1 ðtÞ⊙y2 ðtÞgH a2 ðtÞ⊙y1 ðtÞ We then arrive at the following system of fuzzy first order differential equations:  0 y1 ðt0 Þ ¼ A1 y1gH ðtÞ ¼ y2 ðtÞ, 0 y2gH ðtÞ ¼ f ðtÞgH a1 ðtÞ⊙y2 ðtÞgH a2 ðtÞ⊙y1 ðtÞ, y2 ðt0 Þ ¼ A2 Note that the types of differentiability of x(t) and y1(t) and also xgH0 (t) and y2(t) are the same. Example—Fuzzy forced harmonic oscillator problem Consider the following fuzzy second order differential equation that is denoted as a forced harmonic oscillator problem: x00gH ðtÞxðtÞ ¼ f ðtÞ, xð0Þ ¼ x0 R , x0gH ð0Þ ¼ x1 R

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Soft Numerical Computing in Uncertain Dynamic Systems

By using the same procedure, it is concluded that:  0 y 1 ðt 0 Þ ¼ x0 y1gH ðtÞ ¼ y2 ðtÞ, 0 y2gH ðtÞ ¼ f ðtÞgH y1 ðtÞ, y2 ðt0 Þ ¼ x1 in the matrix form:  A¼

     0 1 0 x , bðtÞ ¼ , y ð 0Þ ¼ 0 x1 1 0 f ðt Þ

and: At

e ¼



   cos t sin t cos t  sin t At , e ¼  sin t cos t sin t cos t

The i-differentiable solution is:   ðt At At Aτ e bðτÞdτ yðtÞ ¼ e x0 e 0



In another form:       ðt    y1 ðtÞ cos t sin t x0 cos t sin t cos τ  sin τ 0 ¼  dτ  sin t cos t x1  sin t cos t 0 sin τ cos τ y2 ðtÞ f ðτ Þ

and: y1 ðtÞ ¼ xðtÞ ¼ x0 cos tx1 sin t

ðt

f ðτÞ sin ðt  τÞdτ

0

So the fuzzy i-differentiable solution is: xðtÞ ¼ x0 cos tx1 sin t

ðt

f ðτÞsin ðt  τÞdτ

0

and the fuzzy ii-differentiable solution is: xðtÞ ¼ x0 ⊙ cos tx1 ⊙ sin tH ð1Þ

ðt

f ðτÞ⊙ sin ðt  τÞdτ

0

As a case, consider that: x00gH ðtÞxðtÞ ¼ ð1, 5, 15Þe2t ð2; 10; 30Þe3t , xð0Þ ¼ ð0:4; 2, 6Þ,x0gH ð0Þ ¼ ð1, 5, 15Þ the i-solution: xðtÞ ¼ ð0:4; 2, 6Þ⊙ cos tð1, 5, 15Þ⊙ sin t

ðt 0



 ð1, 5, 15Þe2τ ð2; 10; 30Þe3τ ⊙ sin ðt  τÞdτ

Continuous numerical solutions of uncertain differential equations 197 The final solution is obtained by fuzzy calculus: xðtÞ ¼ ð0:2; 1, 3Þe2t ð0:2; 1, 3Þe3t

4.4 Z-differential equations In Chapter 2, we discussed the Z-numbers and calculations on these uncertain numbers. In the real world, most phenomena are based on doubt, and the information that we have from various subjects such as economics, politics, and physics is evaluated according to verbal values. Here, we try to formulate and investigate the mentioned information to the initial value problem while our initial data are Z-numbers. Now, in this section, first the differential operator will be introduced on the Z-number valued functions under gH-differentiability, then the differential equation with Z-number initial value will be introduced and discussed. To this purpose, we should first define a Z-process or a Z-valued function. Definition—Z-process A function x is called a Z-valued function if for any t  Dx the value x(t) is a Z-number. Definition—Continuity A Z-valued function is continuous if and only if its first part is continuous. Indeed, if x ¼ (xA, xN) is a Z-process or Z-valued then the continuity of x is equivalent to continuity of xA at the point t0 and it is: 8E > 09δ > 08t ðjt  t0 j < δ ) DH ðxðtÞ, xðt0 ÞÞ < EÞ Note. In this definition, we assume that the probability part is always continuous. We shall now define the derivative of a Z-process under gH-differentiability and then its corresponding differential equations. Note. As we mentioned in Chapter 2, the special case of a Z-number occurs when the probability part is also a fuzzy number. In other words, it can be approximated by a trapezoidal or triangular fuzzy number. In this case, two parts of the Z-process are fuzzy numbers. Example Assume that X ¼ R is the set of real numbers, A is close to 10, and N is quite sure. Then the Z-number can be defined as: Z ¼ ðclose to 10, quite sureÞ The Z-valued function or process can be defined as x(t) ¼ Zt. The membership function of A also can be defined as:

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Soft Numerical Computing in Uncertain Dynamic Systems

1 A (t)

N (t) 6

Fig. 4.22

7

8

9

10

11

12

13

14

Z-number Z ¼ (close to 10, quite sure).

 1 AðtÞ ¼ 1 + ðt  10Þ2 and the membership function for N: ! 1 ðt  μÞ2 N ðtÞ ¼ pffiffiffiffiffi exp  , μ ¼ 10, σ ¼ 1 2σ 2 σ 2π See Fig. 4.22. The level-wise form of this Z-number is: Z ½r  ¼ ðA, N Þ½r  ¼ ½A½r , N ½r , 0  r  1, where: A½r  ¼ ½Al ðr Þ, Au ðr Þ ¼ ½5 + 5r, 15  5r , N ½r  ¼ ½Nl ðr Þ, Nu ðr Þ ¼ ½6 + 4r, 14  4r : Moreover, it is clear both parts are continuous and immediately the continuity of the Z-process x(t) ¼ Zt can be concluded. Definition—gH-differentiability of Z-process A Z-process x is gH-differentiable on [a, b] if for all t  [a, b]:   x0gH ðtÞ ¼ x0AgH ðtÞ, x0NgH ðtÞ 

xA ðt + hÞgH xA ðtÞ xN ðt + hÞgH xN ðtÞ , lim ¼ lim h!0 h!0 h h



Note. The necessary and sufficient conditions for the gH-differentiability of a Z-process are both parts gH-differentiable. Note. The Z-process x is i  gH differentiable if xA and xN both are i  gH differentiable:   x0igH ðtÞ ¼ x0AigH ðtÞ, x0NigH ðtÞ

Continuous numerical solutions of uncertain differential equations 199 The level-wise form: x0igH ðtÞ½r  ¼

 0



 xl, A ðt, r Þ, x0u, A ðt, r Þ , x0l, N ðt, r Þ, x0u, N ðt, r Þ

Note. The Z-process x is ii  gH differentiable if xA and xN both are ii  gH differentiable:   x0iigH ðtÞ ¼ x0AiigH ðtÞ, x0NiigH ðtÞ The level-wise form: x0iigH ðtÞ½r  ¼





 x0u, A ðt, r Þ, x0l, A ðt, r Þ , x0u,N ðt, r Þ, x0l, N ðt, r Þ

Z-number initial value problem Consider the following differential equation with Z-number initial value: x0gH ðtÞ ¼ f ðt, xðtÞÞ, xðt0 Þ ¼ x0 , t½t0 , T , T > 0 where x0 is a Z-number and x0 ¼ (x0, A, x0, N) then the solution is also a Z-process, and it can be considered in the following form: xðtÞ ¼ ðxA ðtÞ, xN ðtÞÞ Here, we also have two cases without considering the switching points. Case 1. The Z-process x is i  gH differentiable, then: 



 x0igH ðtÞ½r  ¼ x0l, A ðt, r Þ, x0u, A ðt, r Þ , x0l, N ðt, r Þ, x0u, N ðt, r Þ ¼ f ðt, xðtÞÞ½r where: f ðt, xðtÞÞ½r  ¼ ððfA ðt, xA ðtÞ, xN ðtÞÞÞ½r , ðfN ðt, xA ðtÞ, xN ðtÞÞÞ½r Þ½r, ðfA ðt, xA ðtÞ, xN ðtÞÞÞ½r  ¼ ½fl, A ðt, xA ðtÞ, xN ðtÞ, r Þ, fu, A ðt, xA ðtÞ, xN ðtÞ, r Þ, ðfN ðt, xA ðtÞ, xN ðtÞÞÞ½r  ¼ ½fl, N ðt, xA ðtÞ, xN ðtÞ, r Þ, fu, N ðt, xA ðtÞ, xN ðtÞ, r Þ: It is concluded that: 0

xl, A ðt, r Þ, x0u, A ðt, r Þ ¼ ½fl,A ðt, xA ðtÞ, xN ðtÞ, r Þ, fu, A ðt, xA ðtÞ, xN ðtÞ, r Þ, 0

xl, N ðt, r Þ, x0u, N ðt, r Þ ¼ ½fl,N ðt, xA ðtÞ, xN ðtÞ, r Þ, fu, N ðt, xA ðtÞ, xN ðtÞ, r Þ: Finally: 8 0 xl, A ðt, r Þ ¼ fl, A ðt, xA ðtÞ, xN ðtÞ, r Þ > > > 0 > x > u, A ðt, r Þ ¼ fu, A ðt, xA ðtÞ, xN ðtÞ, r Þ > <  x0 ðt, r Þ ¼ f ðt, x ðtÞ, x ðtÞ, r Þ l, N

l, N

A

N

x0u, N ðt, r Þ ¼ fu, N ðt, xA ðtÞ, xN ðtÞ, r Þ > >  > > > xl, A ðt0 , r Þ ¼ xl,0,A ðr Þ,xl, N ðt0 , r Þ ¼ xl,0,N ðr Þ > : xu, A ðt0 , r Þ ¼ xu,0,A ðr Þ,xu, N ðt0 , r Þ ¼ xu, 0,N ðr Þ

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Soft Numerical Computing in Uncertain Dynamic Systems

Case 2. The Z-process x is ii  gH differentiable, then: 8 0 x A ðt, r Þ ¼ fl, A ðt, xA ðtÞ, xN ðtÞ, r Þ > > u, > > x0l, A ðt, r Þ ¼ fu, A ðt, xA ðtÞ, xN ðtÞ, r Þ > > > <  x0 ðt, r Þ ¼ f ðt, x ðtÞ, x ðtÞ, r Þ l, N A N u, N > x0l, N ðt, r Þ ¼ fu, N ðt, xA ðtÞ, xN ðtÞ, r Þ > >  > > > xl,A ðt0 , r Þ ¼ xl,0, A ðr Þ, xl,N ðt0 , r Þ ¼ xl,0,N ðr Þ > : xu, A ðt0 , r Þ ¼ xu,0, A ðr Þ, xu, N ðt0 , r Þ ¼ xu,0,N ðr Þ where: fl, A ðt, xA ðtÞ, xN ðtÞ, r Þ ¼ min ffA ðt, u, vÞj uxA ðtÞ½r , vxN ðtÞ½r g fu, A ðt, xA ðtÞ, xN ðtÞ, r Þ ¼ max ffA ðt, u, vÞj uxA ðtÞ½r , vxN ðtÞ½r g fl, N ðt, xA ðtÞ, xN ðtÞ, r Þ ¼ min ffN ðt, u, vÞj uxA ðtÞ½r , vxN ðtÞ½r g fu, N ðt, xA ðtÞ, xN ðtÞ, r Þ ¼ max ffN ðt, u, vÞj uxA ðtÞ½r , vxN ðtÞ½rg Therefore, in those cases the Z-number initial value problem is transformed to the system of real differential equations and the characteristic theorem is necessary to find the solution of the original Z-valued differential equations. Characteristic theorem If the Z-process f is a continuous and gH-differentiable that satisfies the following fuzzy differential equation: x0gH ðtÞ ¼ f ðt, xðtÞÞ, xðt0 Þ ¼ x0 , t½t0 , T , T > 0 and also suppose the following conditions: • • •

fA(t, x(t), r) ¼ [fl, A(t, xA(t), xN(t), r), fu, A(t, xA(t), xN(t), r)] fN(t, x(t), r) ¼ [fl, N(t, xA(t), xN(t), r), fu, N(t, xA(t), xN(t), r)] fl, A(t, xl(t, r), xu(t, r)), fu, A(t, xl(t, r), xu(t, r)), fl, N(t, xA(t), xN(t), r) and fu, N(t, xA(t), xN(t), r) are equicontinuous. It means, for any E > 0 and any (t, u, v)  [t0, T] R2 if k(t, u, v)  (t, u1, v1) k < δ we have the following inequalities for 8 r  [0, 1]: j fl, A ðt, xl ðt, r Þ, xu ðt, r ÞÞ  fl, A ðt, xl ðt, r Þ, xu ðt, r ÞÞj < E j fl, A ðt, xl ðt, r Þ, xu ðt, r ÞÞ  fu, A ðt, xl ðt, r Þ, xu ðt, r ÞÞj < E j fu, A ðt, xl ðt, r Þ, xu ðt, r ÞÞ  fl, A ðt, xl ðt, r Þ, xu ðt, r ÞÞj < E j fu, A ðt, xl ðt, r Þ, xu ðt, r ÞÞ  fu, A ðt, xl ðt, r Þ, xu ðt, r ÞÞj < E j fl, N ðt, xl ðt, r Þ, xu ðt, r ÞÞ  fl, N ðt, xl ðt, r Þ, xu ðt, r ÞÞj < E j fl, N ðt, xl ðt, r Þ, xu ðt, r ÞÞ  fu, N ðt, xl ðt, r Þ, xu ðt, r ÞÞj < E

Continuous numerical solutions of uncertain differential equations 201 j fu, N ðt, xl ðt, r Þ, xu ðt, r ÞÞ  fl, N ðt, xl ðt, r Þ, xu ðt, r ÞÞj < E j fu, N ðt, xl ðt, r Þ, xu ðt, r ÞÞ  fu, N ðt, xl ðt, r Þ, xu ðt, r ÞÞj < E

• •

fl, A(t, xl(t, r), xu(t, r)), fu, A(t, xl(t, r), xu(t, r)), fl, N(t, xA(t), xN(t), r) and fu, N(t, xA(t), xN(t), r) are uniformly bounded on any bounded set. Lipschitz property. There exists L > 0 such that: j fl, A ðt, u1 , v1 , r Þ  fl, A ðt, u2 , v2 , r Þj < L max fju1  u2 j, jv1  v2 jg j fl, A ðt, u1 , v1 , r Þ  fu, A ðt, u2 , v2 , r Þj < L max fju1  u2 j, jv1  v2 jg j fu, A ðt, u1 , v1 , r Þ  fl, A ðt, u2 , v2 , r Þj < L max fju1  u2 j, jv1  v2 jg j fu, A ðt, u1 , v1 , r Þ  fu, A ðt, u2 , v2 , r Þj < L max fju1  u2 j, jv1  v2 jg j fl, N ðt, u1 , v1 , r Þ  fl, N ðt, u2 , v2 , r Þj < Lmax fju1  u2 j, jv1  v2 jg j fl, N ðt, u1 , v1 , r Þ  fu, N ðt, u2 , v2 , r Þj < L max fju1  u2 j, jv1  v2 jg j fu, N ðt, u1 , v1 , r Þ  fl, N ðt, u2 , v2 , r Þj < L max fju1  u2 j, jv1  v2 jg j fu, N ðt, u1 , v1 , r Þ  fu, N ðt, u2 , v2 , r Þj < Lmax fju1  u2 j, jv1  v2 jg

for any r  [0, 1]. Then the Z-differential equation is equivalent to the each of following real differential equations in the cone: 8 0 xl, A ðt, r Þ ¼ fl, A ðt, xA ðtÞ, xN ðtÞ, r Þ > > > > x0u, A ðt, r Þ ¼ fu, A ðt, xA ðtÞ, xN ðtÞ, r Þ > > <  x0 ðt, r Þ ¼ f ðt, x ðtÞ, x ðtÞ, r Þ l, N

l, N

A

N

x0 ðt, r Þ ¼ fu, N ðt, xA ðtÞ, xN ðtÞ, r Þ > >  u, N > > > x ðt , r Þ ¼ xl,0,A ðr Þ,xl, N ðt0 , r Þ ¼ xl,0,N ðr Þ > : l, A 0 xu, A ðt0 , r Þ ¼ xu,0,A ðr Þ,xu, N ðt0 , r Þ ¼ xu, 0,N ðr Þ

or 8 0 xu, A ðt, r Þ ¼ fl, A ðt, xA ðtÞ, xN ðtÞ, r Þ > > > x0 ðt, r Þ ¼ f ðt, x ðtÞ, x ðtÞ, r Þ > u, A A N > l, A > <  x0 ðt, r Þ ¼ f ðt, x ðtÞ, x ðtÞ, r Þ l, N A N u, N 0 ð t, r Þ ¼ f ð t, x ð t Þ, x x > u, N A N ðtÞ, r Þ l, N >  > > > xl, A ðt0 , r Þ ¼ xl,0,A ðr Þ,xl, N ðt0 , r Þ ¼ xl,0,N ðr Þ > : xu, A ðt0 , r Þ ¼ xu,0,A ðr Þ,xu, N ðt0 , r Þ ¼ xu, 0,N ðr Þ The proof is very similar to the same theorem that has been proved in this chapter previously.

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Remark. An immediate conclusion of the theorem is, the solution of the Z-differential equation is unique (Pirmuhammadi et al., 2017). Example—Population biology Differential equations whose solutions involve exponential growth or decay were discussed earlier. Consider the differential equation: x0gH ðtÞ ¼ λ⊙xðtÞ, xðt0 Þ ¼ x0 Clearly, the population does have the following statuses: 8 < stationary or xðtÞ ¼ xðt0 Þ when λ ¼ 0 population≔ growth or xðtÞ ! ∞ when λ > 0 : decay or xðtÞ ! 0 when λ < 0 First, suppose that λ ¼ 1 and we have the growth model. The initial value is defined as a Z-number on the set of “the births in some country last month” and it is: x0 ¼ ðconsiderably close to 20; 000, not sureÞ such that: x0,A ¼ considerably close to 20; 000, x0, B ¼ not sure with the membership functions (19,20,21): 8 0, t < 19 > > > > < t  1, 19  t < 20 x0, A ðtÞ ¼ 1, t ¼ 20 > > > 1  t, 20 < t < 21 > : 0, t > 21 and x0, N(t) presents the normal probability density function: ! 1 ðt  μÞ2 N ðtÞ ¼ pffiffiffiffiffi exp  , μ ¼ 20, σ ¼ 2 2σ 2 σ 2π It is very easy to show that:   xðtÞ ¼ ðxA ðtÞ, xN ðtÞÞ ¼ x0,A ⊙eðtt0 Þ , x0, N ⊙eðtt0 Þ x0,A ðt, r Þ ¼ ½19 + r, 21  r , x0,N ðt, r Þ ¼ ½4 + 16r, 36  16r  Now decay the model; assume λ ¼  1. The initial value is defined as a Z-number on the set of “the deaths in some country last month” and it is: x0 ¼ ðconsiderably close to 2 million, not sureÞ

Continuous numerical solutions of uncertain differential equations 203 such that: x0, A ¼ considerably close to 2 million,x0, B ¼ not sure With the same membership function (19,20,21), and x0, N(t) presents the same normal probability density function. It is also very easy to show that: xl, A ðt, r Þ ¼

xl,0, A ðr Þ  xu,0,A ðr Þ ðtt0 Þ xl, 0,A ðr Þ + xu,0,A ðr Þ ðtt0 Þ e e + 2 2

xu, A ðt, r Þ ¼

xu,0,A ðr Þ  xl,0,A ðr Þ ðtt0 Þ xl,0,A ðr Þ + xu,0,A ðr Þ ðtt0 Þ + e e 2 2

xl, N ðt, r Þ ¼

xl,0, N ðr Þ  xu,0,N ðr Þ ðtt0 Þ xl,0,N ðr Þ + xu,0,N ðr Þ ðtt0 Þ + e e 2 2

xu, N ðt, r Þ ¼

xu,0,N ðr Þ  xl,0,N ðr Þ ðtt0 Þ xl,0,N ðr Þ + xu,0,N ðr Þ ðtt0 Þ e e + 2 2

where: x0, A ðt, r Þ ¼ ½19 + r, 21  r , x0, N ðt, r Þ ¼ ½4 + 16r, 36  16r , 0  r  1: Note. According to the solution of the growth model, we find that by increasing time, the fuzziness and distribution are increased as well. In other words, uncertainty tends to infinity and probability tends to zero. The same is true for the decay model. So, if at the initial point: x0 ¼ ðthe births in some country last month, considerably close to 20;000, not sureÞ it will be at other time points like: xðtÞ ¼ ðthe births in some country last month, considerably close to 20; 000, not at allÞ Example—Medicine The rate of a certain drug that is eliminated from the bloodstream is proportional to the amount of the drug in the bloodstream. A patient now has about 10 mg deficiency of the drug in his or her bloodstream, quite sure. The drug is being administered to the patient intravenously at a constant rate of 5 mg/h. A differential equation reflecting the situation is: x0gH ðtÞ ¼ 5gH k⊙xðtÞ where x(t) is the amount of the drug in the patient’s bloodstream at time t, and: xðt0 Þ≔ðdeficiency of the drug, about 10 mg, quite sureÞ such that x0, A ≔ (about 10 mg) is a triangular fuzzy number (8, 10, 12) and:

204

Soft Numerical Computing in Uncertain Dynamic Systems

x0, N ≔ (quite sure) presents the normal probability density function: ! 1 ðt  μÞ2 , μ ¼ 10, σ ¼ 0 N ðtÞ ¼ pffiffiffiffiffi exp  2σ 2 σ 2π and k is the proportionality constant determined by the drug. The solutions are: ½xl, A ðt, r Þ, xu, A ðt, r Þ ¼ ½et ð3 + 5et + 2r Þ, et ð7 + 5et  2r Þ ½xl, N ðt, r Þ, xu, N ðt, r Þ ¼ ½5et ð1 + 5et Þ, 5et ð1 + 5et Þ Note. According to the solutions of this example, we find that by increasing t the fuzziness (or uncertainty) tends to infinity and distribution (or probability) is fixed: xðt0 Þ≔ðdeficiency of the drug, about 10 mg, quite sureÞ ! xðtn Þ≔ðdeficiency of the drug, not close to 10 mg, quite sureÞ: Example—Economics Let us suppose the highest saved money by an employee in a country is significantly over 1000 dollars per month, very likely, which is deposited in a bank account, and assume x(t) is the amount of money in a bank account at time t, given in years, and we have: x0gH ðtÞ ¼ 13000:04⊙xðtÞ  50t where: xðt0 Þ≔ðhigh saved money, significantly over 1000 dollars, very likelyÞ such that x0, A ≔ (significantly over 1000 dollars) is a triangular fuzzy number (999, 1000, 1001) and x0, N ≔ (very likely) presents the normal probability density function: ! 1 ðt  μÞ2 N ðtÞ ¼ pffiffiffiffiffi exp  , μ ¼ 100, σ ¼ 1 2σ 2 σ 2π The term 0.04 ⊙ x(t) must reflect the rate of growth due to interest. It is possible that the interest is nominally 4% per year, compounded continuously. The term 1300 reflects deposits into the account at a constant rate of 1300 dollars per year. The term 50t accounts for the contribution to the decrease in money, which represents the rate of withdrawal. The rate at which money is being withdrawn is increasing with time. The solutions are: xl, A ðt, r Þ ¼ 1250 + 2249e0:04t + e0:04t r + 1250t xu, A ðt, r Þ ¼ 1250 + 2251e0:04t  e0:04t r + 1250t

Continuous numerical solutions of uncertain differential equations 205 xl, N ðt, r Þ ¼ 1250 + 2247e0:04t + e0:04t r + 1250t xu, N ðt, r Þ ¼ 1250 + 2253e0:04t  e0:04t r + 1250t for any 0  r  1. Note. According to the solutions of this model, we can find out by increasing t that the fuzziness and distribution are fixed.

References Allahviranloo, T., 2020. Uncertain information and linear systems. In: Studies in Systems, Decision and Control. 254 Springer. Allahviranloo, T., Chehlabi, M., 2015. Solving fuzzy differential equations based on the length function properties. Soft Comput. 19, 307–320. https://doi.org/10.1007/s00500-014-1254-4. Allahviranloo, S., Abbasbandy, S., Behzadi, S., 2014. Solving nonlinear fuzzy differential equations by using fuzzy variational iteration method. Soft Comput. 18, 2191–2200. https://10.1007/s00500013-1193-5. Allahviranloo, T., Gouyandeh, Z., Armand, A., Hasanoglu, A., 2015. On fuzzy solutions for heat equation based on generalized Hukuhara differentiability. Fuzzy Set. Syst. 265, 1–23. Bede, B., Gal, S.G., 2005. Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Set Syst. 151, 581–599. Bede, B., Rudas, I.J., Bencsik, A.L., 2007. First order linear fuzzy differential equations under generalized differentiability. Inform. Sci. 177, 1648–1662. Chehlabi, M., Allahviranloo, T., 2018. Positive or negative solutions to first-order fully fuzzy linear differential equations under generalized differentiability. Appl. Soft Comput. 70, 359–370. Gouyandeha, Z., Allahviranloob, T., Abbasbandyb, S., Armand, A., 2017. A fuzzy solution of heat equation under generalized Hukuhara differentiability by fuzzy Fourier transform. Fuzzy Set. Syst. 309, 81–97. Liu, B., 2015. Uncertain Theory. Springer-Verlag, Berlin. Pirmuhammadi, S., Allahviranloo, T., Keshavarz, M., 2017. The parametric form of Z-number and its application in Z-number initial value problem. Int. J. Intell. Syst. 00, 1–32. Salahshour, S., Allahviranloo, T., 2013. Applications of fuzzy Laplace transforms. Soft Comput. 17, 145–158. https://doi.org/10.1007/s00500-012-0907-4. Stefanini, L., Bede, B., 2009. Generalized Hukuhara differentiability of interval-valued functions and interval differential equations. Nonlinear Anal. 71, 1311–1328. Tofigh Allahviranloo, M., Ahmadi, B., 2010. Fuzzy Laplace transforms. Soft Comput. 14, 235–243. https://doi.org/10.1007/s00500-008-0397-6. Yao, K., 2016. Uncertain Differential Equations. Springer Uncertainty Research https://doi.org/ 10.1007/978-3-662-52729-0_6.

5.1

Chapter 5

Discrete numerical solutions of uncertain differential equations Introduction

In the previous chapters, we talked about the fuzzy differential equations and their solutions. Despite the uniqueness of the fuzzy solutions of the equations, usually finding the exact solution of the equations is not an easy problem and perhaps the solution cannot even be found. In this situation, we must approximate the solutions. In this chapter, we introduce some numerical methods to approximate the exact solutions of fuzzy differential equations. The fuzzy Taylor method is one of the numerical methods to find the fuzzy approximate and unique solution of fuzzy differential equations. This expansion was expressed in Chapter 3 up to the order three of differentiability, as follows, because of computing the complicity of the expansion involved with the type of differentiability.

207 Soft Numerical Computing in Uncertain Dynamic Systems. https://doi.org/10.1016/B978-0-12-822855-5.00005-7 © 2020 Elsevier Inc. All rights reserved.

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Soft Numerical Computing in Uncertain Dynamic Systems

To introduce the fuzzy Taylor method (Gouyandeha et al., 2017), we should discuss several cases. In each case, we have options in choosing the first two terms or the first three terms of the expansion to find and investigate the several numerical methods, such as the fuzzy Euler method, the fuzzy Tylor method of order two and three, etc. Since we are not able to compute and, as a matter of fact, we cannot evaluate the expansion with more terms or differentials, we thus claim that the fuzzy Taylor method with the first order of differentiability is known as a fuzzy Euler method, and the fuzzy modified Euler method is obtained with considering the first and the second derivative of the fuzzy function. The cases are as follows. Case 1. Let us assume that x(t) is a continuous fuzzy number valued function and all the derivatives are (i
gH)-differentiable for n ¼ 1, 2, …, m without changing the type of differentiability. Then, based on the previous case, we have the following relation: ð1Þ

xðsÞ ¼ xðaÞxi
gH ðaÞ⊙ðs
aÞ ð2Þ

xi
gH ðaÞ⊙

ð s
aÞ 2 ðs
aÞm
1 ðm
1Þ ⋯xi
gH ðaÞ⊙ Rn ða, sÞ 2! ðm
1Þ!

where Rn(a, s) is noted as a reminder term of the expansion and it is:   ð s ð s1 ð sn
1 ðnÞ Rn ða, sÞ ¼ FR ⋯ xi
gH ðsn Þdsn dsn
1 ⋯ ds1 a

a

a

Case 2. Let us now assume all the derivatives are (ii
gH)-differentiable for n ¼ 1, 2, …, m without changing the type of differentiability. Then, based on the previous case, we have the following relation: ð1Þ

ð2Þ

xðsÞ ¼ xðaÞH ð
1Þxii
gH ðaÞ⊙ðs
aÞH ð
1Þxii
gH ðaÞ⊙ ðm
1Þ

H ð
1Þ⋯H ð
1Þxii
gH ðaÞ⊙

ð s
aÞ 2 2!

ðs
aÞm
1 H ð
1ÞRn ða, sÞ ðm
1Þ!

where again Rn(a, s) is noted as a reminder term of the expansion and it is:   ð s ð s1 ð sn
1 ðnÞ Rn ða, sÞ ¼ FR ⋯ xii
gH ðsn Þdsn dsn
1 ⋯ ds1 a

a

a

Case 3. Suppose that the same function is i
gH differentiable for n ¼ 2k
1, k  ℕ and it is ii
gH differentiable for n ¼ 2k, k  ℕ [ {0}: ð1Þ

xðsÞ ¼ xðaÞH ð
1Þxii
gH ðaÞ⊙ðs
aÞ m
1

ð2Þ

xi
gH ðaÞ⊙

ð s
aÞ 2 ðs
aÞ 2 ðm
1 2 Þ   H ð
1Þ⋯H ð
1Þxii
gH ðaÞ⊙  m
1 2! ! 2

Discrete numerical solutions of uncertain differential equations

209

m

ð s
aÞ 2 ðm2 Þ xi
gH ðaÞ⊙ m H ð
1Þ⋯H ð
1ÞRn ða, sÞ ! 2 where again Rn(a, s) is noted as: Rn ða, sÞ ¼ FR

ð s  ð s1 a

⋯

ð sn
1

a

a

  ðnÞ xi
gH ðsn Þdsn dsn
1 ⋯ ds1

Case 4. Suppose that the same function is i
gH differentiable in interval [a, ξ] and ξ is the switching point. So: ð1Þ

ð2Þ

xðsÞ ¼ xðaÞxi
gH ðaÞ⊙ðξ
aÞH xii
gH ðaÞ⊙ða
ζ 1 Þ⊙ðξ
aÞ ð2Þ

xi
gH ðζ 1 Þ⊙

ðξ
ζ 1 Þ2 2

!   ð ξ ð ζ1 ð s2 ða
ζ 1 Þ2 ð3Þ
xii
gH ðs4 Þds4 ds2 ds1 H ð
1ÞFR 2 a a a FR

ð ξ ð s1 ð s3 a

ζ1

ζ1

ð2Þ

ð3Þ xi
gH ðs5 Þds5

⊙ðs
ξÞxii
gH ðξÞ⊙ FR

ð s ð t ð t2 ξ

ξ

a



  ð1Þ ds3 ds1 H ð
1Þ xii
gH ðξÞ

ðs
ξÞ2 2!

   ð3Þ xii
gH ðt3 Þdt3 dt2 dt1

As we discussed before, we should rest assured that the solution exists and is unique. In Chapter 4 we proved the following theorem, in which, under the following conditions, the fuzzy solution exists, and it is unique. 1. The function f : I1  I2 ! R is continuous where I1 is a closed interval containing the initial value x0 and I2 is any other area such that f is bounded on it. 2. The function f is bounded. It means 9 M < 0, DH(f(t, x), 0)  M, 8 (t, x)  I1  I2. 3. The real function g : I1  I3 ! R such that g(t, u) ≡ 0 is bounded on I1  I3, 9 M1 > 0, 0  g(t, u)  M1, 8 (t, u)  I1  I3 where I3 is another closed interval containing u. Moreover g(t, u) is nondecreasing in u and its corresponding initial value problem: u0 ðtÞ ¼ gðt, uðtÞÞ, uðt0 Þ ¼ 0 has only the solution u(t) ≡ 0 on I1.

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Soft Numerical Computing in Uncertain Dynamic Systems

4. Also: DH ðf ðt, xÞ, f ðt, yÞÞ  gðt, jx
yjÞ, 8ðt, xÞ8ðt, yÞI1  I2 , x,yI3 Then the fuzzy initial value problem has both (i
gH)-solution xi
gH(t) and (ii
gH)-solution xii
gH(t) and the following successive iterations converge to these two solutions: ðt xi, n + 1 ðtÞ ¼ xi,0  f ðt, xi, n ðzÞÞdz, xi,0 ðtÞ ¼ x0 t0

and: xii, n + 1 ðtÞ ¼ xii,0 ð
1Þ⊙

ðt

f ðt, xii, n ðzÞÞdz, xii, 0 ðtÞ ¼ x0

t0

5.2

Fuzzy Euler method

Consider the following fuzzy initial value problem: x0gH ðtÞ ¼ f ðt, xÞ, xðt0 Þ ¼ x0 R , t0  t  T As we know, the solution x(t) is an unknown fuzzy number valued function and f : ½0, T   R ! R is a fuzzy continuous function. To derive the fuzzy Euler method, let a partition IN ¼ {0 ¼ t0 < t1 < ⋯ < tN ¼ T} of the interval [0, T], where tk ¼ kh, k ¼ 0, 1, …, N. The step size is h ¼ NT and the points are equidistant points. As a first case, suppose that the unique solution x(t) is (i
gH)differentiable and xðtÞC2gH ð½0, T , R Þ (Allahviranloo et al., 2015, 2007; Allahviranloo & Salahshour, 2011; Armand et al., 2019; Gouyandeha et al., 2017). Case 1. The fuzzy number valued solution function x(t) is (i
gH)-differentiable and the type of differentiability does not change on [0, T]. So the fuzzy Taylor expansion is: xðtk + 1 Þ ¼ xðtk Þx0i
gH ðtk Þ⊙ðtk + 1
tk Þx00i
gH ðηk Þ⊙

ðtk + 1
tk Þ2 2!

for some point ηk  [tk, tk+1]. Since h ¼ tk+1
tk then it gives: xðtk + 1 Þ ¼ xðtk Þh⊙x0i
gH ðtk Þ

h2 00 ⊙x ðη Þ 2! i
gH k

From the equation, we have: x0i
gH ðtÞ ¼ f ðt, xðtÞÞ

Discrete numerical solutions of uncertain differential equations

211

then: xðtk + 1 Þ ¼ xðtk Þh⊙f ðtk , xðtk ÞÞ

h2 00 ⊙x ðη Þ 2! i
gH k

Using the Hausdorff distance and approaching it to zero, we will find the fuzzy Euler method with its error function. So we have:   h2 0 0 DH xðtk + 1 Þ, xðtk Þh⊙f ðtk , xðtk ÞÞ ⊙xi
gH ðηk Þ  2!   h2 00 DH ðxðtk + 1 Þ, xðtk Þh⊙f ðtk , xðtk ÞÞÞ + DH 0, ⊙xi
gH ðηk Þ ! 0 2! Hence we find:   h2 0 0 DH ðxðtk + 1 Þ, xðtk Þh⊙f ðtk , xðtk ÞÞÞ ! 0, DH 0, ⊙xi
gH ðηk Þ ! 0 2! We know that the step size h is small enough and goes to zero. Then: xðtk + 1 Þ  xðtk Þh⊙f ðtk , xðtk ÞÞ It is clear that the right-hand side approximates the left-hand side. Based on this approximate method, we can introduce a similar iterative method to find or define the sequence of {xk}N
1 k¼0 such that xk  x(tk):  xk + 1 ¼ xk h⊙f ðtk , xk Þ, k ¼ 0, 1,…, N
1 x0 R Case 2. The solution x(t) is (ii
gH)-differentiable and the type of differentiability does not change on [0, T]. So the fuzzy Taylor expansion is: xðtk + 1 Þ ¼ xðtk ÞH ð
1Þh⊙x0ii
gH ðtk ÞH ð
1Þ

h2 00 ⊙x ðη Þ 2! ii
gH k

Since xii
gH0 (t) ¼ f(t, x(t)) then: xðtk + 1 Þ ¼ xðtk ÞH ð
1Þh⊙f ðtk , xðtk ÞÞH ð
1Þ

h2 00 ⊙x ðη Þ 2! ii
gH k

Thus by a process similar to the previous one, the fuzzy Euler method takes the form:  xk + 1 ¼ xk H ð
1Þh⊙f ðtk , xk Þ, k ¼ 0, 1, …,N
1 x0 R

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Soft Numerical Computing in Uncertain Dynamic Systems

Case 3. Let us suppose that x(t) has a switching point type (1) on the interval [0, T] as like as ξ, and also assume that t0, t1, …, tj, ξ, tj+1, …, tN. Thus the fuzzy Euler method is obtained as follows: 8 k ¼ 0,1, …, j < xk + 1 ¼ xk h⊙f ðtk , xk Þ, k ¼ j + 1, j + 2,…, N
1 xk + 1 ¼ xk H ð
1Þh⊙f ðtk , xk Þ, : x0 R Case 4. If ξ is a type (2) switching point, the fuzzy Euler method takes the form as follows: 8 < xk + 1 ¼ xk H ð
1Þh⊙f ðtk , xk Þ, k ¼ 0,1, …, j ¼ xk h⊙f ðtk , xk Þ, k ¼ j + 1, j + 2, …,N
1 x : k+1 x0 R

5.2.1

ANALYSIS

OF THE FUZZY

EULER

METHOD

Since error in the numerical method is inevitable, so we are going to define local truncation and global truncation errors. Using this concept, the consistency of the Euler method and its convergence and stability are studied and proved (Armand et al., 2019).

5.2.1.1 Local truncation error and consistency For all abovementioned cases of the fuzzy Euler method, we can define the local truncation error by using a residual value as follows: Rk ¼ xðtk + 1 ÞgH ðxðtk Þf ðtk , xðtk ÞÞÞ ¼

h2 00 ⊙xi
gH ðηk Þ 2

or: Rk ¼ xðtk + 1 ÞgH ðxðtk ÞH ð
1Þf ðtk , xðtk ÞÞÞ ¼ H ð
1Þ and the mentioned error is defined as: 1 h τk ¼ Rk ¼ ⊙x00i
gH ðηk Þ h 2 or: 1 h τk ¼ Rk ¼ H ð
1Þ ⊙x00ii
gH ðηk Þ h 2

h2 00 ⊙xii
gH ðηk Þ 2

Discrete numerical solutions of uncertain differential equations

213

Note. In four cases, the residual values are in two mentioned forms. Now we claim that the methods are said to be consistent if: lim max DH ðτk , 0Þ ¼ 0

h!0 0tk T

To this end, we suppose that the functions xi
gH00 (ηk) and xii
gH00 (ηk) are bounded and this means:     9M1 > 0; DH x00i
gH ðηk Þ, 0  M1 , 9M2 > 0; DH x00ii
gH ðηk Þ, 0  M2 Thus the methods will be consistent, because: lim max DH ðτk , 0Þ ¼ lim max DH

h!0 0tk T

h!0 0tk T

  h 00 ⊙xi
gH ðηk Þ, 0 2

  h h max DH x00i
gH ðηk Þ, 0  lim M1 ¼ 0 h!0 2 0tk T h!0 2

¼ lim

The same process is true for the ii
gH differentiability:   h lim max DH ðτk , 0Þ ¼ lim max DH H ð
1Þ ⊙x00ii
gH ðηk Þ, 0 h!0 0tk T h!0 0tk T 2        h h ¼ lim ð
1Þ  max DH H x00ii
gH ðηk Þ, 0 ¼ lim max DH x00ii
gH ðηk Þ, 0 h!0 h!0 2 0tk T 2 0tk T h  lim M2 ¼ 0 h!0 2 So the fuzzy Euler method in each case is consistent so long as the fuzzy number valued solution is in C2gH ð½0, T , R Þ:

5.2.1.2 Global truncation error and convergence The global truncation error is the agglomeration of the local truncation error over all the iterations, assuming perfect knowledge of the true solution at the initial time step. Formally, for both case (1) and (2) the global truncation error is noted by ek+1 at the point tk+1 and defined as: ek + 1 ¼ xðtk + 1 ÞgH xk + 1 ¼ ¼ xðtk + 1 ÞgH ðx0 h⊙f ðt0 , x0 Þh⊙f ðt1 , x1 Þ⋯h⊙f ðtk , xk ÞÞ and in the case (2): ek + 1 ¼ xðtk + 1 ÞgH xk + 1 ¼

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Soft Numerical Computing in Uncertain Dynamic Systems

¼ xðtk + 1 ÞgH ðx0 H ð
1Þh⊙f ðt0 , x0 ÞH ð
1Þh⊙ f ðt1 , x1 ÞH ð
1Þ⋯H ð
1Þh⊙f ðtk , xk ÞÞ for any k ¼ 0, 1, …, N
1. We claim that the fuzzy Euler method is convergent if the global truncation error goes to zero as the step size goes to zero; in other words, the numerical solution converges to the exact solution if:  lim max DH ðek + 1 , 0Þ ¼ 0 ) lim max DH xðtk + 1 ÞgH xk + 1 , 0 h!0

k

h!0

k

¼ lim max DH ðxðtk + 1 Þ, xk + 1 Þ ¼ 0 h!0

k

5.2.1.3 Theorem—Convergence Suppose that xgH00 (t) exists and also f(t, x(t)) satisfies the Lipschitz condition that was defined in Chapter 4. Then the fuzzy Euler method converges to the fuzzy solution of the fuzzy initial value problem: x0gH ðtÞ ¼ f ðt, xÞ, xðt0 Þ ¼ x0 R , t0  t  T To prove the theorem, consider that x(t) is i
gH differentiable and also suppose that: dk ≔

h2 00 ⊙xi
gH ðtk Þ 2

By this assumption, the exact solution satisfies the following relation: xðtk + 1 Þ ¼ xðtk Þh⊙f ðtk , xðtk ÞÞdk On the other hand, we have: xk + 1 ¼ xk h⊙f ðtk , xk Þ Now let us consider the distances of both sides of the abovementioned relations, so: DH ðxðtk + 1 Þ, xk + 1 Þ ¼ DH ðxðtk Þ, xk Þ + h½DH ðf ðtk , xðtk Þ, f ðtk , xk ÞÞ + DH ðdk , 0Þ Since the function f(t, x(t)) satisfies the Lipschitz condition, then: DH ðf ðtk , xðtk Þ, f ðtk , xk ÞÞ  Lk DH ðxðtk Þ, xk Þ

Discrete numerical solutions of uncertain differential equations

215

Using the triangular inequality and Lipschitz condition, we arrive at: DH ðxðtk + 1 Þ, xk + 1 Þ  ð1 + hLk ÞDH ðxðtk Þ, xk Þ + DH ðdk , 0Þ, k ¼ 0,1, …, N
1 Also consider that: L ¼ max Lk , d ¼ max DH ðdk , 0Þ 0kN
1

0kN
1

then we have: DH ðxðtk + 1 Þ, xk + 1 Þ  ð1 + hLÞDH ðxðtk Þ, xk Þ + d, k ¼ 0,1, …,N
1 By backward substituting: DH ðxðtk + 1 Þ, xk + 1 Þ  ð1 + hLÞ½ð1 + hLÞDH ðxðtk
1 Þ, xk
1 Þ + d  + d ¼ ð1 + hLÞ2 DH ðxðtk
1 Þ, xk
1 Þ + d½1 + ð1 + hLÞ Continuing this process: h i DH ðxðtk + 1 Þ, xk + 1 Þ  ð1 + hLÞk + 1 DH ðxðt0 Þ, x0 Þ + d 1 + ð1 + hLÞ + ⋯ + ð1 + hLÞk The last term does have the general formula and actually it is a geometric series: ð1 + hLÞ + ⋯ + ð1 + hLÞk ¼

ð1 + hLÞk + 1
1 hL

By substituting: "

DH ðxðtk + 1 Þ, xk + 1 Þ  ð1 + hLÞ

k+1

ð1 + hLÞk + 1
1 DH ðxðt0 Þ, x0 Þ + d hL

 ð1 + hLÞk + 1 DH ðxðt0 Þ, x0 Þ +

#

i dh ð1 + hLÞk + 1
1 hL

On the other hand:  k + 1 1 + hL  ehL ) ð1 + hLÞk + 1  ehL  eLT , hðk + 1Þ  T thus: DH ðxðtk + 1 Þ, xk + 1 Þ  eLT DH ðxðt0 Þ, x0 Þ +

 d LT e
1 hL

By substituting: d ¼ max DH ðdk , 0Þ ¼ 0kN
1

  h max DH x00i
gH ðtÞ, 0 , DH ðxðt0 Þ, x0 Þ ¼ 0 2 0tT

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Soft Numerical Computing in Uncertain Dynamic Systems

Finally: DH ðxðtk + 1 Þ, xk + 1 Þ 

   h LT e
1 max DH x00i
gH ðtÞ, 0 0tT 2L

Taking the limit of two sides: lim DH ðxðtk + 1 Þ, xk + 1 Þ ¼ 0

h!0

Now consider that x(t) is ii
gH differentiable without any switching point and: dk ≔H ð
1Þ

h2 00 ⊙xii
gH ðtk Þ 2

By this assumption, the exact solution satisfies the following relation: xðtk + 1 Þ ¼ xðtk ÞH ð
1Þh⊙f ðtk , xðtk ÞÞdk On the other hand, we have: xk + 1 ¼ xk H ð
1Þh⊙f ðtk , xk Þ Now let us consider the distances of both sides of the abovementioned relations, so: DH ðxðtk + 1 Þ, xk + 1 Þ ¼ DH ðxðtk Þ, xk Þ
h½DH ðf ðtk , xðtk ÞÞ, f ðtk , xk ÞÞ + DH ðdk , 0Þ Since the function f(t, x(t)) satisfies the Lipschitz condition, then: DH ðf ðtk , xðtk ÞÞ, f ðtk , xk ÞÞ  Lk DH ðxðtk Þ, xk Þ Using the triangular inequality and Lipschitz condition, we arrive at: DH ðxðtk + 1 Þ, xk + 1 Þ  ð1
hLk ÞDH ðxðtk Þ, xk Þ + DH ðdk , 0Þ, k ¼ 0,1, …,N
1 Also consider that: L ¼ max Lk , d ¼ max DH ðdk , 0Þ 0kN
1

0kN
1

then we have: DH ðxðtk + 1 Þ, xk + 1 Þ  ð1
hLÞDH ðxðtk Þ, xk Þ + d, k ¼ 0,1, …,N
1 By backward substituting: DH ðxðtk + 1 Þ, xk + 1 Þ  ð1
hLÞ½ð1
hLÞDH ðxðtk
1 Þ, xk
1 Þ + d  + d ¼ ð1
hLÞ2 DH ðxðtk
1 Þ, xk
1 Þ + d ½1 + ð1
hLÞ Continuing this process: h i DH ðxðtk + 1 Þ, xk + 1 Þ  ð1
hLÞk + 1 DH ðxðt0 Þ, x0 Þ + d 1 + ð1
hLÞ + ⋯ + ð1
hLÞk

Discrete numerical solutions of uncertain differential equations

217

The last term does have the general formula and actually it is a geometric series: ð1
hLÞ + ⋯ + ð1
hLÞk ¼

1
ð1
hLÞk + 1 hL

By substituting: "

1
ð1
hLÞk + 1 DH ðxðtk + 1 Þ, xk + 1 Þ  ð1
hLÞk + 1 DH ðxðt0 Þ, x0 Þ + d hL  ð1
hLÞk + 1 DH ðxðt0 Þ, x0 Þ +

#

i dh 1
ð1
hLÞk + 1 hL

On the other hand:  k + 1  e
LT , hðk + 1Þ  T 1 + hL  ehL ) ð1
hLÞk + 1  e
hL thus: DH ðxðtk + 1 Þ, xk + 1 Þ  e
LT DH ðxðt0 Þ, x0 Þ +

 d

1
e
LT hL

By substituting:   h d ¼ max DH ðdk , 0Þ ¼
max DH x00ii
gH ðtÞ, 0 , DH ðxðt0 Þ, x0 Þ ¼ 0 0kN
1 2 0tT Finally: DH ðxðtk + 1 Þ, xk + 1 Þ 

  
h

1
e
LT max DH x00ii
gH ðtÞ, 0 0tT 2L

Taking the limit of two sides: lim DH ðxðtk + 1 Þ, xk + 1 Þ ¼ 0

h!0

5.2.1.4 Stability In addition, to ensure that the problem has a solution, we must be sure that the method is stable. This means that small perturbations in the initial data will only lead to small changes in the solutions (Armand et al., 2019). The stability can be defined as follows. Let xk+1, k + 1 0 be the fuzzy solution of the fuzzy Euler method with fuzzy initial value x0 R and let zk+1 be the solution of the same numerical method with a perturbed fuzzy initial condition z0 ¼ x0 δ0 R . The fuzzy Euler method is stable

if there exist positive constants h and κ such that: DH ðzk + 1 , yk + 1 Þ  κδ, 8ðk + 1Þh  T, k  N
1

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Soft Numerical Computing in Uncertain Dynamic Systems

whenever DH(δ0, 0)  δ. Now suppose that the solution x(t) is i
gH differentiable and consider the perturbed problem as follows: zk + 1 ¼ zk h⊙f ðtk , zk Þ, z0 ¼ x0 δ0 and also consider the mentioned iterative method in the case (1): xk + 1 ¼ xk h⊙f ðtk , xk Þ Using the distance, we will get: DH ðzk + 1 , xk + 1 Þ  DH ðzk , xk Þ + hDH ðf ðtk , zk Þ, f ðtk , xk ÞÞ Using the properties of the distance and also the Lipschitz condition: DH ðzk + 1 , xk + 1 Þ  ð1 + hLÞDH ðzk , xk Þ and also: DH ðzk + 1 , xk + 1 Þ  ð1 + hLÞ½ð1 + hLÞDH ðzk
1 , xk
1 Þ ¼ ð1 + hLÞ2 DH ðzk
1 , xk
1 Þ Iterating this inequality and the previous similar process, we will find: DH ðzk + 1 , xk + 1 Þ  ð1 + hLÞk DH ðz0 , x0 Þ  ehLðk + 1Þ DH ðz0 H x0 , 0Þ  eLT DH ðδ0 , 0Þ  κδ where κ ¼ eLT. So based on the previous illustrations, the method is stable. In the case that the solution x(t) is ii
gH differentiable, we consider the perturbed problem as follows: zk + 1 ¼ zk H ð
1Þh⊙f ðtk , zk Þ, z0 ¼ x0 δ0 and also consider the mentioned iterative method in case (2): xk + 1 ¼ xk H ð
1Þh⊙f ðtk , xk Þ Using the distance, we will get: DH ðzk + 1 , xk + 1 Þ  DH ðzk , xk Þ
hDH ðf ðtk , zk Þ, f ðtk , xk ÞÞ Using the properties of the distance and also the Lipschitz condition: DH ðzk + 1 , xk + 1 Þ  ð1
hLÞDH ðzk , xk Þ and also: DH ðzk + 1 , xk + 1 Þ  ð1
hLÞ½ð1
hLÞDH ðzk
1 , xk
1 Þ ¼ ð1
hLÞ2 DH ðzk
1 , xk
1 Þ

Discrete numerical solutions of uncertain differential equations

219

Iterating this inequality and the previous similar process, we will find: DH ðzk + 1 , xk + 1 Þ  ð1
hLÞk + 1 DH ðz0 , x0 Þ  e
hLðk + 1Þ DH ðz0 H x0 , 0Þ  e
LT DH ðδ0 , 0Þ  κδ where κ ¼ e
LT. So based on the previous illustrations, the method is stable. Note. The Euler method is not stable generally. By this example, we show that the fuzzy Euler method is not stable generally. To this end, we explain the method by using the concept of the tangent line of a fuzzy number valued solution function. Consider the following example: x0 ðtÞ ¼ xðtÞ, xð0Þ ¼ ð1, 2, 3Þ, t 0 The trivial solution of this fuzzy initial value problem is x(t) ¼ (0.9,1.2,1.4)et (see Fig. 5.1). As can be seen in the figure: DH ðxðt1 Þ, x1 Þ < DH ðxðt2 Þ, x2 Þ < ⋯ This is the main reason to claim that the fuzzy Euler method is unstable. In other words, the stability area is not so extended.

2.2

xu (t1) xl (t2) 2

1.8

x2 1.6

1.4

xl (t1) x1

1.2

1

0

x0

0.1

0.2

0.3

0.4

0.5

Fig. 5.1 Instability of the fuzzy Euler method.

0.6

0.7

0.8

0.9

220

Soft Numerical Computing in Uncertain Dynamic Systems

Example. The following fuzzy initial value problem is assumed:  0 xi
gH ðtÞ ¼ xðtÞð1:3; 2, 2:1Þ, 0  t  1 xð0Þ ¼ ð0:82; 1, 1:2Þ where (1.3,2, 2.1) and (0.82,1, 1.2) are triangular fuzzy numbers. The exact solution is obtained very easily, and it is: xðtÞ ¼ ðxð0Þð1:3; 2, 2:1ÞÞet H ð1:3; 2, 2:1Þ The fuzzy Euler method in the sense of i
gH differentiability does have the following form:  xk + 1 ¼ xk h⊙ðxk ð1:3; 2, 2:1ÞÞ, xð0Þ ¼ ð0:82; 1, 1:2Þ or: 

xk + 1 ¼ ð1 + hÞ⊙xk h⊙ð1:3;2, 2:1Þ, xð0Þ ¼ ð0:82; 1, 1:2Þ

Since the fuzzy number valued solution function is i
gH differentiable, the parametric form of the solution is: xl ðt, r Þ ¼ ð0:82 + 0:18r + 1:3 + 0:7rÞet
1:3
0:7r ¼ ð2:12 + 0:88r Þet
1:3
0:7r xu ðt, r Þ ¼ ð1:2
0:2r + 2:1
0:1r Þet
2:1 + 0:1r ¼ ð3:3
0:3r Þet
2:1 + 0:1r These lower and upper functions are the solutions of the following parametric equations of fuzzy equations:  0 xl ðt, r Þ ¼ xl ðtÞ + 1:3 + 0:7r, 0  t  1 xl ð0Þ ¼ 0:82 + 0:18r  0 xu ðt, r Þ ¼ xu ðtÞ + 2:1
0:1r, 0  t  1 xl ð0Þ ¼ 1:2
0:2r where: ð1:3; 2, 2:1Þ½r  ¼ ½1:3 + 0:7r, 2:1
0:1r , ð0:82;1, 1:2Þ½r  ¼ ½0:82 + 0:18r, 1:2
0:2r Fig. 5.2 shows the i
gH differentiable solution of the fuzzy initial value problem. In addition, the global truncation errors have been reported for h ¼ 0.025 and h ¼ 0.005 in Table 5.1. It is seen that the results do have less error when the value of the step size is less. Moreover, for a fixed step size, the errors are going to increase when the time increases as well. This process is called error propagation.

Discrete numerical solutions of uncertain differential equations

6

5

4

3

2

1 0.1

0.2

0.3

0.4

0.5

0.6

Fig. 5.2 The fuzzy i
gH differentiable solution.

TABLE 5.1 Global error results.

0.7

0.8

0.9

221

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Soft Numerical Computing in Uncertain Dynamic Systems

Example. Let us consider the following fuzzy initial value problem:  0 xii
gH ðtÞ ¼
xðtÞt⊙ð0:7; 1, 1:8Þ, 0  t  1 xð0Þ ¼ ð0, 1, 2:2Þ where (0.7,1, 1.8) and (0, 1, 2.2) are triangular fuzzy numbers and their parametric forms are, respectively: ð0:7; 1, 1:8Þ½r  ¼ ½0:7 + 0:3r, 1:8
0:8r , ð0, 1, 2:2Þ½r  ¼ ½r, 2:2
1:2r  The exact solution of the fuzzy initial value problem can be obtained easily by using the parametric form of the problem and considering the ii
gH differentiability of the solution. The parametric form is as follows:  0 xl ðt, r Þ ¼
xl ðt, r Þ + tð1:8
0:8r Þ, 0  t  1 x l ð 0Þ ¼ r  0 xu ðt, r Þ ¼
xu ðt, r Þ + tð0:7 + 0:3r Þ, 0  t  1 xu ð0Þ ¼ 2:2
1:2r The solutions of these parametric initial value problems are: xl ðt, r Þ ¼ ð1:8 + 0:2r Þe
t + ð1:8
0:8r Þðt
1Þ xu ðt, r Þ ¼ ð2:9
0:9r Þe
t + ð0:7 + 0:3r Þðt
1Þ The fuzzy number valued solution function is in the form of: xðtÞ ¼ C⊙e
t ð0:7; 1, 1:8Þ⊙ðt
1Þ, C ¼ ð0, 1, 2:2ÞH ð
1Þð0:7; 1, 1:8Þ In Fig. 5.3, the ii
gH differentiable solution function is plotted and as we know, the length of the function is decreasing. With the same explanation about the global error and its propagation of the previous example, Table 5.2 shows us the global error propagation with the same step sizes. Example. In this example, there is a switching point and we will see that the type of differentiability is changed before and after the switching point. Consider the following fuzzy initial value problem:  0 xgH ðtÞ ¼ ð1
tÞ⊙xðtÞ, 0t2 xð0Þ ¼ ð0, 1, 2Þ where (0, 1, 2) is again a triangular fuzzy number with the parametric form of: ð0, 1, 2Þ½r  ¼ ½r, 2
r  It is clear that in the interval 0  t < 1 the solution is i
gH differentiable and on another subinterval 1 < t  2 it is ii
gH differentiable. So considering these

Discrete numerical solutions of uncertain differential equations

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

0

0.1

0.2

0.3

0.4

0.5

0.6

Fig. 5.3 The fuzzy ii
gH differentiable solution.

TABLE 5.2 Global error results.

0.7

0.8

0.9

223

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Soft Numerical Computing in Uncertain Dynamic Systems

explanations, the switching point is t ¼ 1 and the parametric form of the fuzzy initial value problem can be explained as the following forms: •

In the case of i
gH differentiability:  8 ð1
tÞxl ðt, r Þ, 0  t  1 > 0 > ð t, r Þ ¼ x > > ð < l  1
tÞxu ðt, r Þ, 1  t  2 ð1
tÞxu ðt, r Þ, 0  t  1 0 > xu ðt, r Þ ¼ > > ð1
tÞxl ðt, r Þ, 1  t  2 > : xð0, r Þ ¼ ½r, 2
r 

The i
gH solution is obtained as follows (Figs. 5.4 and 5.5): t2

t2

xl ðt, r Þ ¼ r  et
2 , xu ðt, r Þ ¼ ð2
r Þ  et
2 , 0  t  1 t2

t2

1

1

xl ðt, r Þ ¼ et
2 + ðr
1Þe2tðt
2Þ , xu ðt, r Þ ¼ et
2
ðr
1Þe2tðt
2Þ , 1  t  2 •

In the case of ii
gH differentiability: 8  ð1
tÞxu ðt, r Þ, 0  t  1 > 0 > x ð t, r Þ ¼ > > < l  ð1
tÞxl ðt, r Þ, 1  t  2 ð1
tÞxl ðt, r Þ, 0  t  1 0 > xu ðt, r Þ ¼ > > ð1
tÞxu ðt, r Þ, 1  t  2 > : xð0, r Þ ¼ ½r, 2
r 

3

2.5

2

1.5

1

0.5 0

Fig. 5.4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

The fuzzy i
gH differentiable solution in 0  t  1.

1.6

1.8

2

Discrete numerical solutions of uncertain differential equations

225

2.5

2

1.5

1

0.5

0

0

1

1.1

1.2

1.3

1.4

1.5

1.5

1.7

1.8

1.9

2

Fig. 5.5 The fuzzy i
gH differentiable solution in 1  t  2.

It is clear that the parametric initial value problems in two subintervals 0  t  1 and 1  t  2 have symmetric displacements. The solutions and figures show the same behavior as well (Fig. 5.6). The fuzzy Euler method for this problem is: 8 N > > > < xk + 1 ¼ xk h⊙ð1
tk Þ⊙xk , 0  tk  1, k ¼ 0, 1, …, 2
1 N xk + 1 ¼ xk H ð
1Þh⊙ð1
tk Þ⊙xk , 1 < tk  2, k ¼ ,…, N > > > 2 : x0 ¼ ð0, 1, 2Þ

2

1

0 0.2

0

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

–2

–2

Fig. 5.6 The gH-differential of the solution with switching point in t ¼ 1

2

2.2

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TABLE 5.3 Global error results.

The numerical results for this example with the same step sizes are presented in Table 5.3 and it can be seen that if the step size is going to be decreased then the errors are going to be decreased as well.

5.3

Fuzzy modified Euler method

In this section, we intend to introduce a modified approach for solving fuzzy differential equations. The modified Euler method is obtained by improving the Euler method discussed in the previous section. The proposed method can estimate the solution by using a two-stage predictor–corrector algorithm with local truncation error of order two. The consistency, convergence, and stability of the proposed method are also investigated in detail (Allahviranloo et al., 2007; Armand et al., 2019). To start, again consider the following fuzzy initial value problem: x0gH ðtÞ ¼ f ðt, xÞ, xðt0 Þ ¼ x0 R , t0  t  T where the fuzzy solution x(t) is an unknown fuzzy number valued function and f : ½0, T   R ! R is a fuzzy continuous function. Let us have the same partition on the interval [t0, T]. As was the case previously, two cases are considered for xðtÞC3gH ð½0, T , R Þ.

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227

Case 1. The fuzzy number valued solution function x(t) is (i
gH)-differentiable and the type of differentiability does not change on [0, T]. So the fuzzy Taylor expansion is: xðtk + 1 Þ ¼ xðtk Þx0i
gH ðtk Þ⊙ðtk + 1
tk Þx00i
gH ðtk Þ⊙ x000 i
gH ðηk Þ⊙

ðtk + 1
tk Þ2 2!

ðtk + 1
tk Þ3 3!

for some point ηk  [tk, tk+1]. Since h ¼ tk+1
tk then it gives: xðtk + 1 Þ ¼ xðtk Þh⊙x0i
gH ðtk Þ

h2 00 h3 ⊙xi
gH ðtk Þ ⊙x000 ðη Þ 2! 3! i
gH k

We know that: x00i
gH ðtk Þ ¼

x0i
gH ðtk + 1 ÞH x0i
gH ðtk Þ h

since we have: x0i
gH ðtk Þ ¼ f ðtk , xðtk ÞÞ so: x00i
gH ðtk Þ ¼

f ðtk + 1 , xðtk + 1 ÞÞH f ðtk , xðtk ÞÞ h

Then by using this second derivative in the abovementioned iterative equation, we have:   h2 f ðtk + 1 , xðtk + 1 ÞÞH f ðtk , xðtk ÞÞ xðtk + 1 Þ ¼ xðtk Þh⊙f ðtk , xðtk ÞÞ ⊙ 2! h 

h3 000 ⊙x ðη Þ 3! i
gH k

After simplification, we have:   3 f ðtk + 1 , xðtk + 1 ÞÞf ðtk , xðtk ÞÞ h ðη Þ xðtk + 1 Þ ¼ xðtk Þh⊙  ⊙x000 3! i
gH k 2 Using the Hausdorff distance and approaching it to zero, we will find the fuzzy Euler method with its error function. So we have:   3   f ðtk + 1 , xðtk + 1 ÞÞf ðtk , xðtk ÞÞ h 00 DH xðtk + 1 Þ, xðtk Þh⊙  ⊙xi
gH ðηk Þ  3! 2

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Soft Numerical Computing in Uncertain Dynamic Systems

     f ðtk + 1 , xðtk + 1 ÞÞf ðtk , xðtk ÞÞ h3 0 0 0 + DH 0, ⊙xi
gH ðηk Þ DH xðtk + 1 Þ, xðtk Þh⊙ 3! 2 !0 Hence we find:    f ðtk + 1 , xðtk + 1 ÞÞf ðtk , xðtk ÞÞ DH xðtk + 1 Þ, xðtk Þh⊙ ! 0, 2 and:   h3 00 0 DH 0, ⊙xi
gH ðηk Þ ! 0 3! We know that the step size h is small enough and goes to zero. Then:   f ðtk + 1 , xðtk + 1 ÞÞf ðtk , xðtk ÞÞ xðtk + 1 Þ  xðtk Þh⊙ 2 It is clear that the right-hand side approximates the left-hand side. Based on this approximate method, we can introduce a similar iterative method to find or define the sequence of {xk}N
1 k¼0 such that xk  x(tk): 8   f ðtk + 1 , xk + 1 Þf ðtk , xk Þ < , k ¼ 0,1, …, N
1 xk + 1 ¼ xk h⊙ 2 : x0 R It is observed that the value x(tk+1) appears in both sides of the iterative equation and this concludes the implicit iterative equation. To have an implicit iterative equation, x(tk+1) can be provided with the fuzzy Euler method in case (1) as x∗(tk+1):  x∗k + 1 ¼ xk h⊙f ðtk , xk Þ, k ¼ 0, 1,…, N
1 x0 R So the final fuzzy modified Euler method is obtained as follows: 8 h   > < xk + 1 ¼ xk  ⊙ f tk + 1 , x∗k + 1 f ðtk , xk Þ , k ¼ 0,1, …, N
1 2 ∗ > : xk + 1 ¼ xk h⊙f ðtk , xk Þ, x0 R Case 2. The solution x(t) is (ii
gH)-differentiable and the type of differentiability does not change on [0, T]. So the fuzzy Taylor expansion is: xðtk + 1 Þ ¼ xðtk ÞH ð
1Þx0ii
gH ðtk Þ⊙ðtk + 1
tk ÞH ð
1Þx00ii
gH ðtk Þ ⊙

ðtk + 1
tk Þ2 ðtk + 1
tk Þ3 H ð
1Þx000 ð η Þ⊙ ii
gH k 2! 3!

Discrete numerical solutions of uncertain differential equations

229

Thus, by a similar process, the fuzzy modified Euler method takes the form: 8 h   > < xk + 1 ¼ xk H ð
1Þ ⊙ f tk + 1 , x∗k + 1 f ðtk , xk Þ , k ¼ 0,1, …, N
1 2 ∗ > : xk + 1 ¼ xk H ð
1Þh⊙f ðtk , xk Þ, x0 R Case 3. Let us suppose that x(t) has a switching point type (1) on the interval [0, T] as like as ξ, and also assume that t0, t1, …, tj, ξ, tj+1, …, tN. So the fuzzy Euler method is obtained as follows: 8 h   > > > xk + 1 ¼ xk  ⊙ f tk + 1 , x∗k + 1 f ðtk , xk Þ , > > 2 > > ∗ > k ¼ 0,1, …, j > < xk + 1 ¼ xk h⊙f ðtk , xk Þ, h   > > xk + 1 ¼ xk H ð
1Þ ⊙ f tk + 1 , x∗k + 1 f ðtk , xk Þ , > > 2 > > > x∗k + 1 ¼ xk H ð
1Þh⊙f ðtk , xk Þ, k ¼ j + 1, j + 2, …, N
1 > > : x0 R Case 4. If ξ is a type (2) switching point, the fuzzy Euler method takes the form as follows: 8 h   > > xk + 1 ¼ xk  ⊙ f tk + 1 , x∗k + 1 f ðtk , xk Þ , > > > 2 > > ∗ > k ¼ j + 1, j + 2, …, N
1 > < xk + 1 ¼ xk h⊙f ðtk , xk Þ, h   >x ∗ > k + 1 ¼ xk H ð
1Þ ⊙ f tk + 1 , xk + 1 f ðtk , xk Þ , > > 2 > > ∗ > k ¼ 0, 1, …, j > >xk + 1 ¼ xk H ð
1Þh⊙f ðtk , xk Þ, : x0 R

5.3.1

ANALYSIS

OF THE FUZZY MODIFIED

EULER

METHOD

As with the fuzzy Euler method, in this section, consistency, convergence, and stability of the modified Euler method are discussed in detail.

5.3.1.1 Local truncation error and consistency For all abovementioned cases of the fuzzy Euler method, we can define the local truncation error by using a residual value as follows:    f ðtk + 1 , x∗ ðtk + 1 ÞÞf ðtk , xðtk ÞÞ Rk ¼ xðtk + 1 ÞgH xðtk Þh⊙ 2

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Soft Numerical Computing in Uncertain Dynamic Systems ¼

h3 000 ⊙x ðη Þ 3! i
gH k

or:    f ðtk + 1 , x∗ ðtk + 1 ÞÞf ðtk , xðtk ÞÞ Rk ¼ xðtk + 1 ÞgH xðtk ÞH ð
1Þ 2 ¼ H ð
1Þ

h3 000 ⊙x ðη Þ 3! ii
gH k

and the mentioned error is defined as: 1 h2 τk ¼ Rk ¼ ⊙x000 ðη Þ 3! i
gH k h or: 1 h2 ðη Þ τk ¼ Rk ¼ H ð
1Þ ⊙x000 3! ii
gH k h Note. In four cases, the residual values are in two mentioned forms. Now we claim that the methods are said to be consistent if: lim max DH ðτk , 0Þ ¼ 0

h!0 0tk T

00 00 (ηk) and xii
gH (ηk) are bounded To this end, we suppose that the functions xi
gH and this means:     000 9M1 > 0; DH x000 i
gH ðηk Þ, 0  M1 , 9M2 > 0; DH xii
gH ðηk Þ, 0  M2

Thus the methods will be consistent, because:  2  h 000 ⊙x ðη Þ, 0 lim max DH ðτk , 0Þ ¼ lim max DH h!0 0tk T h!0 0tk T 3! i
gH k   h h ð η Þ, 0  lim M1 ¼ 0 ¼ lim max DH x000 i
gH k h!0 2 0tk T h!0 2 The same process is true for the ii
gH differentiability:   h2 ð η Þ, 0 lim max DH ðτk , 0Þ ¼ lim max DH H ð
1Þ ⊙x000 h!0 0tk T h!0 0tk T 3! ii
gH k        h2  h2  max DH x000 ¼ lim ð
1Þ  max DH H x000 ii
gH ðηk Þ, 0 ¼ lim ii
gH ðηk Þ, 0 h!0 h!0 3! 0tk T 3! 0tk T h2 M2 ¼ 0 h!0 3!

 lim

Discrete numerical solutions of uncertain differential equations

231

So the fuzzy modified Euler method in each case is consistent so long as the fuzzy number valued solution is in C3gH ð½0, T , R Þ:

5.3.1.2 Global truncation error and convergence The global truncation error is the agglomeration of the local truncation error over all the iterations, assuming perfect knowledge of the true solution at the initial time step. Formally, for each case (1) and (2) the global truncation error is noted by ek+1 at the point tk+1 and defined as: ek + 1 ¼ xðtk + 1 ÞgH xk + 1 ¼        f t0 , x∗0 f ðt0 , x0 Þ f t1 , x∗1 f ðt1 , x1 Þ ¼ xðtk + 1 ÞgH x0 h⊙ h⊙ 2 2    f tk , x∗k f ðtk , xk Þ ⋯h⊙ 2 and in case (2): ek + 1 ¼ xðtk + 1 ÞgH xk + 1 ¼      f t0 , x∗0 f ðt0 , x0 Þ H ð
1Þh⊙ ¼ xðtk + 1 ÞgH x0 H ð
1Þh⊙ 2       f t1 , x∗1 f ðt1 , x1 Þ f tk , x∗k f ðtk , xk Þ H ð
1Þ⋯H ð
1Þh⊙ 2 2 for any k ¼ 0, 1, …, N
1. We claim that the fuzzy Euler method is convergent if the global truncation error goes to zero as the step size goes to zero; in other words, the numerical solution converges to the exact solution if:  lim max DH ðek + 1 , 0Þ ¼ 0 ) lim max DH xðtk + 1 ÞgH xk + 1 , 0 h!0

k

h!0

k

¼ lim max DH ðxðtk + 1 Þ, xk + 1 Þ ¼ 0 h!0

k

5.3.1.3 Theorem—Convergence 00 Suppose that xgH (t) exists and also f(t, x(t)) satisfies the Lipschitz condition. Then the fuzzy modified Euler method converges to the fuzzy solution of fuzzy initial value problem:

x0gH ðtÞ ¼ f ðt, xÞ, xðt0 Þ ¼ x0 R , t0  t  T

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Soft Numerical Computing in Uncertain Dynamic Systems

To prove the theorem, consider that x(t) is i
gH differentiable and also suppose that: dk ≔

h3 000 ⊙x ðt k Þ 3! i
gH

By this assumption, the exact solution satisfies the following relation:   f ðtk + 1 , x∗ ðtk + 1 ÞÞf ðtk , xðtk ÞÞ dk xðtk + 1 Þ ¼ xðtk Þh⊙ 2 On the other hand, we have:    f tk + 1 , x∗k + 1 f ðtk , xk Þ , k ¼ 0, 1,…, N
1 xk + 1 ¼ xk h⊙ 2 x∗k + 1 ¼ xk h⊙f ðtk , xk Þ, Now let us consider the distances of both sides of the abovementioned relations, so: DH ðxðtk + 1 Þ, xk + 1 Þ ¼ DH ðxðtk Þ, xk Þ +

   h  DH ðf ðtk + 1 , x∗ ðtk + 1 ÞÞf ðtk , xðtk ÞÞÞ, f tk + 1 , x∗k + 1 f ðtk , xk Þ 2 + D H ð dk , 0 Þ

Since the function f(t, x(t)) satisfies the Lipschitz condition, then:

DH



DH ðf ðtk , xðtk ÞÞ, f ðtk , xk ÞÞ  Lk DH ðxðtk Þ, xk Þ   f ðtk + 1 , x∗ ðtk + 1 ÞÞ, f tk + 1 , x∗k + 1  Lk + 1 DH x∗ ðtk + 1 Þ, x∗k + 1

Using the triangular inequality and Lipschitz condition, we arrive at:    DH ðf ðtk + 1 , x∗ ðtk + 1 ÞÞf ðtk , xðtk ÞÞÞ, f tk + 1 , x∗k + 1 f ðtk , xk Þ    DH f ðtk + 1 , x∗ ðtk + 1 ÞÞ, f tk + 1 , x∗k + 1 + DH ðf ðtk , xðtk ÞÞ, f ðtk , xk ÞÞ   Lk + 1 DH x∗ ðtk + 1 Þ, x∗k + 1 + Lk DH ðxðtk Þ, xk Þ then: DH ðxðtk + 1 Þ, xk + 1 Þ  DH ðxðtk Þ, xk Þ +

  h

Lk + 1 DH x∗ ðtk + 1 Þ, x∗k + 1 + Lk DH ðxðtk Þ, xk Þ 2

+DH ðdk , 0Þ

Discrete numerical solutions of uncertain differential equations

233

Suppose that: L ¼ max fLk , Lk + 1 g 0kN
1

then:  Lh  ∗ DH x ðtk + 1 Þ, x∗k + 1 + DH ðxðtk Þ, xk Þ 2

DH ðxðtk + 1 Þ, xk + 1 Þ  DH ðxðtk Þ, xk Þ +

On the other hand, we have the Lipschitz condition for the fuzzy Euler method:  DH x∗ ðtk + 1 Þ, x∗k + 1  DH ðxðtk Þ + h⊙f ðtk , xðtk ÞÞ, xk + h⊙f ðtk , xk ÞÞ  ð1 + hLk ÞDH ðxðtk Þ, xk Þ  ð1 + hLÞDH ðxðtk Þ, xk Þ Then it is obtained that: DH ðxðtk + 1 Þ, xk + 1 Þ  DH ðxðtk Þ, xk Þ +

Lh ½ð1 + hLÞDH ðxðtk Þ, xk Þ + DH ðxðtk Þ, xk Þ 2

and: DH ðxðtk + 1 Þ, xk + 1 Þ  DH ðxðtk Þ, xk Þ +

Lh ½ð2 + hLÞDH ðxðtk Þ, xk Þ 2

Finally: ! ðLhÞ2 DH ðxðtk + 1 Þ, xk + 1 Þ  1 + Lh + DH ðxðtk Þ, xk Þ + DH ðdk , 0Þ,k ¼ 0,1, …, N
1 2 Also consider that: d ¼ max DH ðdk , 0Þ 0kN
1

then we have: ! ðLhÞ2 DH ðxðtk + 1 Þ, xk + 1 Þ  1 + Lh + DH ðxðtk Þ, xk Þ + d, k ¼ 0,1, …,N
1 2 By backward substituting: ðLhÞ2 DH ðxðtk + 1 Þ, xk + 1 Þ  1 + Lh + 2 ðLhÞ2 ¼ 1 + Lh + 2

!2

!"

# ! ðLhÞ2 1 + Lh + DH ðxðtk
1 Þ, xk
1 Þ + d + d 2 "

ðLhÞ2 DH ðxðtk
1 Þ, xk
1 Þ + d 1 + 1 + Lh + 2

!#

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Soft Numerical Computing in Uncertain Dynamic Systems

Continuing this process: ðLhÞ2 DH ðxðtk + 1 Þ, xk + 1 Þ  1 + Lh + 2 2

ðLhÞ2 +d 41 + 1 + Lh + 2

!

!k + 1 DH ðxðt0 Þ, x0 Þ

ðLhÞ2 + ⋯ + 1 + Lh + 2

!k 3 5

The last term does have the general formula and actually it is a geometric series: ðLhÞ2 1 + 1 + Lh + 2

!

ðLhÞ2 + ⋯ + 1 + Lh + 2 ¼

!k  1 + eLh + ⋯ + ekLh

eðk + 1ÞLh
1 eLh

since: 1 + Lh +

ðLhÞ2  eLh 2

By substituting: ðLhÞ2 DH ðxðtk + 1 Þ, xk + 1 Þ  1 + Lh + 2 +d

eðk + 1ÞLh
1 eLh

!k + 1 DH ðxðt0 Þ, x0 Þ

thus: ðk + 1ÞLh

e
1 DH ðxðtk + 1 Þ, xk + 1 Þ  e DH ðxðt0 Þ, x0 Þ + d eLh Lh

On the other hand: ðk + 1Þh  T ) eðk + 1ÞLh  eLT By substituting: d ¼ max DH ðdk , 0Þ ¼ 0kN
1

  h3 max DH x000 ð t Þ, 0 , DH ðxðt0 Þ, x0 Þ ¼ 0 k i
gH 3! 0tT

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235

Finally:

  h3 eLT
1 00 max D x ð t Þ, 0 H i
gH 0tT 3! eLh

 M1 h3 eLT
1 ¼ O h2  3! eLh

DH ðxðtk + 1 Þ, xk + 1 Þ 

Taking the limit of the two sides: lim DH ðxðtk + 1 Þ, xk + 1 Þ ¼ 0

h!0

Now consider that x(t) is ii
gH differentiable without any switching point and: dk ≔H ð
1Þ

h3 000 ⊙x ðt k Þ 3! ii
gH

Here we would like to change the strategy to prove the case. By this assumption, the iterative method is as follows: xk + 1 ¼ xk H ð
1Þh⊙ϕðtk , xk , hÞ where: 1 ϕðtk , xk , hÞ ¼ ½ f ðtk , xk Þf ðtk + h, xk H ð
1Þh⊙f ðtk , xk ÞÞ 2 where ϕ(tk, xk, h) is a continuous fuzzy function. Now using the Lipschitz condition on ϕ(tk, xk, h) in terms of the second terms: 1 DH ðϕðtk , xðtk Þ, hÞ, ϕðtk , xk , hÞÞ  DH ðf ðtk , xðtk ÞÞ, f ðtk , xk ÞÞ 2 1 + DH ðf ðtk + h, xðtk ÞH ð
1Þh⊙f ðtk , xðtk ÞÞÞ, f ðtk + h, xk H ð
1Þh⊙f ðtk , xk ÞÞÞ 2 L  DH ðxðtk Þ, xk Þ + 2 L DH ðxðtk ÞH ð
1Þh⊙f ðtk , xðtk ÞÞ, xk H ð
1Þh⊙f ðtk , xk ÞÞ 2 using the properties of the distance: L L hL  DH ðxðtk Þ, xk Þ + DH ðxðtk Þ, xk Þ
DH ðf ðtk , xðtk ÞÞ, f ðtk , xk ÞÞ 2 2 2   hL Lh DH ðxðtk Þ, xk Þ  LDH ðxðtk Þ, xk Þ + LDH ðxðtk Þ, xk Þ ¼ L 1
2 2

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Soft Numerical Computing in Uncertain Dynamic Systems

It should be noted that the L is the maximum value of all Lipschitz constants. Finally: DH ðxðtk + 1 Þ, xk + 1 Þ 

   h2

1
e
L1 T max DH x000 ii
gH ðtÞ, 0 0tT 3!L1

where:   Lh L1 ¼ L 1 + 2 Taking the limit of the two sides: lim DH ðxðtk + 1 Þ, xk + 1 Þ ¼ 0

h!0

5.3.1.4 Stability of the modified fuzzy Euler method With the same illustrations about the fuzzy Euler method, we shall now discuss the stability of the modified method. To do this, again we assume that there is the same perturbation δ0 in the fuzzy initial value and we try to show: DH ðzk + 1 , yk + 1 Þ  κδ, 8ðk + 1Þh  T, k  N
1 whenever DH(δ0, 0)  δ. Now suppose that the solution x(t) is i
gH differentiable and consider the perturbed problem as follows: h zk + 1 ¼ zk  ⊙ðf ðtk , zk h⊙f ðtk , zk ÞÞf ðtk , zk ÞÞ, z0 ¼ x0 δ0 2 and also consider the mentioned iterative method in case (1): h xk + 1 ¼ xk  ⊙ðf ðtk , xk h⊙f ðtk , xk ÞÞf ðtk , xk ÞÞ 2 Using the distance, we will get: D H ðzk + 1 , xk + 1 Þ  DH ðzk , xk Þ +

Lh Lh DH ðzk h⊙f ðtk , zk Þ, xk h⊙f ðtk , xk ÞÞ + DH ðzk , xk Þ 2 2

 D H ðzk , xk Þ +

Lh Lh ðDH ðzk , xk Þ + hLDH ðzk , xk ÞÞ + DH ðzk , xk Þ 2 2   2 2 Lh DH ðzk , xk Þ ¼ 1 + Lh + 2

Discrete numerical solutions of uncertain differential equations

237

Finally:   L2 h2 D H ðzk , xk Þ DH ðzk + 1 , xk + 1 Þ  1 + Lh + 2 and:  k + 1 L2 h2 DH ðz0 , x0 Þ  ehLðk + 1Þ DH ðz0 H x0 , 0Þ DH ðzk + 1 , xk + 1 Þ  1 + Lh + 2  eLT DH ðδ0 , 0Þ  κδ where κ ¼ eLT. So based on the previous illustrations the method is stable. In the case that the solution x(t) is ii
gH differentiable, we consider the perturbed problem as follows: h zk + 1 ¼ zk H ð
1Þ ⊙ðf ðtk , zk h⊙f ðtk , zk ÞÞf ðtk , zk ÞÞ, z0 ¼ x0 δ0 2 and also consider the mentioned iterative method in case (2): h xk + 1 ¼ xk H ð
1Þ ⊙ðf ðtk , xk h⊙f ðtk , xk ÞÞf ðtk , xk ÞÞ 2 Using the distance, we will get: DH ðzk + 1 , xk + 1 Þ  DH ðzk , xk Þ


Lh Lh DH ðzk h⊙f ðtk , zk Þ, xk h⊙f ðtk , xk ÞÞ
DH ðzk , xk Þ 2 2

 DH ðzk , xk Þ


Lh Lh ðDH ðzk , xk Þ + hLDH ðzk , xk ÞÞ
DH ðzk , xk Þ 2 2   L2 h2 DH ðzk , xk Þ ¼ 1
Lh + 2

Finally:   L 2 h2 DH ðzk , xk Þ DH ðzk + 1 , xk + 1 Þ  1
Lh + 2 and:  k + 1 L2 h2 DH ðz0 , x0 Þ  e
hLðk + 1Þ DH ðz0 H x0 , 0Þ DH ðzk + 1 , xk + 1 Þ  1
Lh + 2  e
LT DH ðδ0 , 0Þ  κδ

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Soft Numerical Computing in Uncertain Dynamic Systems

where κ ¼ e
LT. So based on the previous illustrations the method is stable. To compare the fuzzy Euler method and fuzzy modified Euler method, we reconsider the previous examples (Allahviranloo et al., 2007). Example. The following fuzzy initial value problem is assumed:  0 xi
gH ðtÞ ¼ xðtÞð1:3; 2, 2:1Þ, 0  t  1 xð0Þ ¼ ð0:82; 1, 1:2Þ where (1.3,2, 2.1) and (0.82,1, 1.2) are triangular fuzzy numbers. The exact solution is obtained very easily, and it is: xðtÞ ¼ ðxð0Þð1:3; 2, 2:1ÞÞet H ð1:3; 2, 2:1Þ The fuzzy Euler method in the sense of i
gH differentiability has the following form: h  xk + 1 ¼ xk  ⊙ x∗k + 1 ð1:3; 2, 2:1Þxk ð1:3; 2, 2:1Þ , k ¼ 0,1, …, N
1 2 x∗k + 1 ¼ xk h⊙ðxk ð1:3; 2, 2:1ÞÞ, x0 ¼ ð0:82; 1, 1:2Þ After substituting and simplifying:   h2 h3 h xk  ⊙ð1:3; 2, 2:1Þ ð2:6; 4, 4:2Þ xk + 1 ¼ 1 + h + 2 2 2 x0 ¼ ð0:82; 1, 1:2Þ,

k ¼ 0, 1, …,N
1

The global truncation errors have been reported for h ¼ 0.025 and h ¼ 0.005 in Table 5.4 to compare the results of the fuzzy Euler method and modified method. It is seen that the results of the modified method are much better than the Euler method. This is because the order of convergence of the modified method is higher than that of the fuzzy Euler method. Example. Let us consider the following fuzzy initial value problem:  0 xii
gH ðtÞ ¼
xðtÞt⊙ð0:7; 1, 1:8Þ, 0  t  1 xð0Þ ¼ ð0, 1, 2:2Þ where (0.7,1, 1.8) and (0, 1, 2.2) are triangular fuzzy numbers and their parametric forms are, respectively, as: ð0:7; 1, 1:8Þ½r  ¼ ½0:7 + 0:3r, 1:8
0:8r , ð0, 1, 2:2Þ½r  ¼ ½r, 2:2
1:2r  The exact solution of the fuzzy initial value problem can be obtained easily by using the parametric form of the problem and considering the ii
gH differentiability of the solution. Here is the exact fuzzy number valued solution function: xðtÞ ¼ C⊙e
t ð0:7; 1, 1:8Þ⊙ðt
1Þ, C ¼ ð0, 1, 2:2ÞH ð
1Þð0:7; 1, 1:8Þ

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239

TABLE 5.4 Comparison of global errors.

In the previous section, the figure of the exact solution and the process to find it were explained. Now the ii
gH solution can be approximated by using case (2) in the fuzzy modified Euler method: 8 h   > < xk + 1 ¼ xk H ð
1Þ ⊙ f tk + 1 , x∗k + 1 f ðtk , xk Þ , k ¼ 0,1, …, N
1 2 ∗ > : xk + 1 ¼ xk H ð
1Þh⊙f ðtk , xk Þ, x0 ¼ ð0, 1, 2:2Þ For this example, it is renewed as: h  xk + 1 ¼ xk H ð
1Þ ⊙ ð
1Þx∗k + 1 tk + 1 ⊙ð0:7; 1, 1:8Þð
1Þxk tk ⊙ð0:7;1, 1:8Þ , 2 x∗k + 1 ¼ xk H ð
1Þh⊙ðð
1Þxk tk ⊙ð0:7;1, 1:8ÞÞ, x0 ¼ ð0, 1, 2:2Þ, k ¼ 0, 1,…, N
1 This can be simplified in the following format: h xk + 1 ¼ xk H ⊙ð½xk H h⊙ðxk tk ⊙ð
1:8,
1,
0:7ÞÞtk + 1 2 ⊙ð
1:8,
1,
0:7Þxk tk ⊙ð
1:8,
1,
0:7ÞÞ

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With the same explanation about the global error and its propagation, Table 5.5 shows us the comparison of the global errors’ propagations with the same step sizes for the classic fuzzy Euler method and its modification. Here also, the results of modified method are more accurate than the classic one. Example. Consider the following fuzzy initial value problem, which is the same as the previous one:  0 xgH ðtÞ ¼ ð1
tÞ⊙xðtÞ, 0t2 xð0Þ ¼ ð0, 1, 2Þ with a switching point at t ¼ 1. •

In the case of i
gH differentiability, the fuzzy modified Euler method is reformed in the following form, which is defined in the interval 0  tk < 1: h  xk + 1 ¼ xk  ⊙ ð1
tk + 1 Þ⊙x∗k + 1 ð1
tk Þ⊙xk , 2 x∗k + 1 ¼ xk h⊙ð1
tk Þ⊙xk ,

TABLE 5.5 Comparison of global errors.

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By substituting: h xk + 1 ¼ xk  ⊙ðð1
tk + 1 Þ⊙xk hð1
tk + 1 Þð1
tk Þ⊙xk ð1
tk Þ⊙xk Þ, 2  

h2 xk + 1 ¼ 1 + h + ð1
tk + 1 Þ ð1
tk Þ ⊙xk , 2

•

In the case of ii
gH differentiability in the interval 1 < tk  2: h  xk + 1 ¼ xk H ð
1Þ ⊙ ð1
tk + 1 Þ⊙x∗k + 1 ð1
tk Þ⊙xk , 2 x∗k + 1 ¼ xk H ð
1Þh⊙ð1
tk Þ⊙xk By substituting the x∗k+1 in x∗k+1 we will find: h xk + 1 ¼ xk H ð
1Þ ⊙ðð1
tk + 1 Þ⊙½xk H ð
1Þh⊙ð1
tk Þ⊙xk ð1
tk Þ⊙xk Þ, 2

Clearly, it can be simplified more but we skip it to the author. The comparison of the results between the fuzzy Euler and the modified one are shown in Table 5.6. The accuracy of the modified method can be observed in the results.

TABLE 5.6 Global error results.

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5.4 Fuzzy Euler method for fuzzy hybrid differential equations The quadratic perturbations of nonlinear differential equations have attracted much attention recently. Such differential equations hybrid differential equations can be called hybrid differential equations. Here we are going to discuss the fuzzy version. In this section, the fuzzy Euler method is presented to solve hybrid fuzzy differential equations with fuzzy initial value. To start the discussion, we define the fuzzy hybrid differential equations in the following: x0gH ðtÞ ¼ f ðt, xðtÞ, λk ðxk ÞÞ, xðtk Þ ¼ xk , t½tk , tk + 1  where 0  t0 < t1 < t2 < ⋯ < tk ,tk ! 0, f C½R +  R  R , R , λk C½R  R : These mean that the function f is a fuzzy continuous function and also λk is the fuzzy parameter of the fuzzy initial value. In this problem, the differential equation is defined on each subinterval [tk, tk+1] and indeed we have piecewise differential equations or piecewise fuzzy initial value problems. In each subinterval, the fuzzy differential equation has a unique fuzzy solution. It is shown as the following piecewise equations: 8 f ðt, xðtÞ, λ0 ðx0 ÞÞ, xðt0 Þ ¼ x0 , t½t0 , t1  > > > > < f ðt, xðtÞ, λ1 ðx1 ÞÞ, xðt1 Þ ¼ x1 , t½t1 , t2  x0gH ðtÞ ¼ ⋮ > > ð ð x Þ Þ, x ð tk Þ ¼ xk , t½tk , tk + 1  f t, x ð t Þ, λ > k k > : ⋮ But in general, we denote the fuzzy solution function as x(t) ≔ xk(t) that is a function of t and the initial value, which is defined at the subinterval [tk, tk+1], and show it as: 8 x0 ðtÞ, t½t0 , t1  > > < x1 ðtÞ, t½t1 , t2  xðtÞ ¼ xðt, t0 , x0 Þ ¼ ⋮ > > : xk ðtÞ, t½tk , tk + 1  As was explained earlier, for the existence and uniqueness of the solution we need the Lipschitz condition; and here, for the fuzzy hybrid differential equation, there is a unique fuzzy solution if and only if each piece does have a unique solution and if and only if each piece satisfies the Lipschitz condition. Now the fuzzy Euler method is used for the hybrid problem and this means that the fuzzy Euler method can be used for each piece even with different step size. Again, it is discussed in four cases as before. Case 1. The fuzzy number valued solution function x(t) is (i
gH)-differentiable and the type of differentiability does not change on [0, T]:  xk, n + 1 ¼ xk,n hk ⊙f ðt, xk,n , λk ðxk, n ÞÞ, n ¼ 0,1, …,N
1 x0 R

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Case 2. The solution x(t) is (ii
gH)-differentiable and the type of differentiability does not change on [0, T]: 

xk,n + 1 ¼ xk, n H ð
1Þh⊙f ðt, xk,n , λk ðxk,n ÞÞ, n ¼ 0, 1,…, N
1 x0 R

Case 3. Let us suppose that x(t) has a switching point type (1) on the interval [0, T] as like as ξ, and also assume that t0, t1, …, tj, ξ, tj+1, …, tN. So the fuzzy Euler method is obtained as follows: 8 n ¼ 0,1, …, j < xk, n + 1 ¼ xk,n h⊙f ðt, xk, n , λk ðxk,n ÞÞ, xk, n + 1 ¼ xk, n H ð
1Þh⊙f ðt, xk,n , λk ðxk,n ÞÞ, n ¼ j + 1, j + 2, …, N
1 : x0 R Case 4. If ξ is a type (2) switching point, the fuzzy Euler method takes the form as follows: 8 < xk, n + 1 ¼ xk, n H ð
1Þh⊙f ðt, xk,n , λk ðxk, n ÞÞ, ¼ xk, n h⊙f ðt, xk,n , λk ðxk, n ÞÞ, x : k, n + 1 x0 R

n ¼ 0, 1, …, j n ¼ j + 1, j + 2, …, N
1

We have the same properties of the fuzzy Euler method for these fuzzy hybrid problems. We shall now solve some numerical examples. Example. Consider the following fuzzy hybrid differential equations: x0gH ðtÞ ¼ ð
1Þ⊙xðtÞmðtÞ⊙λk ðxðtk ÞÞ, tk ¼ k, k ¼ 0,1, 2, … xð0Þ ¼ ð0:75; 1, 1:25Þ where:  mðtÞ ¼

2ðtð mod 1ÞÞ, tð mod 1Þ < 0:5 , λ ðμÞ ¼ 2ð1
tð mod 1ÞÞ, tð mod 1Þ > 0:5 k



0, k¼0 μ, kf1, 2, …g

The level-wise form of the triangular fuzzy initial value is as: xð0, r Þ ¼ ½0:75 + 0:25r, 1:125
0:125r : The abovementioned fuzzy hybrid differential equation is equivalent to the following problem:  0 x0, gH ðtÞ ¼ ð
1Þ⊙x0 ðtÞ, 0  t  1 xð0, r Þ ¼ ½0:75 + 0:25r, 1:125
0:125r   0 xk, gH ðtÞ ¼ ð
1Þ⊙xk ðtÞmðtÞ⊙xk ðtk Þ, tk  t  tk + 1 xk ðtk Þ ¼ xk
1 ðtk Þ, k ¼ 1, 2, …

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To this purpose, first of all the first fuzzy differential equation should be solved to find the fuzzy solution x0(t). It is obtained by the fuzzy Euler method, but for comparing the approximate solution and the exact solution, we need to investigate the exact solution and it is in the level-wise [x0, l(r), x0, u(r)] form and does have the following relations, which were discussed in Chapter 4 It should be observed that we have these notations here: x0 ðt, r Þ ¼ ½x0, l ðr Þ, x0, u ðr Þ, x0 ðr Þ ¼ ½xl,0 ðr Þ, xu,0 ðr Þ To find the exact solution in level-wise form, we use the parametric form or levelwise form of the derivative. If type (1) of the differentiability is used, we will see that the fuzzy solution will not satisfy our differential equations, which means the type of differentiability is changed to type (2). So in the following, we use the routine levelwise form and we find a system of two lower and upper form differential equations and the solutions in lower and upper case are as follows:

where A ¼ 12 exp

2

t
t20 2

x0,l ðt, r Þ ¼

A
B A+B xu,0 ðr Þ + xl,0 ðr Þ 2 2

x0, u ðt, r Þ ¼

A+B A
B xu,0 ðr Þ + xl,0 ðr Þ 2 2

 , B ¼ A1 .

At the point t ¼ 1, t0 ¼ 0 we have: 0 0 1 1     C C 1B 1 2 1B 1 2  C  C exp + exp
x0, l ð1, r Þ ¼ B xu,0 ðr Þ + B @ A @ Axl,0 ðr Þ 1 1 2 2 2 2 exp exp 2 2 0 0 1 1     C C 1B 1 2 1B 1 2  C  C exp
exp + x0, u ð1, r Þ ¼ B xu,0 ðr Þ + B xl, 0 ðr Þ @ @ A 1 1 A 2 2 2 2 exp exp 2 2 xl,0 ðr Þ ¼ 0:75 + 0:25r, xu,0 ðr Þ ¼ 1:125
0:125r Now in the sense of ii-differentiability we will have the right solution and it is: x0 ðtÞ ¼ x0 ð0Þexp ðtÞ, x0 ðt, r Þ ¼ ½ð0:75 + 0:25r Þexp ð
tÞ, ð1:125
0:125r Þ exp ð
tÞ and at the point t ¼ 1 it is: x0 ð1, r Þ ¼ ½ð0:75 + 0:25r Þexp ð
1Þ, ð1:125
0:125r Þ exp ð
1Þ

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To find the approximate solution, we use the fuzzy Euler method in the sense of ii-differentiability and we have: x0,n + 1 ¼ x0, n H ð
1Þh⊙f ðtn , x0,n Þ ¼ x0,n H ð
1Þh⊙ð
1Þ⊙x0, n ¼ x0,n H h⊙x0, n ¼ ð1
hÞ⊙x0, n ¼ ð1
hÞ2 ⊙x0, n
1 ¼ ⋯   1 n+1 n+1 ⊙x0,0 ¼ ð1
hÞ ⊙x0,0 ¼ 1
n+1 In the level-wise form it is: 

 1 n+1 ð0:75 + 0:25r Þ x0,n + 1,l ðr Þ ¼ ð1
hÞ x0,0,l ðr Þ ¼ 1
n+1   1 n+1 ðð1:125
0:125r ÞÞ x0, n + 1,u ðr Þ ¼ ð1
hÞn + 1 x0, 0,u ðr Þ ¼ 1
n+1 n+1

For instance, for nine iterations we have: "     # 1 10 1 10 x0 ð1, r Þ  x0, n ðr Þ ¼ ð0:75 + 0:25r Þ 1
, ð1:125
0:125r Þ 1
10 10 for any 0  r  1. Now we use the fuzzy Euler method to solve the second part of the hybrid differential equation:  0 xk, gH ðtÞ ¼ ð
1Þ⊙xk ðtÞmðtÞ⊙xk ðtk Þ, tk  t  tk + 1 xk ðtk Þ ¼ xk
1 ðtk Þ, k ¼ 1, 2, … If we choose k ¼ 1 then x1(t1) ¼ x0(t1) and from the first part we found the approximate values of x0(t) at any points including t1, so we already have computed the x0(t1) and also the same for x1(t1). Since t(mod 1) < 0.5 for any tk  [0, 1] with the step size 0.1 then m(tk) ¼ 2(tk(mod 1)) is computed easily. Then using the fuzzy Euler method is easier because all information is provided. Indeed, in each piece we use the fuzzy Euler method at 11 points. Now the fuzzy Euler method is used in the sense of ii-differentiability and we have: For k ¼ 1:  x1, n + 1 ¼ x1, n H ð
1Þh⊙ð
1Þ⊙x1, n mðtn Þ⊙x1 ðt1 Þ, n ¼ 0, 1, …,9 x0 ð0, r Þ ¼ ½ð0:75 + 0:25r Þ, ð1:125
0:125r Þ or: 

x1, n + 1 ¼ x1,n H h⊙x1, n mðtn Þ⊙x1 ðt1 Þ, n ¼ 0, 1,…, 9 x0 ð0, r Þ ¼ ½ð0:75 + 0:25r Þ, ð1:125
0:125r Þ

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since: "

x1 ðt1 Þ ¼ x0 ðt1 Þ ¼ x0 ð0:1Þ ¼ x0,1 ðr Þ

    # 1 2 1 2 ¼ ð0:75 + 0:25r Þ 1
, ð1:125
0:125r Þ 1
2 2 Then in the subinterval [t1, t2], the fuzzy Euler method has been shown as the following form: 8 ðtn Þ⊙x1 ðt1 Þ, n ¼ 0, 1,…, 9 > < x1, n + 1 ¼ x"1, n H h⊙x1,n m    # 1 2 1 2 > : x0,1 ðr Þ ¼ ð0:75 + 0:25r Þ 1
2 , ð1:125
0:125r Þ 1
2 After finding the x1,10 then we choose k ¼ 2, repeat the same procedure, and find the x2,10  x(t2). By repeating this process on each subinterval and using the Euler method in each subinterval with n ¼ 0, 1, …, 9, the solution in each subinterval is found (Allahviranloo & Salahshour, 2011).

5.5 Fuzzy Euler method for fuzzy impulsive differential equations In this section we shall find the numerical solution of fuzzy impulsive differential equations by using both the fuzzy Euler and fuzzy modified Euler methods. Generally, the analytical solution of these fuzzy impulsive equations is a complex solution and usually it is impossible to obtain, and therefore the numerical solution is suggested. The fuzzy impulsive differential equation is introduced in the following form: 8 0 < xgH ðtÞ ¼ f ðt, xðtÞÞ, tJ ¼ ½0, T , t 6¼ tk ,k ¼ 1,2,…, N Δxjt¼tk ¼ Ik ðxðtk ÞÞ, t ¼ tk , k ¼ 1, 2, …,N : xðt0 Þ ¼ x0 where: x0 R ,

f : J  R ! R ,

Ik : R ! R ,

k ¼ 1, 2, …,N

and:   Δxjt¼tk ¼ x tk+ gH x t
k , k ¼ 1, 2,…, N where:   + lim
xðtÞ ¼ x t
k , lim+ xðtÞ ¼ x tk

t!tk

t!tk

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Indeed, it seems that we have some points like tk, k ¼ 1, 2, …, N, and at these points the fuzzy derivative cannot be defined and there is only a fuzzy difference, but between these points or in the subinterval a fuzzy differential equation is defined. So in the arbitrary subinterval (tk, tk+1], the fuzzy differential equations is solved numerically or analytically, then the obtained solution is determined at the point tk+1, again at this point we have only difference (one can call it by a jump from the left limit to the right limit), the difference is a fuzzy difference and it can be defined in two cases. For more clarity, the procedure of finding the numerical solution of the impulsive equation is explained by the following steps: Step 1. We use the fuzzy Euler method for the impulsive equation in the interval (tk, tk+1]. It should be noted that in each half segment or subinterval the initial value for the fuzzy Euler method is as x0 ¼ x(t+k ). It is clear that the Euler method can be used in two cases of differentiability. Step 2. The numerical solution obtained from step 1, is used in the difference function, at the point t ¼ tk+1:   ¼ Ik ðxðtk Þh⊙f ðtk , xðtk ÞÞÞ Δxjt¼tk + 1 ¼ Ik x t
k+1 On the other hand:    
x tk++ 1 gH x t
k + 1 ¼ Ik ðxðtk Þh⊙f ðtk , xðtk ÞÞÞ ¼ I x tk + 1 The x(tk) is a scalar and we know:  xðtÞ ¼ x t
lim k+1 ,
t!tk + 1

 xðtÞ ¼ x tk++ 1 lim +

t!tk + 1

Now if I(x(t
k+1)) is a fuzzy number then the gH-difference is defined and:    
x tk++ 1 gH x t
k + 1 ¼ I x tk + 1   +  

+ 
 
¼ I x tk + 1 x tk +1 x t x tk + 1 gH x tk + 1 ¼ I x tk + 1 ¼  k
+ 1 x tk + 1 ¼ x tk++ 1 ð
1ÞI x t
k+1 Now in each subinterval, (tk, tk+1], x(t+k ) is the initial point for the Euler method and by applying the fuzzy Euler method and even the modified version, we will find + the x(t
k+1), and this satisfies the jump, so we have obtained the x(tk+1). At each stage, we determine the left limit point and right limit point, and the jumps are skipped, so to find the final solution, all the jumps must be added to the approximate solution. This process is continued in every subinterval. Step 3. The steps 1 and 2 are repeated while tk+1  tz where tz is known and fixed time. Step 4. All the jumps are added to the solution to introduce the general solution of the fuzzy impulsive differential equations. See the abovementioned explanations.

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Soft Numerical Computing in Uncertain Dynamic Systems

ERROR

ANALYSIS

In the numerical investigation of the problem, we know that we need the convergence solution, consistency of the problem, and the stability of the method. The presented numerical methods have been analyzed before in accordance with convergency and consistency, and now we only discuss the stability of the fuzzy impulsive problem.

5.5.1.1 Stability To investigate the stability of the problem, we add a small perturbation δ3 in the fuzzy initial value. Clearly, it causes some small perturbations like δ2, δ1 in the other equations. It is expected that the created or forced perturbation δ in the solution of the problem should be small as well. To this end, consider the perturbed fuzzy impulsive differential equation in the following form: 8 0 < zgH ðtÞ ¼ f ðt, zðtÞÞðδ1 ⊙f ðt, zðtÞÞÞ, tJ ¼ ½0, T , t 6¼ tk , k ¼ 1, 2, …,N Δzj ¼ Ik ðzðtk ÞÞðδ2 ⊙Ik ðzðtk ÞÞÞ, t ¼ tk , k ¼ 1,2,…, N : t¼tk zðt0 Þ ¼ δ3 ⊙x0 The solution of this perturbed problem is z(t) and we expect that the perturbation in the solution will be δ and: zðtÞ ¼ xðtÞðxðtÞ⊙δÞ where: 8t9E > 0;δN ðxðtÞ, EÞ, 9E2 > 0;δN ðIk ðzðtk ÞÞ, E2 Þ,

9E3 > 0;δN ðz0 , E3 Þ, 8t9E1 > 0;δN ðf ðt, zðtÞÞ, E1 Þ

E ¼ min fE1 , E2 , E3 g Now consider the following relation, which means the perturbation in the Ik(x(tk)) concludes the Ik(z(tk)):    + 
z tk+ gH z t
k ¼ x tk gH x tk ðδ2 ⊙Ik ðxðtk ÞÞÞ Based on the definition of the gH-difference, we have:   +     gH x tk+ gH x t
¼ δ2 ⊙Ik ðxðtk ÞÞ z tk gH z t
k k since:     

z tk+ ¼ x tk+ δ⊙x tk+ , z t
k ¼ x tk δ⊙x tk By substituting:   +    
   gH x tk+ gH x t
x tk δ⊙x tk+ gH x t
k δ⊙x tk k ¼ δ2 ⊙Ik ðxðtk ÞÞ

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249

Assuming all gH-differences exist, for instance, suppose that the:   +    
x tk δ⊙x tk+ gH x t
k δ⊙x tk exists in case (1):   +    
x tk δ⊙x tk+ i
gH x t
k δ⊙x tk then:   +    
xl tk δ⊙xl tk+
xl t
k δ⊙xl tk    
¼ xl tk+
δ⊙xl tk+
xl t
k
δ⊙xl tk and:   +    
xu tk δ⊙xu tk+
xu t
k δ⊙xu tk    
¼ xu tk+
δ⊙u tk+
xu t
k
δ⊙xu tk then the gH-difference in the interval form is as:   +    
x tk δ⊙x tk+ i
gH x t
k δ⊙x tk

 +   
 +  + 

 xl tk
δ⊙xl tk+
xl t
k
δ⊙xl tk , xu tk
δ⊙u tk
xu tk
δ⊙xu tk and also, in case (2) of gH-difference, the interval form is as:   +    
x tk δ⊙x tk+ ii
gH x t
k δ⊙x tk

 +   
 +  + 

 xu tk
δ⊙u tk+
xu t
k
δ⊙xu tk , xl tk
δ⊙xl tk
xl tk
δ⊙xl tk All these mean that in each case of the difference we have the following relations immediately:   +    
   gH x tk+ gH x t
¼ x tk δ⊙x tk+ gH x t
k δ⊙x tk k   + 
  + 
  + 
x tk gH x tk  δ⊙x tk gH δ⊙x tk gH x tk gH x tk   ¼ δ⊙x tk+ gH δ⊙x t
k then:   δ⊙x tk+ gH δ⊙x t
k ¼ δ2 ⊙Ik ðxðtk ÞÞ since δ is positive:

   δ⊙ x tk+ gH x t
¼ δ2 ⊙Ik ðxðtk ÞÞ k

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Soft Numerical Computing in Uncertain Dynamic Systems

or:       δ⊙ x tk+ gH x t
¼ δ2 ⊙ x tk+ gH x t
k k This means the perturbations in the difference equation corresponding to the jump are the same: δ ¼ δ2. What about other perturbations? In the case that we use the fuzzy Euler method and it does not matter which case of Euler we use, then: zk + 1 ¼ zk h⊙ðf ðtk , zk Þðδ1 ⊙f ðtk , zk ÞÞÞ, z0 ¼ δ3 ⊙x0 By backward substituting: zk ¼ zk
1 h⊙ðf ðtk
1 , zk
1 Þðδ1 ⊙f ðtk
1 , zk
1 ÞÞÞ Since δ1 is positive we have: zk + 1 ¼ zk
1 h⊙½f ðtk
1 , zk
1 Þf ðtk , zk Þδ1 ⊙½f ðtk
1 , zk
1 Þf ðtk , zk Þ Finally, by continuing this process we get: zk + 1 ¼ z0 h⊙

k X

f ðti , zi Þδ1 ⊙

k X

i¼0

f ðti , zi Þ

i¼0

Since z0 ¼ δ3 ⊙ x0 and zi ¼ zi  δ ⊙ zi then: zk + 1 δ⊙zk + 1 ¼ δ3 ⊙x0 h⊙

k X

f ðti , zi Þδ1 ⊙

i¼0

k X

f ðt i , zi Þ

i¼0

If you look at to both sides, the perturbation on the left-hand side is δ ⊙ zk+1 and the perturbations on the right-hand side are: δ3 ⊙x0 δ1 ⊙

k X

f ðti , zi Þ

i¼0

Now we should claim that: δ⊙zk + 1  δ3 ⊙x0 δ1 ⊙

k X

f ðti , zi Þ

i¼0

This inequality or ordering of fuzzy numbers in the level-wise form is: δ⊙zl, k + 1  δ3 ⊙xl,0 δ1 ⊙

k X

fl ðti , zi Þ

i¼0

δ⊙zu, k + 1  δ3 ⊙xu,0 δ1 ⊙

k X i¼0

f u ðt i , zi Þ

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thus: δ

δ

δ3 ⊙xl, 0 δ1 ⊙

Xk

f ðt , z Þ i¼0 l i i

zl, k + 1 Xk δ3 ⊙xu, 0 δ1 ⊙ f ðt , z Þ i¼0 u i i zu, k + 1

Considering δ ¼ δ2 the maximum error or perturbation is: 9 8 Xk =

> < s1 ðtÞ, t½t1 , t2  sðtÞ ¼ ⋮ > > : t½tn
1 , tn  sn
1 ðtÞ,

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Soft Numerical Computing in Uncertain Dynamic Systems

where si(t)  Ck
1[ti, ti+1], i ¼ 0, 1…, n
1 are the fuzzy polynomials of the degree k with fuzzy number coefficients, such that: ðk
1Þ

ðk
1Þ

si
1 ðti Þ ¼ si

ðti Þ, i ¼ 1, …, n
1

si
1 ðti Þ ¼ xi , i ¼ 1,…, n, si ðti Þ ¼ xi , i ¼ 0,1, …, n
1 Now let us consider the triple form of a triangular fuzzy number as xi ¼ (xli, xci , xri ) and suppose that f(t) ¼ ( f l(t), f c(t), f r(t)) is a fuzzy function that we want to approximate in the [t0, tn]. The mentioned spline should be defined for each element of the fuzzy number separately, and we have: X X 8 > f l ðtÞ ¼ si ðtÞxli + si ðtÞxri > > > > si ðtÞ 0 si ðtÞ > n < X f c ðt Þ ¼ si ðtÞxci > > i¼0 X X > > > > f r ðt Þ ¼ si ðtÞxri + si ðtÞxli > : si ðtÞ 0

si ðtÞ > xl ðti + 2 , r Þ ¼ xl ðti
1 , r Þ + ½fl ðti
1 , xðti
1 Þ, r Þ + fl ðti , xðti Þ, r Þ + 4fl ðti + 1 , xðti + 1 Þ, r Þ > > 2 > > < h xu ðti + 2 , r Þ ¼ xu ðti
1 , r Þ + ½fu ðti
1 , xðti
1 Þ, r Þ + fu ðti , xðti Þ, r Þ + 4fu ðti + 1 , xðti + 1 Þ, r Þ 2 > > > > x ð t , r Þ ¼ x ð r Þ, x ð > l i+2 l, i + 2 l ti + 1 , r Þ ¼ xl, i + 1 ðr Þ, xl ðti , r Þ ¼ xl, i ðr Þ > : xu ðti + 2 , r Þ ¼ xu, i + 2 ðr Þ, xu ðti + 1 , rÞ ¼ xu, i + 1 ðr Þ, xu ðti , r Þ ¼ xu, i ðr Þ Note. It should be observed that: fl ðti
1 , xðti
1 Þ, r Þ≔fl ðti
1 , xl ðti
1 , rÞ, xu ðti
1 , rÞÞ fu ðti
1 , xðti
1 Þ, r Þ≔fu ðti
1 , xl ðti
1 , rÞ, xu ðti
1 , rÞÞ

5.6.4

FUZZY

IMPLICIT TWO STEPS METHOD

To introduce the method, we need the points x(ti
1), x(ti) and f(ti
1, x(ti
1)), f(ti, x(ti)), which are triangular fuzzy numbers, and:  l f ðti
1 , xðti
1 ÞÞ, f c ðti
1 , xðti
1 ÞÞ, f r ðti
1 , xðti
1 ÞÞ  l f ðti , xðti ÞÞ, f c ðti , xðti ÞÞ, f r ðti , xðti ÞÞ By using the same fuzzy linear splines for f(ti
1, x(ti
1)), f(ti, x(ti)) and f(ti+1, x(ti+1)) at the same intervals, and considering the following integral equation: ð ti + 1 f ðt, xðtÞÞdt xðti + 1 Þ ¼ xðti
1 Þ ti
1

the level-wise form of this integral equation is as follows: ð ti  c rsi
1 ðtÞ + ð1
r Þsli
1 ðtÞ dt xl ðti + 1 , rÞ ¼ xl ðti
1 , rÞ + ti
1

ð ti + 1 + ti



rsci ðtÞ + ð1
r Þsli ðtÞ dt

260

Soft Numerical Computing in Uncertain Dynamic Systems xu ðti + 1 , r Þ ¼ xu ðti
1 , r Þ +

ð ti ti
1

ð ti + 1 + ti





rsci
1 ðtÞ + ð1
r Þsri
1 ðtÞ dt

rsci ðtÞ + ð1
r Þsri ðtÞ dt

The same process as that above is done here and finally we have: h xl ðti + 1 , r Þ ¼ xl ðti
1 , r Þ + ½fl ðti
1 , xðti
1 Þ, r Þ + 2fl ðti , xðti Þ, r Þ + fl ðti + 1 , xðti + 1 Þ, r Þ 2 h xu ðti + 1 , rÞ ¼ xu ðti
1 , rÞ + ½fu ðti
1 , xðti
1 Þ, r Þ + 2fu ðti , xðti Þ, r Þ + fu ðti + 1 , xðti + 1 Þ, r Þ 2 It is indeed: h xðti + 1 Þ ¼ xðti
1 Þ ½f ðti
1 , xðti
1 ÞÞ + 2⊙f ðti , xðti ÞÞ + f ðti + 1 , xðti + 1 ÞÞ 2 Therefore, the two steps fuzzy implicit method is defined as: 8 h > > > xl ðti + 1 , rÞ ¼ xl ðti
1 , rÞ + ½fl ðti
1 , xðti
1 Þ, r Þ + 2fl ðti , xðti Þ, r Þ + fl ðti + 1 , xðti + 1 Þ, r Þ > 2 > < h xu ðti + 1 , rÞ ¼ xu ðti
1 , r Þ + ½fu ðti
1 , xðti
1 Þ, r Þ + 2fu ðti , xðti Þ, r Þ + fu ðti + 1 , xðti + 1 Þ, r Þ > 2 > > xl ðti + 1 , rÞ ¼ xl, i + 1 ðr Þ, xl ðti , r Þ ¼ xl, i ðr Þ > > : xu ðti
1 , r Þ ¼ xu, i
1 ðr Þ, xu ðti , r Þ ¼ xu, i ðr Þ

5.6.5

FUZZY

PREDICTOR AND CORRECTOR THREE STEPS METHODS

We have introduced two methods: the fuzzy three steps explicit method and the fuzzy two steps implicit method. In the following method, we shall introduce a predictor and corrector method to obtain an accurate result. Consider the same fuzzy differential equations in the form of: x0gH ðtÞ ¼ f ðtxðtÞÞ, xðt0 Þ ¼ x0 R In this method, we need more initial values as initial steps. To do this, we can use the fuzzy Euler method to find the x(t1)  x1, x(t2)  x2. Then we have three initial values. Now the algorithm is introduced as follows: 0 h ¼ T
t N The initial values x1, x2 are obtained by the fuzzy Euler method. We have x0, x1, x2. ð0Þ xi + 2 ¼ xi
1  h2 ⊙½f ðti
1 , xi
1 Þf ðti , xi Þ4⊙f ðti + 1 , xi + 1 Þ   ð0Þ 5. xi + 2 ¼ xi  h2 ⊙f ðti , xi Þh⊙f ðti + 1 , xi + 1 Þ h2 ⊙f ti + 2 , xi + 2 6. Repeat until i > N
2.

1. 2. 3. 4.

Discrete numerical solutions of uncertain differential equations

261

Convergency To discuss convergency, we must consider the uniform convergence and especially should show that at the end point of the interval [t0, T] we have convergence. Because the approximate values, at the end point, T, do have more error propagation: lim DH ðxðT Þ, xN Þ ¼ 0

h!0

using the definition of the Hausdorff distance, we should discuss in the level-wise form: lim xl ðT, r Þ ¼ xl, N ðr Þ, lim xu ðT, r Þ ¼ xu, N ðr Þ

h!0

h!0

where: h¼

T
t0 , t0 < t1 < ⋯ < tN , ti ¼ t0 + ih, i ¼ 1,2, …,N N

To prove the convergence we need the following lemma, which is proved very easily by using mathematical induction. Lemma Let us consider that the sequence of numbers {wi}Ni¼0 satisfy: jwi + 1 j  Ajwi j + Bjwi
1 j + C, A:B, C > 0, 0  i  N Then, if i is odd: 0

1 i B i
1 C i
3 i
5 2 2 jwi j  @A + β1 A B + β2 A B + ⋯ + βs B Ajw1 j 1 i  B i
2 C i
4 2 2 + @A + γ 1 A B + ⋯ + βt AB Ajw0 j + Ai
2 + Ai
3 + ⋯1 C 0

 + δ1 Ai
4 + δ2 Ai
5 + ⋯ + δm A + 1 BC   + ζ 1 Ai
6 + ζ 2 Ai
7 + ⋯ + ζ l A + 1 B2 C + λ1 Ai
8 + λ2 Ai
9 + λp A + 1 B3 C + ⋯ Then, if i is even: 

i  i
1 i
3 i
5 2 jwi j  A + β1 A B + β2 A B + ⋯ + βs B 2
1 jw1 j 1 i  B i
2 C + @A + γ 1 Ai
4 B2 + ⋯ + βt AB 2 Ajw0 j + Ai
2 + Ai
3 + ⋯1 C 0

262

Soft Numerical Computing in Uncertain Dynamic Systems  + δ1 Ai
4 + δ2 Ai
5 + ⋯ + δm A + 1 BC   + ζ 1 Ai
6 + ζ 2 Ai
7 + ⋯ + ζ l A + 1 B2 C + λ1 Ai
8 + λ2 Ai
9 + λp A + 1 B3 C + ⋯

where βs, γ t, δm, ζ l, λp are constants for all s, t, m, j, p. Convergence theorem For any arbitrary but fixed 0  r  1 the implicit two steps approximations converge to xl(T, r), xu(T, r) for xl, xu  C3[t0, T]. To prove this, it is enough to show that: lim xl ðT, r Þ ¼ xl,N ðr Þ, lim xu ðT, r Þ ¼ xu, N ðr Þ

h!0

h!0

The approximate solution satisfies the recursive method for the implicit two steps method exactly, while the exact solution satisfies it approximately. This means: h xl ðti + 1 , r Þ  xl ðti
1 , r Þ + ½fl ðti
1 , xðti
1 Þ, r Þ + 2fl ðti , xðti Þ, r Þ + fl ðti + 1 , xðti + 1 Þ, r Þ 2 and: h xl, i + 1 ðr Þ ¼ xl, i
1 ðr Þ + ½fl ðti
1 , xl, i
1 ðr ÞÞ + 2fl ðti , xl, i ðr ÞÞ + fl ðti + 1 , xl, i + 1 ðr ÞÞ 2 To have equality for the exact solution, we must add the error term to the recursive relation, so we have: h xl ðti + 1 , rÞ ¼ xl ðti
1 , r Þ + ½fl ðti
1 , xðti
1 Þ, r Þ + 2fl ðti , xðti Þ, r Þ + fl ðti + 1 , xðti + 1 Þ, r Þ 2 h3 000 + xl ðηi Þ 6 It should be remembered that the lower and upper functions f in the recursive relations are in terms of lower and upper functions of x: fl ðti
1 , xðti
1 Þ, r Þ≔fl ðti
1 , xl ðti
1 , r Þ, xu ðti
1 , r ÞÞ fu ðti
1 , xðti
1 Þ, r Þ≔fu ðti
1 , xl ðti
1 , r Þ, xu ðti
1 , r ÞÞ Now consider the following recursive relations for the lower and upper values: h xl ðti + 1 , rÞ ¼ xl ðti
1 , r Þ + ½fl ðti
1 , xðti
1 Þ, r Þ + 2fl ðti , xðti Þ, r Þ + fl ðti + 1 , xðti + 1 Þ, r Þ 2 h3 000  + xl ηl, i 6 h xl, i + 1 ðr Þ ¼ xl, i
1 ðr Þ + ½fl ðti
1 , xl, i
1 ðr ÞÞ + 2fl ðti , xl, i ðr ÞÞ + fl ðti + 1 , xl, i + 1 ðr ÞÞ 2

Discrete numerical solutions of uncertain differential equations

263

and: h xu ðti + 1 , rÞ ¼ xu ðti
1 , rÞ + ½fu ðti
1 , xðti
1 Þ, r Þ + 2fu ðti , xðti Þ, r Þ + fu ðti + 1 , xðti + 1 Þ, r Þ 2 h3 000  + xu ηu, i 6 h xu, i + 1 ðr Þ ¼ xu, i
1 ðr Þ + ½fu ðti
1 , xl,i
1 ðr ÞÞ + 2fu ðti , xl, i ðr ÞÞ + fu ðti + 1 , xl, i + 1 ðr ÞÞ 2 for ηl,i, ηu,i  [ti, ti+1]. Consider the Hausdorff distance of two values xl(ti+1) and xl, i+1 or the real absolute values in the level-wise form: h xl ðti + 1 , r Þ
xl, i + 1 ðr Þ ¼ xl ðti
1 , rÞ
xl, i
1 ðr Þ + ½fl ðti
1 , xðti
1 Þ, r Þ
fl ðti
1 , xl, i
1 ðr ÞÞ 2 h h½fl ðti , xðti Þ, r Þ
fl ðti , xl, i ðr ÞÞ + ½fl ðti + 1 , xðti + 1 Þ, r Þ
fl ðti + 1 , xl, i + 1 ðr ÞÞ 2 +

h3 000  x η 6 l l, i

and: xu ðti + 1 , rÞ
xu, i + 1 ðr Þ ¼ xu ðti
1 , r Þ
xu, i
1 ðr Þ h + ½fu ðti
1 , xðti
1 Þ, r Þ
fu ðti
1 , xl, i
1 ðr ÞÞ 2 h h½fu ðti , xðti Þ, r Þ
fu ðti , xl, i ðr ÞÞ + ½fu ðti + 1 , xðti + 1 Þ, r Þ
fu ðti + 1 , xl, i + 1 ðr ÞÞ 2 +

h3 000  x η 6 u u, i

Let us assume that wi ¼ xl(ti, r)
xl,i (r) and vi ¼ xu(ti, r)
xu,i (r), then the following relations are concluded immediately:     hL2 hL3 h3 jwi + 1 j  hL1 jwi j + 1 + jwi
1 j + jwi + 1 j + Ml 2 2 6     hL5 hL6 h3 jvi + 1 j  hL4 jvi j + 1 + jvi
1 j + j vi + 1 j + M u 2 2 6 where:   000 Ml ¼ max x000 l ηl, i , Mu ¼ max xu ηu, i t0 tT

t0 tT

264

Soft Numerical Computing in Uncertain Dynamic Systems

where L1, L2, L3, L4, L5, L6 are Lipschitz constants and: L ¼ max fL1 , L2 , L3 , L4 , L5 , L6 g
> > > < xl ð0, r Þ ¼ bl,0 ðr Þ x0l ð0, r Þ ¼ bl,1 ðr Þ > > > ⋮ > : ðn
1Þ ð0, r Þ ¼ bl, n
1 ðr Þ xl 8 xðnÞ ðtÞ + an
1 ðtÞxðn
1Þ ðtÞ + ⋯ + a1 ðtÞx0 ðtÞ + a0 ðtÞ u ðr Þ ¼ gu ðt, r Þ > > > > < xu ð0, r Þ ¼ bu,0 ðr Þ x0u ð0, r Þ ¼ bu,1 ðr Þ > > > ⋮ > : ðn
1Þ xu ð0, r Þ ¼ bu, n
1 ðr Þ To investigate the lower and upper functions in the first equation of the abovementioned nth-order differential equations, we need to discuss the signs of the coefficients. Case 1. All the coefficients ai(t), i ¼ 0, 1, …, n
1 are nonnegative. Then we have: 8 ðnÞ ðn
1Þ > xl ðt, r Þ + an
1 ðtÞxl ðt, r Þ + ⋯ + a1 ðtÞx0l ðt, r Þ + a0 ðtÞ ¼ gl ðt, r Þ > > > < xl ð0, r Þ ¼ bl,0 ðr Þ x0l ð0, r Þ ¼ bl,1 ðr Þ > > > > : ðn
1Þ ⋮ ð0, r Þ ¼ bl, n
1 ðr Þ xl 8 xðunÞ ðt, r Þ + an
1 ðtÞxðun
1Þ ðt, r Þ + ⋯ + a1 ðtÞx0u ðt, r Þ + a0 ðtÞ ¼ gu ðt, r Þ > > > > < xu ð0, r Þ ¼ bu,0 ðr Þ x0u ð0, r Þ ¼ bu,1 ðr Þ > > > ⋮ > : ðn
1Þ xu ð0, r Þ ¼ bu, n
1 ðr Þ Now, by using these equations: ðiÞ

xl,N ðt, r Þ ¼

N X

ð iÞ

ðiÞ

αl, k ðr Þϕk ðtÞ, xu, N ðt, r Þ ¼

k¼0

N X

ðiÞ

αu, k ðr Þϕk ðtÞ, i ¼ 0,1, …, n

k¼0

we will have the following system of equations: N X

ðnÞ

αl, k ðr Þϕk ðtÞ + an
1 ðtÞ

k¼0

N X

ðn
1Þ

αl, k ðr Þϕk

ðtÞ + ⋯ + a1 ðtÞ

k¼0

+ a0 ðtÞ

N X k¼0

N X k¼0

αl, k ðr Þϕk ðtÞ ¼ gl ðt, r Þ

αl, k ðr Þϕ0k ðtÞ

Discrete numerical solutions of uncertain differential equations N X

αl, k ðr Þϕk ð0Þ ¼ bl,0 ,

k¼0

N X

N X

αl,k ðr Þϕ0k ð0Þ ¼ bl,1 ðr Þ, …,

k¼0

ðnÞ

k¼0

N X

ðn
1Þ

αu, k ðr Þϕk

ðtÞ + ⋯ + a1 ðtÞ

k¼0

+ a0 ðtÞ

N X

ðn
1Þ

αl, k ðr Þϕk

ð 0Þ

k¼0

¼ bl, n
1 ðr Þ

αu, k ðr Þϕk ðtÞ + an
1 ðtÞ

N X

269

N X

αu, k ðr Þϕ0k ðtÞ

k¼0

αu, k ðr Þϕk ðtÞ ¼ gu ðt, r Þ

k¼0 N X

αu, k ðr Þϕk ð0Þ ¼ bu,0 ,

k¼0

N X

αu, k ðr Þϕ0k ð0Þ ¼ bu,1 ðr Þ,…,

k¼0

N X

ðn
1Þ

αu, k ðr Þϕk

ð 0Þ

k¼0

¼ bu, n
1 ðr Þ

By setting: ðnÞ

ðn
1Þ

ϕk ðtÞ + an
1 ðtÞϕk

ðn
2Þ

ðtÞ + an
2 ðtÞ + ϕk

ðtÞ + ⋯ + a0 ðtÞϕk ðtÞ ¼ βk

ð jÞ

ϕk ð0Þ ¼ σ jk , j ¼ 0,1,…, n
1, k ¼ 0,1, …, N and substituting, we get: 8 N X > > > αl, k ðr Þβk ¼ gl ðt, r Þ > > > > k¼0 > > N >

k¼0 > > ⋮ > > > N > X > > > αl, k ðr Þσ n
1, k ¼ bl, n
1 ðr Þ : k¼0

8 N X > > > αu, k ðr Þβk ¼ gu ðt, r Þ > > > > k¼0 > >X > < N αu, k ðr Þσ 0k ¼ bu,0 ðr Þ > k¼0 > > ⋮ > > > N > X > > > αu, k ðr Þσ n
1, k ¼ bu, n
1 ðr Þ : k¼0

The matrix form of these system of linear equations can be formed as the following block-wise linear system SX ¼ Y, where: 2 3 ⋯ βN β0 β1

6 σ 0,0 S S σ 0,1 ⋯ σ 0, N
1 7 7 S¼ 1 2 , S1 ¼ 6 4 5 S2 S1 2ðN + 1ÞðN + 1Þ ⋮ ⋮ ⋯ ⋮ σ n
1,0 σ n
1,1 ⋯ σ n
1, N
1 ðN + 1ÞðN + 1Þ

270

Soft Numerical Computing in Uncertain Dynamic Systems 2

0 60 S1 ¼ 6 4⋮ 0

0 0 ⋮ 0

3 0 07 7 ⋮5 0 ðN + 1ÞðN + 1Þ

⋯ ⋯ ⋯ ⋯

X ¼ ½αl,0 ðr Þ, αl,1 ðr Þ, αl,2 ðr Þ, …, αl, N ðr Þ, αu,0 ðr Þ, αu,1 ðr Þ, αu, 2 ðr Þ, …, αu, N ðr ÞT Y ¼ ½gl ðt, r Þ, bl,0 ðr Þ, …, bl,n
1 ðr Þ, gu ðt, r Þ, bu, 0 ðr Þ, …, bu, n
1 ðr Þ For more explanation on this topic, see Allahviranloo (2020). Now we can use the collocation points like t ¼ a  [0, T]: ðnÞ

ðn
1Þ

ϕk ðaÞ + an
1 ðaÞϕk

ðn
2Þ

ðaÞ + an
2 ðaÞϕk

ðaÞ + ⋯ + a0 ðaÞϕk ðaÞ ¼ βk

ð jÞ

ϕk ð0Þ ¼ σ jk , j ¼ 0, 1,…, n
1, k ¼ 0, 1,…, N 8 N 8 N X X > > > > > > αl, k ðr Þβk ¼ gl ða, r Þ αu, k ðr Þβk ¼ gu ða, r Þ > > > > > > > > k¼0 k¼0 > > N > > N > >

k¼0 k¼0 > > >⋮ >⋮ > > > > > > N N > > X X > > > > > > α ð r Þσ ¼ b ð r Þ αu, k ðr Þσ n
1,k ¼ bu, n
1 ðr Þ l, k n
1, k l,n
1 : : k¼0

k¼0

Then all: αl,0 ðr Þ, αl,1 ðr Þ, αl,2 ðr Þ, …, αl, N ðr Þ,αu,0 ðr Þ,αu,1 ðr Þ,αu,2 ðr Þ,…, αu, N ðr Þ are determined very easily for arbitrary but fixed r  [0, 1]. Therefore, the solutions xl(t, r) and xu(t, r) are approximated. Example Consider the following second-order fuzzy linear differential equation: 8 00 x ðtÞ  xðtÞ ¼
t, 0  t  1 > < gH xð0Þ ¼ ð0:1r
0:1;0:1
0:1r Þ > : 0 xgH ð0Þ ¼ ð0:088 + 0:1r, 0:288
0:1r Þ The fuzzy exact solution of the problem is: xl ðt, r Þ ¼ ð0:1r
0:1Þ cos t + ð1:088 + 0:1r Þ sin t
t xu ðt, r Þ ¼ ð0:1
0:1r Þ cos t + ð1:288
0:1r Þ sin t
t If the basis functions is ϕk(t) ¼ tk, k ¼ 0, 1, 2 and we have: xl,2 ðt, r Þ ¼ αl,0 ðr Þ + αl,1 ðr Þt + αl,2 ðr Þt2 xu,2 ðt, r Þ ¼ αu,0 ðr Þ + αu,1 ðr Þt + αu,2 ðr Þt2

Discrete numerical solutions of uncertain differential equations

271

the linear system to determine the coefficients is: 8 < 2αl,2 ðr Þ + αl,0 ðr Þ + αl,1 ðr Þt + αl,2 ðr Þt2 ¼
t α ðr Þ ¼ 0:1
0:1r : l,0 αl,1 ðr Þ ¼ 0:088 + 0:1r 8 < 2αu,2 ðr Þ + αu,0 ðr Þ + αu,1 ðr Þt + αu,2 ðr Þt2 ¼
t α ðr Þ ¼ 0:1
0:1r : u,0 αl,1 ðr Þ ¼ 0:288
0:1r SX ¼ Y 2

S¼

S1 S2

3 αl,0 ðr Þ 2 3 2 3 6 αl,1 ðr Þ 7

0 0 0 1 t 2 + t2 6 7 S2 6 α ðr Þ 7 4 5 4 5 , S1 ¼ 1 0 0 , S2 ¼ 0 0 0 , X ¼ 6 l,1 7, S1 6 αu,0 ðr Þ 7 0 0 0 0 1 1 4 5 αu,1 ðr Þ αu, 2 ðr Þ 2 3
t 6 0:1
0:1r 7 6 7 6 0:088 + 0:1r 7 Y¼6 7 6
t 7 4 5 0:1
0:1r 0:288
0:1r

Choosing the collocation point as ¼ 12:

3 1
7 6 2 3 2 6 0:1
0:1r 7 1 9 7 6 1 7 6 6 + 0:1r 7 S1 ¼ 4 1 20 40 5, Y ¼ 6 0:088 7 7 6
1 7 6 0 1 1 5 4 2 0:1
0:1r 0:288
0:1r 2 1 1=2 9=4 0 0 0 61 0 0 0 0 0 6 0 0 60 1 1 0 S¼6 1=2 9=4 60 0 0 1 40 1 0 0 0 0 0 0 0 0 1 1 2

3 7 7 7 7 7 5

the matrix S is nonsingular, and by solving the SX ¼ Y the vector solution X can be determined easily and the solution in level-wise form is: xl ðt, r Þ ¼ 0:1r
0:1 + ð0:88e
1 + 0:1r Þt + ð
0:1973333334
0:6666666666e
1r Þt2

272

Soft Numerical Computing in Uncertain Dynamic Systems xu ðt, r Þ ¼ 0:1
0:1r + ð0:288
0:1r Þt + ð
0:3306666666 + 0:6666666666e
1r Þt2

For more illustration, Tables 5.13–5.15 show the comparison of the exact and approximate solutions at t ¼ 0, 0.001, 0.01 for r  [0, 1] with step size 0.1. Case 2. All the coefficients ai(t), i ¼ 0, 1, …, n
1 are negative. Then we have: 8 ðnÞ > xl ðt, r Þ + an
1 ðtÞxðun
1Þ ðt, r Þ + ⋯ + a1 ðtÞx0u ðt, r Þ + a0 ðtÞ ¼ gl ðt, r Þ > > > < xl ð0, r Þ ¼ bl,0 ðr Þ x0l ð0, r Þ ¼ bl,1 ðr Þ > > > > : ðn
1Þ ⋮ ð0, r Þ ¼ bl, n
1 ðr Þ xl 8 ðn
1Þ > xðunÞ ðt, r Þ + an
1 ðtÞxl ðt, r Þ + ⋯ + a1 ðtÞx0l ðt, r Þ + a0 ðtÞ ¼ gu ðt, r Þ > > > < xu ð0, r Þ ¼ bu, 0 ðr Þ x0u ð0, r Þ ¼ bu, 1 ðr Þ > > > > : ðn
1Þ ⋮ xu ð0, r Þ ¼ bu, n
1 ðr Þ Now by using the same previously mentioned linear combinations, we will have the following system of equations:

TABLE 5.13 Comparison of the exact and approximate solutions at t ¼ 0. r

xl,

xl

Error

xu,

xu

Error

0


0.1


0.1

0

0.1

0.1

0

0.1


0.09


0.09

0

0.09

0.09

0

0.2


0.08


0.08

0

0.08

0.08

0

0.3


0.07


0.07

0

0.07

0.07

0

0.4


0.06


0.06

0

0.06

0.06

0

0.5


0.05


0.05

0

0.05

0.05

0

0.6


0.04


0.04

0

0.04

0.04

0

0.7


0.03


0.03

0

0.03

0.03

0

0.8


0.02


0.02

0

0.02

0.02

0

0.9


0.01


0.01

0

0.01

0.01

0

1

0

0

0

0

0

0

N

N

TABLE 5.14 Comparison of the exact and approximate solutions at t ¼ 0.001. r

xl,

xl

Error

xu,

xu

Error

0

0.9991219733e
1

0.9991195018e
1

0.24715e
6

0.1002876693

0.1002879498

0.2805e
6

0.1

0.8990220400e
1

0.8990195518e
1

0.24882e
6

0.9027767600e
1

0.9027795479e
1

0.27879e
6

0.2

0.7989221067e
1

0.7989196018e
1

0.25049e
6

0.8026768267e
1

0.8026795979e
1

0.27712e
6

0.3

0.6988221733e
1

0.6988196519e
1

0.25214e
6

0.7025768933e
1

0.7025796479e
1

0.27546e
6

0.4

0.5987222400e
1

0.5987197019e
1

0.25381e
6

0.6024769600e
1

0.6024796979e
1

0.27379e
6

0.5

0.4986223067e
1

0.4986197519e
1

0.25548e
6

0.5023770267e
1

0.5023797479e
1

0.27212e
6

0.6

0.3985223733e
1

0.3985198019e
1

0.25714e
6

0.4022770933e
1

0.4022797980e
1

0.27047e
6

0.7

0.2984224400e
1

0.2984198519e
1

0.25881e
6

0.3021771600e
1

0.3021798480e
1

0.26880e
6

0.8

0.1983225067e
1

0.1983199020e
1

0.26047e
6

0.2020772267e
1

0.2020798980e
1

0.26713e
6

0.9

0.9822257333e
2

0.9821995196e
2

0.262137e
6

0.1019772933e
1

0.1019799480e
1

0.26713e
6

1

0.1877360000e
3

0.187999802e
3

0.2638020e
6

0.1877360000e
3

0.187999802e
3

0.2638020e
6

N

N

TABLE 5.15 Comparison of the exact and approximate solutions at t ¼ 0.01. r

xl,

xl

Error

xu,

xu

Error

0


0.9913973333e
1


0.9911518137e
1

0.2455196e
4

0.1028469333

0.1028747854

0.278521e
4

0.1


0.8904040000e
1


0.8901568304e
1

0.2471696e
4

0.9274760000e
1

0.927752870e
1

0.2768700e
4

0.2


0.7894106667e
1


0.7891618470e
1

0.2488197e
4

0.8264826667e
1

0.8267578870e
1

0.2752203e
4

0.3


0.6884173333e
1


0.6881668636e
1

0.2504697e
4

0.7254893333e
1

0.7257629036e
1

0.2735703e
4

0.4


0.5874240000e
1


0.5871718802e
1

0.2521198e
4

0.6244960000e
1

0.6247679202e
1

0.2719202e
4

0.5


0.4864306667e
1


0.4861768969e
1

0.2537698e
4

0.5235026667e
1

0.5237729369e
1

0.2702702e
4

0.6


0.3854373333e
1


0.3851819135e
1

0.2554198e
4

0.4225093333e
1

0.4227779535e
1

0.2686202e
4

0.7


0.2844440000e
1


0.2841869301e
1

0.2570699e
4

0.3215160000e
1

0.3217829701e
1

0.2669701e
4

0.8


0.1834506667e
1


0.1831919468e
1

0.2587199e
4

0.2205226667e
1

0.2207879868e
1

0.2653201e
4

0.9


0.8245733333e
2


0.8219696334e
2

0.26036999e
4

0.1195293333e
1

0.1197930033e
1

0.2636700e
4

1

0.1853600000e
2

0.187980200e
2

0.26202000e
4

0.1853600000e
2

0.187980200e
2

0.26202000e
4

N

N

Discrete numerical solutions of uncertain differential equations N X

ðnÞ

αl, k ðr Þϕk ðtÞ + an
1 ðtÞ

k¼0

N X

ðn
1Þ

αu, k ðr Þϕk

ðtÞ + ⋯ + a1 ðtÞ

k¼0 N X

+ a0 ðtÞ

N X

275

αu, k ðr Þϕ0k ðtÞ

k¼0

αu, k ðr Þϕk ðtÞ ¼ gl ðt, r Þ

k¼0 N X

αl, k ðr Þϕk ð0Þ ¼ bl,0 ,

k¼0

N X

N X

αl,k ðr Þϕ0k ð0Þ ¼ bl,1 ðr Þ, …,

k¼0

ðnÞ

k¼0

N X

ðn
1Þ

αu, k ðr Þϕk

ðtÞ + ⋯ + a1 ðtÞ

k¼0

+ a0 ðtÞ

N X

ðn
1Þ

αl, k ðr Þϕk

ð 0Þ

k¼0

¼ bl, n
1 ðr Þ

αu, k ðr Þϕk ðtÞ + an
1 ðtÞ

N X

N X

αu, k ðr Þϕ0k ðtÞ

k¼0

αu, k ðr Þϕk ðtÞ ¼ gl ðt, r Þ

k¼0 N X

αu, k ðr Þϕk ð0Þ ¼ bu,0 ,

k¼0

N X

αu, k ðr Þϕ0k ð0Þ ¼ bu,1 ðr Þ,…,

k¼0

N X

ðn
1Þ

αu, k ðr Þϕk

ð 0Þ

k¼0

¼ bu, n
1 ðr Þ

By setting: ðnÞ

ðn
1Þ

ϕk ðtÞ + an
1 ðtÞϕk ðnÞ

ðn
2Þ

ðtÞ + an
2 ðtÞ + ϕk

ðtÞ + ⋯ + a0 ðtÞϕk ðtÞ ¼ δk

ð jÞ

ϕk ðtÞ ¼ γ nk , ϕk ð0Þ ¼ σ jk , j ¼ 0, 1,…, n
1, k ¼ 0, 1, …,N and substituting, we get: 8 N X > > > ðαl, k ðr Þγ nk + αu, k ðr Þδk Þ ¼ gl ðt, r Þ > > > > k¼0 > > N >

k¼0 > > ⋮ > > > N > X > > > αl, k ðr Þσ n
1, k ¼ bl, n
1 ðr Þ : k¼0

8 N X > > > ðαu, k ðr Þγ nk + αl, k ðr Þδk Þ ¼ gu ðt, r Þ > > > > k¼0 > > N >

k¼0 > > ⋮ > > > N > X > > > αu, k ðr Þσ n
1, k ¼ bu, n
1 ðr Þ : k¼0

The matrix form of these system of linear equations can be formed as the following block-wise linear system SX ¼ Y, where: 2 3 γ n1 ⋯ γ nN γ n0

6 σ 0,0 S S σ 0,1 ⋯ σ 0, N
1 7 7 S¼ 1 2 , S1 ¼ 6 4 5 S2 S1 2ðN + 1ÞðN + 1Þ ⋮ ⋮ ⋯ ⋮ σ n
1,0 σ n
1,1 ⋯ σ n
1, N
1 ðN + 1ÞðN + 1Þ

276

Soft Numerical Computing in Uncertain Dynamic Systems 2

δ0 6 0 S1 ¼ 6 4⋮ 0

δ1 0 ⋮ 0

⋯ ⋯ ⋯ ⋯

3 δN 0 7 7 ⋮5 0 ðN + 1ÞðN + 1Þ

X ¼ ½αl,0 ðr Þ, αl,1 ðr Þ, αl,2 ðr Þ, …, αl, N ðr Þ, αu,0 ðr Þ, αu,1 ðr Þ, αu, 2 ðr Þ, …, αu, N ðr ÞT Y ¼ ½gl ðt, r Þ, bl,0 ðr Þ, …, bl,n
1 ðr Þ, gu ðt, r Þ, bu, 0 ðr Þ, …, bu, n
1 ðr Þ By the same process that we have discussed in Case 1, we can use the collocation points like t ¼ a  [0, T] to realize the system of linear equations to find the coefficients: 8 N 8 N X > >X > > > ðαl, k ðr Þγ nk + αu, k ðr Þδk Þ ¼ gl ða, r Þ > ðαu, k ðr Þγ nk + αl, k ðr Þδk Þ ¼ gu ða, r Þ > > > > > > > > k¼0 k¼0 > > > > N N > >

k¼0 k¼0 > > > > ⋮ ⋮ > > > > > > N N > > X X > > > > > > αl, k ðr Þσ n
1, k ¼ bl, n
1 ðr Þ αu, k ðr Þσ n
1, k ¼ bu, n
1 ðr Þ : : k¼0

k¼0

Then all: αl,0 ðr Þ, αl,1 ðr Þ, αl,2 ðr Þ, …, αl, N ðr Þ,αu,0 ðr Þ,αu,1 ðr Þ,αu,2 ðr Þ,…, αu, N ðr Þ are determined very easily for arbitrary but fixed r  [0, 1]. Therefore, the solutions xl(t, r) and xu(t, r) are approximated. Example Consider the following second-order fuzzy linear differential equation: 8 2 2 > x00gH ðtÞH 2 ⊙xðtÞ ¼ , t 1 > > < t t xð1Þ ¼ ð0:1r
0:1; 0:1
0:1r Þ > > > : 0 xgH ð1Þ ¼ ð
0:25 + 0:25r, 0:25
0:25r Þ The fuzzy exact solution of the problem is:   1 xl ðt, r Þ ¼ ð
0:500000001e
r + 0:3833333334Þ + ð0:15r + 0:5166666666Þt2
t t xu ðt, rÞ ¼

  1 ð0:500000003e
r + 0:2833333337Þ + ð
0:15r + 0:8166666666Þt2
t t

If the basis functions is ϕk(t) ¼ tk, k ¼ 0, 1, 2 and we have:

Discrete numerical solutions of uncertain differential equations

277

xl,2 ðt, r Þ ¼ αl,0 ðr Þ + αl,1 ðr Þt + αl,2 ðr Þt2 xu,2 ðt, r Þ ¼ αu,0 ðr Þ + αu,1 ðr Þt + αu,2 ðr Þt2 the linear system to determine the coefficients is SX ¼ Y where: 2 3 2 3 2 2

0 0 2


2 2 S S 6 7 S ¼ 1 2 , S2 ¼ 4 0 t 0 t 0 5 , S1 ¼ 4 1 1 1 5 , S2 S1 0 1 2 0 0 0 2 3 2 3 αl,0 ðr Þ 2=t 6 αl,1 ðr Þ 7 6 0:1
0:1r 7 6 7 6 7 6 αl,1 ðr Þ 7 6
0:25 + 0:25r 7 X¼6 7, Y ¼ 6 7 6 αu,0 ðr Þ 7 6 2=t 7 4 5 4 5 αu,1 ðr Þ 0:1
0:1r 0:25
0:25r αu,2 ðr Þ Choosing the collocation point as ¼ 32: 2

3 4 6 3 7 2 3 6 0:1
0:1r 7 1 9 6 7 1 6 7 6 7 S1 ¼ 4 1 20 40 5, Y ¼ 6 4
0:25 + 0:25r 7, 6 7 63 7 0 1 1 4 5 0:1
0:1r 0:25
0:25r 2

0 0 6 1 1 6 1 6 0 S¼6
8=9
4=3 6 4 0 0 0 0

2 1 2
2 0 0


8=9 0 0 0 1 0


4=3 0 0 0 1 1


2 0 0 2 1 2

3 7 7 7 7 7 5

the matrix S is nonsingular and by solving the SX ¼ Y the vector solution X can be determined easily and the solution in level-wise form is: xl ðt, r Þ ¼ 0:99
0:24r + ð
1:93 + 0:43r Þt + ð0:84
0:8999999996e
1r Þt2 xl ðt, r Þ ¼ 0:51
0:24r + ð
1:07 + 0:43r Þt + ð0:66
0:8999999996e
1r Þt2 For more illustration, Tables 5.16–5.18 show the comparison of the exact and approximate solutions at t ¼ 1, 1.001, 1.01 for r  [0, 1] with step size 0.1.

278

Soft Numerical Computing in Uncertain Dynamic Systems

TABLE 5.16 Comparison of the exact and approximate solutions at t ¼ 1. r

xl,

xl

Error

xu,

xu

Error

0


0.1


0.1

0

0.1

0.1

0

0.1


0.09


0.09

0

0.09

0.09

0

0.2


0.08


0.08

0

0.08

0.08

0

0.3


0.07


0.07

0

0.07

0.07

0

0.4


0.06


0.06

0

0.06

0.06

0

0.5


0.05


0.05

0

0.05

0.05

0

0.6


0.04


0.40000000e
1

0.1e
9

0.04

0.04

0

0.7


0.03


0.30000000e
1

0.1e
9

0.03

0.03

0

0.8


0.02


0.02

0

0.02

0.02

0

0.9


0.01


0.10000000e
1

0.1e
9

0.01

0.01

0

1


0.1e
9


0.1e
9

0

0

0

0

N

N

Case 3. Suppose that some coefficients as the following are negative and the rest are nonnegative: an
m
1 ðtÞ, an
m
2 ðtÞ,…, a1 ðtÞ,a0 ðtÞ Then we have: 8 ðnÞ ðn
1Þ ðn
mÞ > ðt, r Þ + ⋯ + an
m ðtÞxl ðt, r Þ xl ðt, r Þ + an
1 ðtÞxl > > ð n
m
1 Þ > > ð t Þx ð t, r Þ + ⋯ + a ð t Þx ðt, r Þ ¼ gl ðt, r Þ + a n
m
1 0 u > u < xl ð0, r Þ ¼ bl,0 ðr Þ 0 > > xl ð0, r Þ ¼ bl,1 ðr Þ > > > > : ðn
1Þ ⋮ ð0, r Þ ¼ bl, n
1 ðr Þ xl 8 ðnÞ xu ðt, r Þ + an
1 ðtÞxðun
1Þ ðt, r Þ + ⋯ + an
m ðtÞxðun
mÞ ðt, r Þ > > > ðn
m
1Þ > > ðt, r Þ + ⋯ + a0 ðtÞxl ðt, r Þ ¼ gu ðt, r Þ + an
m
1 ðtÞxl > < xu ð0, r Þ ¼ bu,0 ðr Þ > x0u ð0, r Þ ¼ bu,1 ðr Þ > > > > > : ðn
1Þ ⋮ xu ð0, r Þ ¼ bu, n
1 ðr Þ Now by using the same previously mentioned linear combinations, we will have the following system of equations:

TABLE 5.17 Comparison of the exact and approximate solutions at t ¼ 1.001. r

xl,

xl

Error

xu,

xu

Error

0

0.1002491600

0.1003491004

0.999404e
4

0.1002506600

0.100351100

0.1004400e
3

0.1

0.902241690e
1

0.903140904e
1

0.899214e
4

0.902256690e
1

0.90316090e
1

0.904210e
4

0.2

0.801991780e
1

0.802790804e
1

0.799024e
4

0.802006780e
1

0.80281080e
1

0.804020e
4

0.3

0.701741870e
1

0.702440704e
1

0.698834e
4

0.701756870e
1

0.70246070e
1

0.703830e
4

0.4

0.601491960e
1

0.602090604e
1

0.598644e
4

0.601506960e
1

0.60211060e
1

0.603640e
4

0.5

0.501242050e
1

0.501740504e
1

0.498454e
4

0.501257050e
1

0.50176050e
1

0.503450e
4

0.6

0.400992140e
1

0.401390405e
1

0.398265e
4

0.401007140e
1

0.40141040e
1

0.403260e
4

0.7

0.300742230e
1

0.301040304e
1

0.298074e
4

0.300757230e
1

0.30106030e
1

0.303070e
4

0.8

0.200492320e
1

0.200690204e
1

0.197884e
4

0.200507320e
1

0.20071020e
1

0.202880e
4

0.9

0.100242410e
1

0.100340104e
1

0.97694e
5

0.100257410e
1

0.10036010e
1

0.102690e
4

1

0.7500e
6

0.1000e
5

0.2500e
6

0.7500e
6

0.1000e
5

0.2500e
6

N

N

TABLE 5.18 Comparison of the exact and approximate solutions at t ¼ 1.01. r

xl,

xl

Error

xu,

xu

Error

0

0.1024160000

0.1034103795

0.99670e
4

0.1025660000

0.103609720

0.10437200e
2

0.1

0.921669000e
1

0.930593746e
1

0.8924746e
3

0.923169000e
1

0.93258715e
1

0.9418150e
3

0.2

0.819178000e
1

0.827083696e
1

0.7905696e
3

0.820678000e
1

0.82907710e
1

0.8399100e
3

0.3

0.614196000e
1

0.723573647e
1

0.6886647e
3

0.718187000e
1

0.72556705e
1

0.7380050e
3

0.4

0.511705000e
1

0.620063597e
1

0.5867597e
3

0.615696000e
1

0.62205700e
1

0.6361000e
3

0.5

0.409214000e
1

0.516553548e
1

0.4848548e
3

0.513205000e
1

0.51854695e
1

0.5341950e
3

0.6

0.306723000e
1

0.413043499e
1

0.3829499e
3

0.410714000e
1

0.41503690e
1

0.4322900e
3

0.7

0.306723000e
1

0.309533450e
1

0.2810450e
3

0.308223000e
1

0.31152685e
1

0.3303850e
3

0.8

0.204232000e
1

0.206023400e
1

0.1791400e
3

0.205732000e
1

0.20801680e
1

0.2284800e
3

0.9

0.101741000e
1

0.102513351e
1

0.772351e
4

0.103241000e
1

0.10450675e
1

0.1265750e
3

1

0.750000e
4

0.99670e
4

0.246700e
4

0.750000e
4

0.99670e
4

0.246700e
4

N

N

281

Discrete numerical solutions of uncertain differential equations N X

ðnÞ

αl, k ðr Þϕk ðtÞ + an
1 ðtÞ

k¼0

N X

ðn
1Þ

αl, k ðr Þϕk

ðtÞ + ⋯ + an
m ðtÞ

k¼0

+an
m
1 ðtÞ

N X

N X

ðn
m
1Þ

αu, k ðr Þϕk

k¼0

ðtÞ + ⋯ + a0 ðtÞ

N X

αu, k ðr Þϕk ðtÞ ¼ gl ðt, r Þ

N X

αl,k ðr Þϕ0k ð0Þ ¼ bl,1 ðr Þ, …,

k¼0

ðnÞ

k¼0

N X

N X

ðn
1Þ

αl, k ðr Þϕk

ð 0Þ

N X

ðn
mÞ

k¼0

¼ bl, n
1 ðr Þ

αu, k ðr Þϕk ðtÞ + an
1 ðtÞ

ðn
1Þ

αu, k ðr Þϕk

ðtÞ + ⋯ + an
m ðtÞ

k¼0 N X

ðn
m
1Þ

αl,k ðr Þϕk

ðtÞ

ðtÞ + ⋯ + a0 ðtÞ

N X

αl, k ðr Þϕk ðtÞ ¼ gl ðt, r Þ

k¼0

αu, k ðr Þϕk ð0Þ ¼ bu,0 ,

k¼0

αu, k ðr Þϕk

k¼0

k¼0 N X

ðt Þ

k¼0

αl, k ðr Þϕk ð0Þ ¼ bl,0 ,

+an
m
1 ðtÞ

ðn
mÞ

αl, k ðr Þϕk

k¼0

k¼0

N X

N X

N X

αu, k ðr Þϕ0k ð0Þ ¼ bu,1 ðr Þ,…,

k¼0

N X

ðn
1Þ

αu, k ðr Þϕk

ð 0Þ

k¼0

¼ bu, n
1 ðr Þ

By setting: ðnÞ

ðn
1Þ

ϕk ðtÞ + an
1 ðtÞϕk ðn
m
1Þ

an
m
1 ðtÞϕk

ðn
mÞ

ðtÞ + ⋯ + an
m ðtÞϕk ðn
m
2Þ

ðtÞ + an
m
2 ðtÞ + ϕk

ðtÞ ¼ ηk

ðtÞ + ⋯ + a0 ðtÞϕk ðtÞ ¼ ξk

ð jÞ

ϕk ð0Þ ¼ σ jk , j ¼ 0,1, …, n
1, k ¼ 0, 1,…, N and substituting, we get: 8 N X > > > ðαl, k ðr Þηk + αu, k ðr Þξk Þ ¼ gl ðt, r Þ > > > > k¼0 > >X > < N αl, k ðr Þσ 0k ¼ bl, 0 ðr Þ > k¼0 > > ⋮ > > > N > X > > > αl, k ðr Þσ n
1, k ¼ bl,n
1 ðr Þ : k¼0

8 N X > > > ðαu, k ðr Þηk + αl, k ðr Þξk Þ ¼ gu ðt, r Þ > > > > k¼0 > >X > < N αu, k ðr Þσ 0k ¼ bu,0 ðr Þ > k¼0 > > ⋮ > > > N > X > > > αu, k ðr Þσ n
1, k ¼ bu, n
1 ðr Þ : k¼0

The matrix form of these systems of linear equations can be formed as the following block-wise linear system SX ¼ Y, where:

282

Soft Numerical Computing in Uncertain Dynamic Systems

S¼

S1 S 2 S2 S1

2

η0

6 σ 0,0 , S1 ¼ 6 4 ⋮ 2ðN + 1ÞðN + 1Þ σ n
1,0 2

ξ0 6 0 S2 ¼ 6 4⋮ 0

ξ1 0 ⋮ 0

⋯ ⋯ ⋯ ⋯

3 η1 ⋯ ηN σ 0,1 ⋯ σ 0, N
1 7 7 5 ⋮ ⋯ ⋮ σ n
1,1 ⋯ σ n
1,N
1 ðN + 1ÞðN + 1Þ 3 ξN 0 7 7 ⋮5 0 ðN + 1ÞðN + 1Þ

X ¼ ½αl,0 ðr Þ, αl,1 ðr Þ, αl,2 ðr Þ, …, αl, N ðr Þ, αu,0 ðr Þ, αu,1 ðr Þ, αu, 2 ðr Þ, …, αu, N ðr ÞT Y ¼ ½gl ðt, r Þ, bl,0 ðr Þ, …, bl,n
1 ðr Þ, gu ðt, r Þ, bu, 0 ðr Þ, …, bu, n
1 ðr Þ By the same process that we have discussed in Case 1, we can use the collocation points like t ¼ a  [0, T] to realize the system of linear equations to find the coefficients: 8 N 8 N X X > > > > > > ð α ð r Þη + α ð r Þξ Þ ¼ g ð a, r Þ ðαu, k ðr Þηk + αl, k ðr Þξk Þ ¼ gu ða, r Þ l, k u, k l > > k k > > > > > > k¼0 k¼0 > > >X >X > > < N < N αl, k ðr Þσ 0k ¼ bl, 0 ðr Þ αu, k ðr Þσ 0k ¼ bu,0 ðr Þ > > k¼0 k¼0 > > > > ⋮ ⋮ > > > > > > N N > > X > >X > > > > αl, k ðr Þσ n
1, k ¼ bl, n
1 ðr Þ αu, k ðr Þσ n
1,k ¼ bu, n
1 ðr Þ : : k¼0

k¼0

Then all: αl,0 ðr Þ, αl,1 ðr Þ, αl,2 ðr Þ, …, αl, N ðr Þ,αu,0 ðr Þ,αu,1 ðr Þ,αu,2 ðr Þ,…, αu, N ðr Þ are determined very easily for arbitrary but fixed r  [0, 1]. Therefore, the solutions xl(t, r) and xu(t, r) are approximated. Example Consider the following second-order fuzzy linear differential equation: 8 00 0 < xgH ðtÞH 4xgH ðtÞ4⊙xðtÞ ¼ 0, t 0 xð0Þ ¼ ð2 + r, 4
r Þ : x0 ð0Þ ¼ ð5 + r, 7
r Þ gH The fuzzy exact solution of the problem is: xl ðt, r Þ ¼ ð2 + r Þe2t + ð1
r Þte2t xu ðt, r Þ ¼ ð4
r Þe2t + ðr
1Þte2t

Discrete numerical solutions of uncertain differential equations

283

If the basis functions is ϕk(t) ¼ tk, k ¼ 0, 1, 2 and we have: xl,2 ðt, r Þ ¼ αl,0 ðr Þ + αl,1 ðr Þt + αl,2 ðr Þt2 xu,2 ðt, r Þ ¼ αu,0 ðr Þ + αu,1 ðr Þt + αu,2 ðr Þt2 the linear system to determine the coefficients is SX ¼ Y where: 2 3 2 3

0
4
8t 4 4t 2 + 4t2 S 1 S2 , S2 ¼ 4 1 0 S¼ 0 5, 0 5 , S1 ¼ 4 0 0 S2 S1 0 0 0 0 1 0 2 3 2 3 αl,0 ðr Þ
0:25 + 0:25r 6 αl,1 ðr Þ 7 62+r 7 6 7 6 7 6 αl,1 ðr Þ 7 6 5+r 7 X¼6 7, Y ¼ 6 7 6 αu,0 ðr Þ 7 6 0:25
0:25r 7 4 5 4 5 αu,1 ðr Þ 4
r 7
r αu,2 ðr Þ Choosing the collocation point as t ¼ 12, we will have a system of linear equations. The solution in level-wise form is: xl ðt, r Þ ¼ ð2 + r Þ + ð5 + r Þt + ð1:42851427
1:428571430r Þt2 xl ðt, r Þ ¼ ð4
r Þ + ð7
r Þt + ð
1:42851429 + 1:428571430r Þt2 For more illustration, Table 5.19 shows the comparison of the exact and approximate solutions at t ¼ 0.01 for r  [0, 1] with step size 0.1. Example. Electrical circuit Consider the electrical circuit in Fig. 5.8, where L ¼ 1h, R ¼ 2Ω, C ¼ 0.25f and E(t) ¼ 20 cos t. If x(t) is the charge on the capacitor at time t > 0, then we have: x00gH ðtÞ2⊙x0gH ðtÞ4⊙xðtÞ ¼ 50 cos t subject to: xð0Þ ¼ ð4 + r, 6
r Þ, x0gH ð0Þ ¼ ðr, 2
r Þ The exact solution in level-wise form is as the following form:   250     pffiffiffi pffiffiffi 150 150 13
t cos 3t + 6
pffiffiffi  sin 3t + cos t 4+r
xl ðt, r Þ ¼ e 13 3   13 100 sin t + 13   250     pffiffiffi pffiffiffi 150 150 13
t p ffiffi ffi xl ðt, r Þ ¼ e cos 3t + 6
cos t 6
r
 sin 3t + 13 13 3   100 + sin t 13

TABLE 5.19 Comparison of the exact and approximate solutions at t ¼ 1.01. r

xl,

xl

Error

xu,

xu

Error

0

2.050142857

2.050604693

0.461836e
3

4.069857143

4.070603347

0.746204e
3

0.1

2.151128571

2.151604626

0.476055e
3

3.968871429

3.969603414

0.731985e
3

0.2

2.252114286

2.252604559

0.490273e
3

3.867885714

3.868603481

0.717767e
3

0.3

2.353100000

2.353604491

0.504491e
3

3.766900000

3.767603549

0.703549e
3

0.4

2.454085714

2.454604424

0.518710e
3

3.665914286

3.666603616

0.689330e
3

0.5

2.555071429

2.555604357

0.532928e
3

3.564928571

3.565603683

0.675112e
3

0.6

2.656057143

2.656604289

0.547146e
3

3.463942857

3.464603751

0.660894e
3

0.7

2.757042857

2.757604222

0.561365e
3

3.362957143

3.363603818

0.646675e
3

0.8

2.858028571

2.858604155

0.575584e
3

3.261971429

3.262603885

0.632456e
3

0.9

2.959014286

2.959604087

0.589801e
3

3.160985714

3.161603953

0.618239e
3

1

3.060

3.060604020

0.604020e
3

3.060

3.060604020

0.604020e
3

N

N

Discrete numerical solutions of uncertain differential equations

285

L

C

E (t)

R Fig. 5.8 Electrical circuit.

If ϕk(t) ¼ ek, k ¼ 0, 1, 2, 3, 4 and using the same process we have the approximate values for xl, 4, xu, 4 in the form of: xl ðt, r Þ ¼ αl,0 ðr Þ + αl,1 ðr Þet + αl,2 ðr Þe2t + αl,3 ðr Þe3t + + αl,4 ðr Þe4t xu ðt, r Þ ¼ αu,0 ðr Þ + αu,1 ðr Þet + αu,2 ðr Þe2t + αu,3 ðr Þe3t + + αu,4 ðr Þe4t It is clear this case is exactly Case 1 of our discussion in this section and we should solve an 8  8 linear system of equations. To this end, we need three collocation points like ¼1, 1.1, 1.2; so finally, the following system is realized: 3 2 4 7e1 12e2 19e3 28e4 6 4 7e1:1 12e2:2 19e3:3 28e4:4 7

7 6 S S S ¼ 1 2 , S1 ¼ 6 4 7e1:2 12e2:4 19e3:6 28e4:8 7 7, S2 ¼ ½0 6 S3 S4 5 41 1 1 1 1 0 1 1 1 1 In conclusion, the approximate solutions are: xl,4 ðt, r Þ ¼ 4 + ð
2:255809230 + 1:889692363r Þet + ð3:042557688
1:104464241r Þe2t + ð0:8603224932 + 0:2321963367r Þe3t + ð0:7357403924e
1
0:1742445864e
1r Þe4t xu,4 ðt, r Þ ¼ 4 + ð1:523575496
1:889692363r Þet + ð0:833629204 + 1:104464241r Þe2t + ð
0:3959298198
0:2321963367r Þe3t + ð0:3872512196e
1 + 0:1742445864e
1r Þe4t Table 5.20 compares the exact solution and the results of approximate solution at the point t ¼ 0.001.

TABLE 5.20 Comparison of the exact and approximate solutions at t ¼ 0.001. r

xl,

xl

Error

xu,

xu

Error

0

4.001544314

4.002014989

0.470675e
3

6.002158904

6.000012991

0.2145913e
2

0.1

4.101575043

4.101914890

0.339847e
3

5.902128174

5.900113092

0.2015082e
2

0.2

4.201605773

4.201814790

0.209017e
3

5.802097444

5.800213192

0.1884252e
2

0.3

4.301636502

4.301714690

0.78188e
4

5.702066714

5.700313292

0.1753422e
2

0.4

4.401667231

4.401614590

0.52641e
4

5.602035984

5.600413392

0.1622592e
2

0.5

4.501697961

4.501514490

0.183471e
3

5.502005255

5.500513492

0.1491763e
2

0.6

4.601728690

4.601414390

0.314300e
3

5.401974526

5.400613591

0.1360935e
2

0.7

4.701759421

4.701314291

0.445130e
3

5.301943796

5.300713691

0.1230105e
2

0.8

4.801790150

4.801214191

0.575959e
3

5.201913068

5.200813791

0.1099277e
2

0.9

4.901820879

4.901114091

0.706788e
3

5.101882337

5.100913891

0.968446e
3

1

5.001851609

5.001013991

0.837618e
3

5.001851607

5.001013991

0.837616e
3

N

N

Discrete numerical solutions of uncertain differential equations

287

References Allahviranloo, T., 2020. Uncertain information and linear systems. In: Studies in Systems, Decision and Control. vol. 254. Springer. Allahviranloo, T., Salahshour, S., 2011. Euler method for solving hybrid fuzzy differential equation. Soft Comput. 15, 1247–1253. Allahviranloo, T., Ahmady, N., Ahmady, E., 2007. Numerical solution of fuzzy differential equations by predictor–corrector method. Inform. Sci. 177, 1633–1647. Allahviranloo, T., Ahmady, E., Ahmady, N., 2008. Nth-order fuzzy linear differential equations. Inform. Sci. 178, 1309–1324. Allahviranloo, T., Abbasbandy, S., Ahmady, N., Ahmady, E., 2009. Improved redictor–corrector method for solving fuzzy initial value problems. Inform. Sci. 179, 945–955. Allahviranloo, T., Gouyandeh, Z., Armand, A., Hasanoglu, A., 2015. On fuzzy solutions for heat equation based on generalized Hukuhara differentiability. Fuzzy Set. Syst. 265, 1–23. Armand, A., Allahviranloo, T., Abbasbandy, S., Gouyandeh, Z., 2019. The fuzzy generalized Taylor’s expansion with application in fractional differential equations. Iran J. Fuzzy Syst. 16 (2), 57–72. Gouyandeha, Z., Allahviranloob, T., Abbasbandyb, S., Armand, A., 2017. A fuzzy solution of heat equation under generalized Hukuhara differentiability by fuzzy Fourier transform. Fuzzy Set. Syst. 309, 81–97.

Further reading Ahmady, N., Allahviranloo, T., Ahmady, E., 2020. A modified Euler method for solving fuzzy differential equations under generalized differentiability. Comput. Appl. Math. 39, 104.

6.1

Chapter 6

Numerical solutions of uncertain fractional differential equations Introduction

The objectives for this chapter are introducing the Caputo fractional derivative on the fuzzy number valued functions and fuzzy fractional differential equations under this Caputo operator. After showing the existence and uniqueness of the fuzzy solution, the fuzzy Taylor method will be illustrated, and its consequence as the fuzzy fractional Euler method will be applied as numerical methods to solve the fuzzy fractional differential equations. To define the Caputo fractional operator, the Riemann-Liouville fractional operator should be defined first.

6.2 Fuzzy Riemann-Liouville Derivative—Fuzzy RL Derivative The fuzzy Riemann-Liouville derivative is defined as: ðs 1 α α I RL xðsÞ≔DRL xðsÞ ¼ ⊙ ðs  tÞα1 ⊙ xðtÞdt, s½t0 , T , ΓðαÞ t0 289 Soft Numerical Computing in Uncertain Dynamic Systems. https://doi.org/10.1016/B978-0-12-822855-5.00006-9 © 2020 Elsevier Inc. All rights reserved.

290

Soft Numerical Computing in Uncertain Dynamic Systems

for any 0 < α < 1. Indeed, it is the antioperator of the RL fractional integral: I αRL xðs, r Þ ¼

1 ⊙ ΓðαÞ

ðs t0

ðs  tÞα1 ⊙ xðt, r Þdt

One of the properties of this derivative entitled combination property can be explained as follows (Mathai & Haubold, 2017; Van Hoa et al., 2019).

6.2.1.1 Note—Combination Property ð α + β Þ

β β α Dα xðtÞ, 0 < α < 1, 0 < β < 1 RL DRL xðtÞ ¼ DRL DRL xðtÞ ¼ DRL ðs 1 ⊙ ðs  tÞα1 ⊙ xðtÞdt, 0 < α < 1: Dα RL xðsÞ ¼ ΓðαÞ t0 ðs 1 Dβ ⊙ ðs  tÞβ1 ⊙ xðtÞdt, 0 < β < 1 RL xðsÞ ¼ Γ ðβ Þ t0

Now: β Dα RL DRL xðsÞ ¼

1 ⊙ ΓðαÞ

ðs t0

ðs  t Þ

α1



1 ⊙ ⊙ Γ ðβ Þ

ðs u

ð t  uÞ

β1

 ⊙ xðuÞdu dt

Since the functions s  t, t  u, Γ(α), Γ(β) are positive, the same procedure for the real cases occurs here. β Dα RL DRL xðsÞ ¼

1 ⊙ Γðα + βÞ

ðs t0

xðuÞdu ⊙

ð s u

ðs  tÞ

α1

ð t  uÞ

β1

 dt

tu ¼ v, where s, u are fixed, then: By changing some variables such as su

1 dt ¼ dv, ðt  uÞβ1 ¼ vβ1 ðs  uÞβ1 , su 1v¼

st ¼)ðs  tÞα1 ¼ ð1  vÞα1 ðs  uÞα1 su

then: ðs  tÞα1 ðt  uÞβ1 ¼ ð1  vÞα1 ðs  uÞα1 vβ1 ðs  uÞβ1 ¼ ðs  uÞα + β2 ð1  vÞα1 vβ1

Numerical Solutions of Uncertain Fractional Differential Equations 291 By substituting: β Dα RL DRL xðsÞ ¼

¼

1 ⊙ Γð α + β Þ

1 ⊙ Γ ðα + β Þ

ðs t0

ðs t0

xðuÞdu ⊙

ð 1

α + β1

ð s  uÞ

ð1  v Þ

α1 β1

v

 dv

0

ð 1  ðs  uÞα + β1 ⊙ xðuÞdu ð1  vÞα1 vβ1 dv 0

Since: ð1

ð1  vÞα1 vβ1 dv ¼ 1

0

then: β Dα RL DRL xðsÞ ¼

1 ⊙ Γð α + β Þ

ðs t0

ðα + βÞ

ðs  uÞα + β1 ⊙ xðuÞdu ¼ DRL

xðsÞ

β α The same procedure can be used to prove D RL DRL x(t).

6.2.1.2 Level-Wise form of Fuzzy Riemann-Liouville Integral Operators Based on the definition of the integral and its level-wise form, we know that the level-wise form of integral is the integrals of end points of the interval in the level-wise form of integrand function.

6.2.1.3 The RL Fractional Integral Operator   I αRL xðs, r Þ ¼ IαRL xl ðs, r Þ, IαRL xu ðs, r Þ , s½t0 , T  where: IαRL xl ðs, r Þ ¼

1 ΓðαÞ

I αRL xu ðs, r Þ ¼

1 ΓðαÞ

because, Γ(α) > 0 and (s  t)α1 > 0.

ðs t0

ðs t0

ðs  tÞα1 xl ðt, r Þdt

ðs  tÞα1 xu ðt, r Þdt

292

Soft Numerical Computing in Uncertain Dynamic Systems

6.2.1.4 The Fuzzy Riemann-Liouville Derivative Operators In the definition of RL fractional integral: ðs 1 Dα x ð s Þ ¼ ðs  tÞα1 ⊙ xðtÞdt, 0 < α < 1 ⊙ RL ΓðαÞ t0 or: DαRL xðsÞ≔

1 ⊙ Γð1  αÞ

ðs t0

ðs  tÞα ⊙ xðtÞdt, 0 < α < 1

assume: ðs

xðtÞ dt ¼ f ðsÞR , xðtÞR ð s  t Þα t0

so: d ds

ðs

xðtÞ 0 dt ¼ fgH ðsÞ, s½t0 , T  ð s  tÞα t0

Remark. The function f(t) is gH-differentiable if and only if fl0 (t, r) and fu0 (t, r) are differentiable with respect to t for all 0  r  1 and:      0 fgH ðtÞ ¼ min fl0 ðt, r Þ, fu0 ðt, r Þ , max fl0 ðt, r Þ, fu0 ðt, r Þ Theorem. The necessary and sufficient condition for RL gH-differentiability of x(t) is gH-differentiability of f(s). This means:      DαRLgH xðt, r Þ ¼ min DαRL xl ðt, r Þ, DαRL xu ðt, r Þ , max DαRL xl ðt, r Þ, DαRL xu ðt, r Þ The proof is very easy, since s  t > 0: ðs ðs xl ðt, r Þ xu ðt, r Þ fl ðs, r Þ ¼ dt, f ð s, r Þ ¼ u α α dt ð s  t Þ ð t0 t0 s  tÞ  1 1 0 0 min f ðt, r Þ, f ðt, r Þ Γð1  αÞ l Γð1  αÞ u  1 1 fl0 ðt, r Þ, fu0 ðt, r Þ  max Γð1  αÞ Γð1  αÞ then:     min DαRL xl ðt, r Þ, DαRL xu ðt, r Þ  max DαRL xl ðt, r Þ, DαRL xu ðt, r Þ This means that the following interval defines an interval for all r  [0, 1]:   DαRLgH xðt, r Þ ¼ DαRL xl ðt, r Þ, DαRL xu ðt, r Þ

Numerical Solutions of Uncertain Fractional Differential Equations 293

6.3

Fuzzy Caputo Fractional Derivative

In this section, another form of the fractional derivative entitled the fuzzy Caputo fractional derivative is expressed (Chehlabi & Allahviranloo, 2016; Garrappa et al., 2019; Mathai & Haubold, 2017; Van Hoa et al., 2019).

6.3.1.1 Caputo gH-Differentiability The fuzzy number valued function x(t) is Caputo gH-differentiable if and only if xl0 (t, r) and xu0 (t, r) are differentiable with respect to t for all 0  r  1 and:      DαCgH xðt, r Þ ¼ min DαC xl ðt, r Þ, DαC xu ðt, r Þ , max DαC xl ðt, r Þ, DαC xu ðt, r Þ where: DαC xl ðs, r Þ ¼

1 Γð1  αÞ

ðs

x0l ðt, r Þ 1 α α dt, DC xu ðt, r Þ ¼ ð s  t Þ Γ ð 1  αÞ t0

ðs

x0u ðt, r Þ α dt t0 ðs  tÞ

The proof is straight: DαCgH xðs, r Þ ¼

1 ⊙ Γð1  αÞ

ðs t0

x0gH ðt, r Þ ðs  tÞα

dt

We have:      x0gH ðs, r Þ ¼ min x0l ðs, r Þ, x0u ðs, r Þ , max x0l ðs, r Þ, x0u ðs, r Þ then: x0gH ðs, r Þ ¼





ðs 0 ðs 0 1 xl ðt, r Þ 1 xu ðt, r Þ min dt, dt , Γð1  αÞ t0 ðs  tÞα Γð1  αÞ t0 ðs  tÞα   ðs 0 ðs 0 1 xl ðt, r Þ 1 xu ðt, r Þ dt, dt max Γð1  αÞ t0 ðs  tÞα Γð1  αÞ t0 ðs  tÞα

This is exactly:      DαCgH xðt, r Þ ¼ min DαC xl ðt, r Þ, DαC xu ðt, r Þ , max DαC xl ðt, r Þ, DαC xu ðt, r Þ Also, in the case where 0 < α < 1, we have two cases of differentiability: •

i  gH (differentiable) x0igH ðtÞ ¼ lim

h!0

xðt + hÞ  H xðtÞ h

Its level-wise form:   DαCi2gH xðs, r Þ ¼ DαC xl ðs, r Þ, DαC xu ðs, r Þ

294 •

Soft Numerical Computing in Uncertain Dynamic Systems

ii  gH (differentiable) x0iigH ðtÞ ¼ lim

h!0

xðtÞ  H xðt + hÞ h

Its level-wise form:   DαCii2gH xðs, r Þ ¼ DαC xu ðs, r Þ, DαC xl ðs, r Þ Example. Consider the same fuzzy exponential c⊙t xðtÞ ¼ e , tðt0 , T ,cR , cðr Þ ¼ ðcl ðr Þ, cu ðr ÞÞ : ðs 1 c ⊙ ec ⊙ t ⊙ DαCgH ec ⊙ s ¼ α dt Γð1  αÞ t0 ðs  t Þ

function

We know that: xl ðt, r Þ ¼ ecl ðrÞt , xu ðt, r Þ ¼ ecu ðrÞt x0l ðt, r Þ ¼ cl ðr Þecl ðrÞt , x0u ðt, r Þ ¼ cu ðr Þecu ðrÞt because t > 0 and the exponential function is increasing. Thus: h i x0gH ðt, r Þ ¼ cl ðr Þecl ðrÞt , cu ðr Þecu ðrÞt Finally, the i-differential is found as follows:   DαCi2gH xðs, r Þ ¼ DαC xl ðs, r Þ, DαC xu ðs, r Þ where: DαC xl ðs, r Þ ¼

1 Γ ð1  α Þ

ðs

cl ðr Þecl ðrÞt cl ðr Þ α dt ¼ ð s  t Þ Γ ð 1  αÞ t0

ðs

ecl ðrÞt α dt t0 ðs  tÞ

Let v ¼ s  t, and then dt ¼  dv, and ec⊙t ¼ ec⊙(sv), s  v 0. By these condiÐ tions, the integral equation t0s(s  t) α ⊙ ec⊙tdt is always i  gH differentiable, and: ð ð cl ðr Þ st0 ecl ðrÞðsvÞ cl ðr Þecl ðrÞs st0 ecl ðrÞv α DC xl ðs, r Þ ¼ dv ¼ dv vα Γð1  αÞ 0 vα Γð1  αÞ 0 The integral equation: ð st0 cl ðrÞv ð st0 1 e 1 dv ¼ vα ecl ðrÞv dv vα Γ ð1  α Þ 0 Γð1  αÞ 0 Ð 0  α  cl(r)v where st e dv is an incomplete γ(v, t) function, thus: 0 v DαC xl ðs, r Þ ¼

cl ðr Þecl ðrÞs γ ðv, tÞ Γ ð1  α Þ

Numerical Solutions of Uncertain Fractional Differential Equations 295 Using the same process: DαC xu ðs, r Þ ¼

cl ðr Þecl ðrÞs γ ðv, tÞ, 0 < α < 1 Γð1  αÞ

Now if for any r  [0, 1], the conditions of the interval are satisfied then the Caputo derivative for this exponential function exists. This is exactly depending on the sign of γ(v, t) and it can be variational. Then it is not easy to claim that: DαC xl ðs, r Þ DαC xu ðs, r Þ, 0  r  1 Remark. For xðtÞR and 0 < α < β < 1 we have: • •

(DRLgHαIαRLx)(t) ¼ x(t) (DRLgHαIβRLx)(t) ¼ I(βα) RL x(t)

To show the first item:

0 ð1αÞ DαRLgH I αRL xðtÞ ¼ I RL I αRL x ðt Þ gH

1 ⊙ ¼ Γ ð1  α Þ

ð t t0

α

ðt  sÞ

⊙ I αRL xðsÞds

0 gH

where: I αRL xðsÞ ¼

1 ⊙ ΓðαÞ

ðs t0

ðt  uÞα1 ⊙ xðuÞdu

By substituting and by using the Dirichlet formula, the known formula for the us : Beta function, and setting ¼ ts DαRLgH I αRL xðtÞ ¼ 1 ⊙ ¼ Γð1  αÞ ¼

ðt

1 1 α ⊙ ð t  s Þ Γ ð αÞ t0

1 ⊙ ΓðαÞΓð1  αÞ

ð t t0

x ðs Þ ⊙

ðs

! !0

1

1α t0 ðs  u Þ

ðt

u  sÞα1 α duds s ð t  uÞ

xðuÞdu ds gH

0 gH

0 10 ðt ð1 B C 1 ⊙B ¼ xðsÞds ⊙ ð1  vÞα vα1 du C @ A ΓðαÞΓð1  αÞ t0 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 1

¼

Bð1  α, αÞ ⊙ ΓðαÞΓð1  αÞ

ð t t0

xðsÞsp1 ds

0 ¼ x ðt Þ gH

gH

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Soft Numerical Computing in Uncertain Dynamic Systems

The second property: DαRLgH I βRL xðtÞ ¼ ðt

1 ⊙ ¼ ΓðβÞΓð1  αÞ 1 ⊙ Γ ð β  α + 1Þ

¼

1

t0 ðt  sÞ

ðt

αβ

1

t0 ðt  sÞ

αβ

⊙ xðsÞds ⊙

ð1

!0 α α1

ð1  v Þ v

0 ðβαÞ

⊙ xðsÞds ¼ IRL

du gH

xðtÞ

Remark. If in the interval (t0, T] the type of differentiability does not change, it means it is either i  gH differentiable or ii  gH differentiable. Then the following relation as a relation of fractional integral and derivative operators is established: I αRL DαRLgH xðtÞ ¼ xðtÞ  gH

ðt  t0 Þα1 ð1αÞ ⊙ IRL xðt0 Þ, tðt0 , T  ΓðαÞ

where I(1α) RL x(t0) exists and: ð1αÞ

ð1αÞ

lim I RL xðtÞ ¼ I RL xðt0 Þ

t!t0 +

To show the assertion, we consider two cases: •

i  gH (differentiability) h i I αRL DαRLi2gH xðt, r Þ ¼ I αRL DαRLgH xl ðt, r Þ, I αRL DαRLgH xu ðt, r Þ

where: I αRL DαRLgH xl ðt, r Þ ¼ xl ðt, r Þ 

ðt  t0 Þα1 ð1αÞ IRL xl ðt0 , r Þ ΓðαÞ

IαRL DαRLgH xu ðt, r Þ ¼ xu ðt, r Þ 

ðt  t0 Þα1 ð1αÞ IRL xu ðt0 , r Þ ΓðαÞ

Now, since xl(t, r) is increasing, then IαRLDRLgHαxl(t, r) is also increasing and is decreasing because xu(t, r) is decreasing.

IαRLDRLgHαxu(t, r)

6.3.1.2 Caputo-Katugampola gH-Fractional Derivative If the RL derivative DRLgHαx(t) exists in [to, T] and 0 < α < 1:   DαCKgH xðtÞ ¼ DαRLgH xðtÞ  gH xðt0 Þ

Numerical Solutions of Uncertain Fractional Differential Equations 297 In the sequel, some relations between fuzzy type Riemann-LiouvilleKatugampola generalized fractional derivative and fuzzy type Caputo-Katugampola fractional derivative are shown (Mathai & Haubold, 2017; Van Hoa et al., 2019). Remark. Assume x(t) is an absolutely continuous fuzzy number valued function that does have increasing or decreasing length, i.e., it is i  gH differentiable or ii  gH differentiable, then: ðt 1 DαCKgH xðtÞ ¼ ðt  sÞα ⊙ x0 ðsÞds, t½t0 , T , 0 < α < 1 Γð1  αÞ t0 To show the relation, we know the fuzzy function I(1α) RL x(t) is absolutely continuous because in its relation: ðt 1 ð1αÞ ⊙ ðt  sÞα ⊙ xðsÞds IRL xðtÞ ¼ Γ ð1  α Þ t0 1 the coefficients ðt  sÞα > 0, Γð1α Þ > 0 and the fuzzy function x(s) is absolutely (1α) 0 continuous; thus (IRL x)gH (t) exists and finally, DRLgHαx(t) exists for t  (t0, T]:

0 ð1αÞ DαRLgH xðtÞ ¼ I RL x ðtÞ: gH

Now, let us consider a constant fuzzy function like yR , which is y(t) ≔ x(t0). Then: ð1αÞ

ð1αÞ

I RL yðtÞ ¼ I RL xðt0 Þ ¼

ðt  t0 Þ1α ⊙ xðt0 Þ Γð2  αÞ

and if α : ! 1 + α: DαRLgH yðtÞ ¼ I α RL yðtÞ ¼

ðt  t0 Þα ⊙ xðt0 Þ Γ ð1  α Þ

This is the reason for gH-differentiability of I(1α) RL x (in two cases) on (t0, T] and it follows that:     DαCKgH xðtÞ ¼ DαRLgH xðtÞ  gH xðt0 Þ ¼ DαRLgH xðtÞ  gH yðtÞ ¼ DαRLgH xðtÞ  gH DαRLgH yðtÞ ¼ DαRLgH xðtÞ  gH

ðt  t0 Þα ⊙ xðt0 Þ Γð1  αÞ

so: DαCKgH xðtÞ ¼ DαRLgH xðtÞ  gH

ðt  t0 Þα ⊙ xðt0 Þ Γð1  αÞ

Two sides in the level-form and based on the first case of gH-difference, we have: Case i  gH difference:

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Soft Numerical Computing in Uncertain Dynamic Systems

DαRLgH xðt, r Þ ¼ DαCKgH xðt, r Þ 

ðt  t0 Þα xðt0 , r Þ Γ ð1  α Þ

In the interval form: h i DαRLgH xl ðt, r Þ, DαRLgH xu ðt, r Þ ¼ h i ðt  t Þα 0 ½xl ðt0 , r Þ, xu ðt0 , r Þ ¼ DαCKgH xl ðt, r Þ, DαCKgH xu ðt, r Þ + Γ ð1  α Þ Therefore, for any r  [0, 1] and t  (t0, T]: DαRLgH xðtÞ ¼ DαCKgH xðtÞ 

ðt  t0 Þα ⊙ xðt0 Þ Γ ð1  α Þ



ðt  t0 Þα ð1αÞ d ⊙ xðt0 Þ x ðtÞ  ¼ I RL Γð1  αÞ ds Substituting in the following: DαCKgH xðtÞ ¼ DαRLgH xðtÞ  gH

ðt  t0 Þα ⊙ xðt 0 Þ Γð1  αÞ

we get: DαCKgH xðtÞ ¼



ð1αÞ d I RL x ðtÞ ds

Based on the definition of: ð1αÞ

I RL xðtÞ ¼

1 ⊙ Γ ð1  α Þ

ðt t0

sp1 ðt  sÞα ⊙ xðsÞds

it is concluded:

ð1αÞ

IRL

ðt d 1 x ðtÞ ¼ ⊙ ðt  sÞα ⊙ x0 ðsÞds ds Γð1  αÞ t0

Thus, the proof is completed in the case of i  gH difference. Case ii  gH difference: DαCKgH xðt, r Þ ¼ DαRLgH xðt, r Þ  ð1Þ

ðt  t0 Þα ⊙ xðt0 , r Þ Γ ð1  α Þ

Because in level-wise form, since the function x(t) is ii  gH differentiable, so: h i DαRLgH xðt, r Þ ¼ DαRLgH xu ðt, r Þ, DαRLgH xl ðt, r Þ ,

Numerical Solutions of Uncertain Fractional Differential Equations 299 h i α,p α, p Dα,p CKgH xðt, r Þ ¼ DCKgH xu ðt, r Þ, DCKgH xl ðt, r Þ h

i DαCKgH xu ðt, r Þ, DαCKgH xl ðt, r Þ ¼

h i ðt  t0 Þα ½xl ðt0 , r Þ, xu ðt0 , r Þ ¼ DαRLgH xu ðt, r Þ, DαRLgH xl ðt, r Þ + ð1Þ Γð1  αÞ Therefore, for any r  [0, 1] and t  (t0, T]: 



 ð1αÞ d ð1αÞ d I RL xl ðt, r Þ, IRL xu ðt, r Þ ¼ ds ds h i ðt  t0 Þα ¼ DαRLgH xu ðt, r Þ, DαRLgH xl ðt, r Þ + ð1Þ ½xl ðt0 , r Þ, xu ðt0 , r Þ Γð1  αÞ so we have:

ð1αÞ IRL

d ðt  t0 Þα ð1Þ ⊙ xðt0 Þ x ðtÞ ¼ DαRLgH xðtÞ  Γ ð1  α Þ ds

Substituting in the following: DαCKgH xðtÞ ¼ DαRLgH xðtÞ  gH

ðt  t0 Þα ⊙ xðt0 Þ Γð1  αÞ

we get: DαCKgH xðtÞ ¼



ð1αÞ IRL

d x ðtÞ ds

Based on the definition of: ð1αÞ

IRL xðtÞ ¼

1 ⊙ Γ ð1  α Þ

ðt t0

ðt  sÞα ⊙ xðsÞds

it is concluded:

ð1αÞ IRL

ðt d 1 x ðt Þ ¼ ⊙ ðt  sÞα ⊙ x0 ðsÞds ds Γð1  αÞ t0

Thus, the proof is also completed in the case of ii  gH difference. Remark. The RL integral operator is bounded: ðt 1 ⊙ ðt  sÞα1 ⊙ xðsÞds I αRL xðtÞ ¼ ΓðαÞ t0

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Soft Numerical Computing in Uncertain Dynamic Systems

It can be shown by Hausdorff distance:   1 sup DH I αRL xðtÞ, 0  DH ðxðsÞ, 0Þ ΓðαÞ t½t0 , T  ¼

ðt t0

ðt  sÞα1 ds ¼

1 DH ðxðsÞ, 0ÞðT  t0 Þα Γðα + 1Þ

As mentioned before, the existence of the Caputo-Katugampola depends on the continuity of x(t) and it has been supposed that it is absolutely continuous. Then we claim the following remark. Remark. DCKgHαx(t) ¼ 0 at t ¼ 0. Since I(1α) is bounded, the Caputo-Katugampola derivative is continuous: RL

ð1αÞ d α DCKgH xðtÞ ¼ I RL x ðtÞ ds To show the assertion, it is enough to show that the upper bound of the derivative goes to zero at the point t ¼ 0. Then:



  ð1αÞ DH DαCKgH xðtÞ, 0 ¼ DH I RL x0 ðtÞ, 0  

1 DH ðx0 , 0Þðt  t0 Þ1α Γ ð2  α Þ

1 sup DH ðx0 , 0Þðt  t0 Þ1α Γð2  αÞ s½t0 , T 

The supremum sups½t0 , T  DH ðx0 , 0Þ is a real number like k, then:

 1 kðt  t0 Þ1α DH DαCKgH xðtÞ, 0  Γð2  αÞ It is clear that, at the point t ¼ 0, the distance goes to zero, and this completes the proof. Remark. If the function x(t) is i  gH or ii  gH differentiable on (t0, T], for 0 < α < 1, we have: • •

IαRLDCKgHαx(t) ¼ x(t)gHx(t0) DCKgHαIαRLx(t) ¼ x(t)

The first item:



ð1αÞ d x ðtÞ I αRL DαCKgH xðtÞ ¼ I αRL I RL ds

ðt d ¼ I 1RL x ðtÞ ¼ x0 ðsÞds ¼ xðtÞ  gH xðt0 Þ ds t0

Numerical Solutions of Uncertain Fractional Differential Equations 301 because in the definition: I αRL xðtÞ ¼

1 ⊙ ΓðαÞ

ðt t0

ðt  sÞα1 ⊙ xðsÞds

If α ¼ 1 then: I 1RL xðtÞ ¼

ðt t0

xðsÞds

To prove the second item, we have the following relations. For x(t0) as a constant fuzzy, we have: ðt 1 ðt  t0 Þ1α ð1αÞ I RL xðt0 Þ ¼ ⊙ xðt0 Þ ⊙ ðt  sÞα ⊙ xðt0 Þds ¼ Γð2  αÞ Γ ð1  α Þ t0 and if α : ! 1 + α: DαRLgH xðt0 Þ ¼ I α RL yðtÞ ¼

ðt  t0 Þα ⊙ xðt0 Þ Γ ð1  α Þ

  ðt  t0 Þα ⊙ xðt0 Þ DαCKgH xðtÞ ¼ DαRLgH xðtÞ  gH xðt0 Þ ¼ DαRLgH xðtÞ  gH Γ ð1  α Þ Now by substituting IαRLx(t) ! x(t), we get: DαCKgH I αRL xðtÞ ¼ DαRLgH I αRL xðtÞ  gH

ðt ¼ t0 Þα  α  ⊙ I RL x ðt0 Þ Γ ð1  α Þ

If we show (IαRLx)(t0) ¼ 0, the proof is completed. To do this, we use the distance: ð  α   1 t ðt  t 0 Þα ðt  sÞα1 ds ¼ k DH I RL x ðt0 Þ, 0  k Γ ð1 + α Þ ΓðαÞ t0 where supt½t0 , T  DH ðxðtÞ, 0Þ ¼ k. Then:    ðt  t 0 Þα !0 DH I αRL x ðt0 Þ, 0  k Γ ð1 + α Þ

6.4 Fuzzy Fractional Differential Equations—CaputoKatugampola Derivative In this section, we are going to cover the fuzzy fractional differential equations that can be defined by each operator, which were defined earlier (Chehlabi & Allahviranloo, 2016; Van Hoa et al., 2019).

302

Soft Numerical Computing in Uncertain Dynamic Systems

6.4.1.1 Definition—Fuzzy Fractional Differential Equations Consider the following fuzzy fractional differential equation with fuzzy initial value: DαCKgH xðtÞ ¼ f ðt, xðtÞÞ, xðt0 Þ ¼ x0 , t½t0 , T , 0 < α < 1: where f : ½t0 , T   R ! R is a fuzzy number valued function, x(t) is a continuous fuzzy set valued solution, and DCKgHα is the fractional Caputo-Katugampola fractional operator. The first discussion is about the relation between the solution of fractional differential equations and solution of its corresponding fuzzy fractional differential equation. This means the fuzzy solution is the common solution of two fractional operators, differential and integral. In the following remark, we suppose that the fuzzy solution is i  gH differentiable or ii  gH differentiable on (t0, T]. Remark. Considering the abovementioned assumptions, the continuous fuzzy function x(t) is the solution of: DαCKgH xðtÞ ¼ f ðt, xðtÞÞ, xðt0 Þ ¼ x0 , t½t0 , T , 0 < α < 1 if, and only if, x(t) satisfies the following integral equation: ð 1 t ðt  sÞα1 f ðs, xðsÞÞds, t½t0 , T  xðtÞ  gH xðt0 Þ ¼ ΓðαÞ t0 Before we show the assertion, it should be noted that the function x(t) can be i  gH differentiable (increasing length) or ii  gH differentiable (decreasing length) on p (t0, T] but the fractional integral operator Iα, RL always has increasing length. To prove the remark, first suppose that x(t) is the solution of the differential equation, and moreover, suppose x(t)gHx(t0) ≔ y(t). The length of y(t) is increasing and it is i  gH differentiable because if x(t) is i  gH differentiable, then: t0 < t¼)xu ðt0 , r Þ  xl ðt0 , r Þ < xu ðt, r Þ  xl ðt, r Þ then: yl ðt, r Þ ¼ xl ðt, r Þ  xl ðt0 , r Þ < xu ðt, r Þ  xu ðt0 , r Þ ¼ yu ðt, r Þ This means y(t, r) is an interval for any r  [0, 1] and we conclude that it satisfies the type (1) of gH-difference. As shown before: I αRL Dα, CKgH xðtÞ ¼ xðtÞ  gH xðt0 Þ and DCKgHαx(t) ¼ f(t, x(t)), then the following relation is concluded: I αRL f ðt, xðtÞÞ ¼ xðtÞ  gH xðt0 Þ On the other hand, based on the definition of the fractional integral: ðt 1 I αRL xðtÞ ¼ ⊙ ðt  sÞα1 ⊙ xðsÞds ΓðαÞ t0

Numerical Solutions of Uncertain Fractional Differential Equations 303 then: I αRL f ðt, xðtÞÞ ¼

1 ⊙ ΓðαÞ

ðt t0

ðt  sÞα1 ⊙ f ðs, xðsÞÞds

The necessary condition is thus proved. Next, the sufficient condition will be proved. Suppose we have: ð 1 t ðt  sÞα1 f ðs, xðsÞÞds xðtÞ  gH xðt0 Þ ¼ ΓðαÞ t0 Effecting the fractional derivative DRLgHα to both sides: ð   1 t ðt  sÞα1 f ðs, xðsÞÞds DαRLgH xðtÞ  gH xðt0 Þ ¼ DαRLgH ΓðαÞ t0 thus:   DαRLgH xðtÞ  gH xðt0 Þ ¼ DαRLgH I αRL f ðt, xðtÞÞ ¼ f ðt, xðtÞÞ therefore:   DαRLgH xðtÞ  gH xðt0 Þ ¼ f ðt, xðtÞÞ Since we had:   DαCKgH xðtÞ ¼ DαRLgH xðtÞ  gH xðt0 Þ thus: DαCKgH xðtÞ ¼ f ðt, xðtÞÞ and the proof is completed. Note. Based on the definition of the gH-difference in the integral equation: ð 1 t ðt  sÞα1 f ðs, xðsÞÞds, t½t0 , T  xðtÞ  gH xðt0 Þ ¼ ΓðαÞ t0 •

In the case where the gH-difference is i  gH difference: ð 1 t xðtÞ ¼ xðt0 Þ  ðt  sÞα1 f ðs, xðsÞÞds, t½t0 , T  ΓðαÞ t0

•

In the case where the gH-difference is ii  gH difference: xðtÞ ¼ xðt0 Þ  H ð1Þ

1 ΓðαÞ

ðt t0

ðt  sÞα1 f ðs, xðsÞÞds, t½t0 , T 

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Soft Numerical Computing in Uncertain Dynamic Systems

Example. Consider the following fuzzy initial value problem:

1 pffi 1 2 D2CKgH xðtÞ ¼ t, pffi , pffi ¼ f ðtÞ, xð0Þ ¼ ð2, 0, 1Þ, tð0, 1 t t Now using the fractional derivative operator on both sides: 1

1

I2RL D2CKgH xðtÞ ¼ xðtÞ  gH xðt0 Þ with: 1 2 f ðtÞ ¼ xðtÞ  gH xðt0 Þ ¼ IRL

1 ΓðαÞ

ðt t0

ðt  sÞα1 f ðs, xðsÞÞds, tð0, 1

and we have: 1 1 I 2RL f ðtÞ ¼ ⊙ 1 Γ 2

ðt

1

ðt  sÞ 2 ⊙

0



pffiffiffi

pffiffi 1 2 π pffiffiffi pffiffiffi t, π , 2 π s, pffiffi , pffiffi ds ¼ 2 s s

then: pffiffiffi

π pffiffiffi pffiffiffi t, π , 2 π xðtÞ  gH xðt0 Þ ¼ 2 By substituting the initial value: pffiffiffi

π pffiffiffi pffiffiffi t, π , 2 π xðtÞ  gH ð2, 0, 1Þ ¼ 2

pffiffi pffiffiffi pffiffiffi  pffiffiffi The length of 2π t, π , 2 π is ð1  r Þ π 2  2t , tð0, 1,r½0, 1: The length function is decreasing because: pffiffiffi

dh t i ð1  r Þ π 2  < 0, r½0, 1 dt 2 In the case of i  gH difference: pffiffiffi



pffiffiffi pffiffiffi π pffiffiffi pffiffiffi π pffiffiffi t, π , 2 π ¼ 2 + t, π , 1 + 2 π xðtÞ ¼ ð2, 0, 1Þ  2 2 The level-wise form of the solution is: pffiffiffi pffiffiffi pffiffiffi pffiffiffi  pffiffiffi π t ðr  1Þ, xu ðt, r Þ ¼ π + 1 + π ð1  r Þ π +2 xl ðt, r Þ ¼ π + 2 and:  pffiffiffi pffiffiffi  pffiffiffi pffiffiffi π π d 3+2 π t ð1  r Þ, t < 0, r½0, 1 lengthðxðtÞÞ ¼ 3 + 2 π  dt 2 2

Numerical Solutions of Uncertain Fractional Differential Equations 305 It can be seen from Figs. 6.1 and 6.2 that both x(t) and x(t)gHx0 have decreasing length and the solution is as follows:

pffiffiffi 1 pffiffiffi π pffiffiffi xðtÞ ¼ x0  I 2RL f ðtÞ ¼ 2 + t, π , 1 + 2 π 2 The figures show that the solution is a fuzzy number at each point, like t ¼ 1. Checking the function will provide the solution to the problem:

pffiffiffi ðt 1 pffiffiffi 0 π pffiffiffi 1 s, π , 1 + 2 π ds DαCKgH xðtÞ ¼ ðt  sÞ 2 ⊙ 2 + 1 0 2 Γ 2 0 pffiffiffi 3 1 pffi 2 pffi

π t2  4 t pffi 2ð1 + 2pffiffiffi pffi 1 2 π Þ tC B pffiffiffi t , pffi , pffi ¼ @3 pffiffiffi , 2 t, A 6¼ π π t t Despite the fuzzy function x(t) is a fuzzy number valued function but it is not the solution of the fuzzy fractional differential equation. This is because the length of x(t)gHx(t0) is not increasing. Example. Consider the following fuzzy initial value problem: 1  pffiffiffi D2CKgH xðtÞ ¼ π t2 , 0, 2t  t2 ¼ f ðtÞ, xð0Þ ¼ ð2, 0, 2Þ, tð0, 1,

Now, using the fractional derivative operator on both sides: 1

1

I 2RL D2CKgH xðtÞ ¼ xðtÞ  gH xðt0 Þ

3

2

1

0 0.5

1

1.5

2

2.5

3

3.5

4

–1

–2

Fig. 6.1 The length of x(t)gHx(t0) is decreasing.

4.5

5.5

6

6.5

306

Soft Numerical Computing in Uncertain Dynamic Systems 2.5 2 1.5 1 0.5 0

5

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

–0.5 –1 –1.5 –2

Fig. 6.2

Length of x(t) is decreasing.

with: 1 2 f ðtÞ ¼ xðtÞ  gH xðt0 Þ ¼ IRL

1 ΓðαÞ

ðt t0

ðt  sÞα1 f ðs, xðsÞÞds, tð0, 1

ðt 1 1  pffiffiffi 1 I2RL f ðtÞ ¼ ⊙ ðt  sÞ 2 ⊙ π s2 , 0, 2s  s2 ds 1 0 Γ 2

16 5 8 3 16 5 ¼  t2 , 0, t2  t2 15 3 15 then (Fig. 6.3):

16 5 8 3 16 5 xðtÞ  gH xðt0 Þ ¼  t2 , 0, t2  t2 15 3 15 Now, the two types of the solution we should have are: •

i  gH (solution) (Fig. 6.4)

16 5 8 3 16 5 2 2 2 xigH ðtÞ ¼ ð2, 0, 2Þ   t , 0, t  t 15 3 15

16 5 8 3 16 5 2 2 2 ¼ 2  t , 0, t  t + 2 15 3 15

Numerical Solutions of Uncertain Fractional Differential Equations 307 2 1.5 1 0.5

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1

1.2

1.4

–0.5 –1 –1.5 –2

Fig. 6.3 Length of x(t)gHx(t0) is increasing.

4

2

0

0 0.2

0.4

0.6

0.8

–2

–4

–6

Fig. 6.4 Length of xigH(t) is increasing.

•

ii  gH (solution) (Fig. 6.5)

16 5 8 3 16 5 xiigH ðtÞ ¼ ð2, 0, 2Þ  H ð1Þ  t2 , 0, t2  t2 15 3 15

8 3 16 5 16 5 t2  t2  2, 0, 2  t2 ¼ 3 15 15



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Soft Numerical Computing in Uncertain Dynamic Systems

2 1.5 1 0.5

0

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

–0.5 –1 –1.5 –2

Fig. 6.5

Length of xiigH(t) is decreasing.

To check the solutions:

1 1 16 5 8 3 16 5 D2CKgH xigH ðtÞ ¼ D2CKgH 2  t2 , 0, t2  t2 + 2 ¼ 15 3 15

1

1 16 5 8 3 16 5 2 2 ¼ DCK 2  t2 , 0, DCK t2  t2 + 2 15 3 15 0 1 1 3 3 ðt ð t 4s2  8 s2 B 1 8s2 1 3 ffi dsC C ffi ds, 0, pffiffiffiffiffiffiffiffi ¼B @ 1 3pffiffiffiffiffiffiffiffi A 1 0 ts ts 0 Γ Γ 2 2  pffiffiffi 2 ¼ π t , 0, 2t  t2 Then the i  gH solution satisfies the fractional problem. The same process can be carried out for checking the ii  gH solution:

1 1 8 3 16 5 16 5 D2CKgH xiigH ðtÞ ¼ D2CKgH t2  t2  2, 0, 2  t2 ¼ 3 15 15

1

1 16 5 8 3 16 5 2 2 ¼ DCK 2  t2 , 0, DCK t2  t2 + 2 15 3 15  pffiffiffi 2 ¼ π t , 0, 2t  t2 Both are i  gH and ii  gH solutions of the problem because the length of x(t)gHx(t0) is increasing.

Numerical Solutions of Uncertain Fractional Differential Equations 309

6.4.1.2 Existence and Uniqueness of the Solution In this section, the existence and uniqueness of the results of the solutions to fuzzy fractional differential equations by using an idea of successive approximations under generalized Lipschitz condition of the right-hand side are investigated. Furthermore, the formula of solution to the linear fuzzy Caputo-Katugampola fractional differential equation is given. Since the real intervals in the level-wise form are used for any arbitrary level, so to reach the aims, we should consider the following theorem in the real case of Caputo fractional derivative (Van Hoa et al., 2019).

6.4.1.3 Theorem—Existence and Uniqueness in Real Fractional Differential Equation Consider the initial value problem as follows: DαCK BðtÞ ¼ gðt, BðtÞÞ, Bðt0 Þ ¼ B0 ¼ 0, t½t0 , T  Let η > 0 be a given constant and BðB0 , ηÞ ¼ fBR; jB  B0 j  ηg is a ball around B0 with radius η. Also assume a real valued function g : [t0, T]  [0, η] ! R+ satisfies the following conditions: for 1. gCð½t0 , T   ½0, η, R + Þ, gðt, 0Þ≡0, 9Mg  0, 0  gðt, BÞ  Mg , ðt, BÞ½t0 , T   ½0, η. 2. gðt, BÞ is a nondecreasing function with respect to x for any t  [t0, T].

all

Then the mentioned fractional problem has at least one solution on [t0, T] and BðtÞBðB0 , ηÞ. Proof. The solution of the mentioned above fractional differential equation is the solution of following fractional integral equation: ð 1 t Bð t Þ ¼ ðt  sÞα1 gðs, BðsÞÞds ΓðαÞ t0 The following successive method for approximation of the solution of fractional differential equation is defined: ð 1 t ðt  sÞα1 gðs, Bn ðsÞÞds, t½t∗ , T ∗  Bn + 1 ðtÞ ¼ ΓðαÞ t0 such that: B0 ð t Þ ¼



M g ð t  aÞ α ηΓðα + 1Þ 1 α + a, T ∗ ¼ min ft∗ , T g , t0 < t  Γðα + 1Þ Mg

For n ¼ 0 and t  [t0, T∗] we have:

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Soft Numerical Computing in Uncertain Dynamic Systems

B1 ðtÞ ¼

1 ΓðαÞ

ðt t0

ðt  sÞα1 gðs, B0 ðsÞÞds 

M g ð t  aÞ α ¼ B0 ðtÞ  η, Γðα + 1Þ

so we found: B1 ðtÞ  B0 ðtÞ ∗]:

for n ¼ 1 and t  [t0, T B2 ðtÞ ¼

1 ΓðαÞ

ðt t0

ðt  sÞα1 gðs, B1 ðsÞÞds 

M g ð t  aÞ α ¼ B0 ðtÞ  η, Γðα + 1Þ

Since g is a nondecreasing function in B and B1 ðtÞ  B0 ðtÞ, then gðs, B1 ðsÞÞ  gðs, B0 ðsÞÞ; thus, B2 ðtÞ  B1 ðtÞ: Now we have: B2 ðtÞ  B1 ðtÞ  B0 ðtÞ Proceeding recursively, we will have: Bn + 1 ðtÞ  Bn ðtÞ  ⋯  B1 ðtÞ  B0 ðtÞ  η, t½t0 , T ∗  It follows that the sequence fBn ðtÞgn is uniformly bounded for n 0. On the other hand:  α  D Bn ðtÞ ¼ jgðt, Bn ðtÞÞj ¼ gðt, Bn ðtÞÞ  Mg CK Then, in the interval [t0, T∗], we can use the mean value theorem for t1, t2  [t0, T∗], n 0: jBn ðt2 Þ  Bn ðt1 Þj ¼

2Mg ðt  aÞα 2Mg ðt2  t1 Þα α  τ , τ½t1 , t2  ½t0 , T ∗  Γ ð α + 1Þ Γ ð α + 1Þ

Therefore, the sequence fBn ðtÞgn is equicontinuous, and then:

EΓðα + 1Þτα 1 α > 0, ðjt2  t1 j < δ ¼)jBn ðt2 Þ  Bn ðt1 Þj < EÞ 8E > 0, 9δ ¼ 2Mg Hence, by the Arzela-Ascoli Theorem and the monotonicity of the sequence we conclude the convergency of the sequence and fBn ðtÞgn , lim n!∞ Bn ðtÞ ¼ BðtÞ, t½t0 , T ∗ :

6.4.1.4 Theorem—Existence and Uniqueness in Fuzzy Fractional Differential Equation Consider the fuzzy initial value problem as follows: DαCKgH xðtÞ ¼ f ðt, xðtÞÞ, xðt0 Þ ¼ x0 , t½t0 , T , 0 < α < 1

Numerical Solutions of Uncertain Fractional Differential Equations 311 Let η > 0 be a given constant and B(x0, η) ¼ {x  R; j x  x0 j  η} is a ball around x0 with radius η. Also assume a fuzzy number valued function f : ½t0 , T   ½0, η ! R satisfies the following conditions: i. f Cð½t0 , T   ½0, η, R Þ, 9Mg  0, DH ðgðt, xÞ, 0Þ  Mg , for all (t, x)  [t0, T]  B(x0, η). ii. For any G, DBðx0 , ηÞ, DH ðf ðt, GÞ, f ðt, DÞÞ  gðt, DH ðG, DÞÞ where g  C([t0, T]  [0, η], R+) in the problem: DαCK xðtÞ ¼ gðt, xðtÞÞ, xðt0 Þ ¼ x0 ¼ 0, t½t0 , T  has only the solution x(t) ≡ 0 on [t0, T] (the previous theorem of existence and uniqueness in real case). Then the following successive approximations given by x0(t) ¼ x0: ð 1 t ðt  sÞα1 ⊙ f ðs, xn1 ðsÞÞds, n ¼ 1,2, …, xn ðtÞ  gH x0 ¼ ΓðαÞ t0 converge to a unique solution of: DαCKgH xðtÞ ¼ f ðt, xðtÞÞ, xðt0 Þ ¼ x0 , 0 < α < 1 ∗

on t  [t0, T∗], T  (t0, T], provided that xn(t)gHx0 does have increasing length. Proof. Let us consider the point t∗:

  ηΓðα + 1Þ 1 α + a, M ¼ max Mf , Mg , T ∗ ¼ min ft∗ , T g t0 < t∗  M Now consider the sequence of fuzzy continuous functions {xn(t)}n, x0(t) ¼ x0, t  [t0, T∗]: ð 1 t ðt  sÞα1 ⊙ f ðs, xn1 ðsÞÞds, n ¼ 1, 2, …, xn ðtÞ  gH x0 ¼ ΓðαÞ t0 First, we prove the xn(t)  C([t0, T∗], B(x0, η)). To this end, assume t1, t2  [t0, T∗] and t1 < t2:   DH xn ðt1 Þ  gH x0 , xn ðt2 Þ  gH x0 ð i 1 t1 h  ðt1  sÞα1  ðt2  sÞα1 DH ðf ðs, xn ðsÞÞ, 0Þds ΓðαÞ t0 ð i 1 t2 h + ðt2  sÞα1 DH ðf ðs, xn ðsÞÞ, 0Þds ΓðαÞ t1

312

Soft Numerical Computing in Uncertain Dynamic Systems

and: 1 ΓðαÞ

ð t2 h t1

1 ΓðαÞ

i ðt2  sÞα1 ds ¼

ð t1 h t0

¼

1 ðt2  t1 Þα Γðα + 1Þ

i ðt1  sÞα1  ðt2  sÞα1 ds ¼

1 ½ðt1  t0 Þα  ðt2  t1 Þα  Γ ð α + 1Þ

and:   DH ðf ðs, xn ðsÞÞ, 0Þ  Mf ,DH xn ðt1 Þ  gH x0 , xn ðt2 Þ  gH x0 ¼ DH ðxn ðt1 Þ, xn ðt2 ÞÞ hence: DH ðxn ðt1 Þ, xn ðt2 ÞÞ 

Mf ½ðt2  t1 Þα + ðt1  t0 Þα + ðt2  t1 Þα  Γ ð α + 1Þ

Finally: DH ðxn ðt1 Þ, xn ðt2 ÞÞ 

2Mf ðt2  t1 Þα  η Γðα + 1Þ

This means, if t2 ! t1 then DH(xn(t1), xn(t2)) ! 0 and follows the function xn(t) is continuous on [t0, T∗]. In addition, it follows for n 1, t  [t0, T∗]: xn ðtÞBðx0 , ηÞ⟺xn ðtÞ  gH x0 Bðx0 , ηÞ Now, if xn1(t)  B(x0, η):   1 DH xn ðtÞ  gH x0 , 0  ΓðαÞ 

ðt t0

ðt  sÞα1 DH ðf ðs, xn1 ðsÞÞ, 0Þds

Mf ðt  t0 Þα  η Γðα + 1Þ

In conclusion, the fuzzy function xn(t)  B(x0, η) for all n 1 and all t  [t0, T∗]. Now our next step is proving the convergence, for xn(t), x(t)  C([t0, T∗], B(x0, η)): lim xn ðtÞ ¼ xðtÞ

n!0

To this purpose, we need some relations:   DH xn + 1 ðtÞ  gH x0 , xn ðtÞ  gH x0 ð 1 t  ðt  sÞα1 DH ðf ðs, xn ðsÞÞ, f ðs, xn1 ðsÞÞds ΓðαÞ t0

Numerical Solutions of Uncertain Fractional Differential Equations 313 

1 ΓðαÞ

M g ð t  aÞ α η ðt  sÞα1 DH gðxn ðsÞ, xn1 ðsÞÞds  Bn ðtÞ  Γðα + 1Þ t0

ðt

since: DαCKgH xðtÞ ¼ xðtÞ  gH x0 Thus, based on the properties of the distance:

 DH DαCKgH xn + 1 ðtÞ, DαCKgH xn ðtÞ  DH ðf ðt, xn ðtÞÞ, f ðt, xn1 ðtÞÞÞ  gðDH ðxn ðtÞ, xn1 ðtÞÞÞ  gðt, Bn1 ðtÞÞ Finally:

 DH DαCKgH xn + 1 ðtÞ, DαCKgH xn ðtÞ  gðt, Bn1 ðtÞÞ Continuing:

 DH DαCKgH xn ðtÞ, DαCKgH xn1 ðtÞ  gðt, Bn2 ðtÞÞ then:

 DH DαCKgH xn + 1 ðtÞ, DαCKgH xn1 ðtÞ



 DH DαCKgH xn + 1 ðt Þ, DαCKgH xn ðtÞ + DH DαCKgH xn ðtÞ, DαCKgH xn1 ðtÞ  gðt, Bn1 ðtÞÞ + gðt, Bn2 ðtÞÞ ⋮ n1

 X gðt, Bi ðtÞÞ ! 0 DH DαCKgH xn ðtÞ, DαCKgH x1 ðtÞ  i¼0

hence: n1   X gðt, Bi ðtÞÞ ! 0 DH xn ðtÞ  gH x0 , x1 ðtÞ  gH x0  i¼0

and: DH ðxn ðtÞ, x1 ðtÞÞ 

n1 X

gðt, Bi ðtÞÞ ! 0

i¼0

In general, assume for m n: DH ðxm ðtÞ, xn ðtÞÞ ! 0

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Soft Numerical Computing in Uncertain Dynamic Systems

then using the definition of the Cauchy sequence the sequence {xn(t)}n ! x(t). Uniqueness. To show this, let us suppose that FðtÞ is another solution of the fuzzy fractional differential equation, and assume:   DH ðxðtÞ, FðtÞÞ ¼ :ðtÞ, DαCK DH ðxðtÞ, FðtÞÞ ¼ DαCK :ðtÞ  DH DαCK xðtÞ, DαCK FðtÞ  DH ðf ðt, xðtÞÞ, f ðt, FðtÞÞÞ  gðt, :ðtÞÞ Finally, we have: DαCK :ðtÞ  gðt, :ðtÞÞ The only solution (because of , the maximal solution) is :ðtÞ≡0. Remark. In conclusion, if the fuzzy function f : ½t0 , T   R ! R in the following fuzzy fractional initial value problem: DαCKgH xðtÞ ¼ f ðt, xðtÞÞ, xðt0 Þ ¼ x0 , t½t0 , T  0 < α < 1 satisfies in the Lipchitz condition: DH ðf ðt, xðtÞÞ, f ðt, yðtÞÞÞ  LDH ðxðtÞ, yðtÞÞ, DH ðf ðt, xðtÞÞ, 0Þ  Mf then the following successive approximations converge uniformly to the unique solution of the problem on [t0, T], subject to xn(t)gHx0 does have increasing length: ð 1 t ðt  sÞα1 ⊙ f ðs, xn1 ðsÞÞds, n ¼ 1, 2,…, xn ðtÞ  gH x0 ¼ ΓðαÞ t0 Example. Consider: DαCKgH xðtÞ ¼ λ ⊙ xðtÞ  hðtÞ, xðt0 Þ ¼ x0 , λR, tðt0 , T  such that xðtÞ, hðtÞCððt0 , T , R Þ: Since IαRLDCKgHαx(t) ¼ x(t)gHx(t0) then: xðtÞ  gH xðt0 Þ ¼ λIαRL xðtÞ  IαRL hðtÞ If we consider the λ 0 then λIαRLx(t)  IαRLh(t) has increasing length because x(t) and h(t) are two fuzzy number valued functions. Now we can use the successive approximation method: xn ðtÞ  gH xðt0 Þ ¼ λIαRL xn1 ðtÞ  I αRL hðtÞ, n ¼ 1, 2,…, if n ¼ 1. Let us consider λ 0 and x is i  gH differentiable (increasing length): x1 ðtÞ  gH xðt0 Þ ¼ x0 ⊙

λ ðt  t 0 Þ α  I αRL hðtÞ, Γðα + 1Þ

Let us consider λ < 0 and x is ii  gH differentiable (decreasing length):

Numerical Solutions of Uncertain Fractional Differential Equations 315   λðt  t0 Þα  I αRL hðtÞ, ð1Þ xðt0 Þ  gH x1 ðtÞ ¼ x0 ⊙ Γðα + 1Þ if n ¼ 2. Let us consider λ 0 and x is i  gH differentiable (increasing length): x2 ðtÞ  gH xðt0 Þ ¼ " # λðt  t0 Þα λðt  t0 Þ2α + ¼ x0 ⊙  I αRL hðtÞ  I 2α RL hðtÞ, Γðα + 1Þ Γð2α + 1Þ Let us consider λ < 0 and x is ii  gH differentiable (decreasing length):   ð1Þ xðt0 Þ  gH x2 ðtÞ ¼ " # λðt  t0 Þα λðt  t0 Þ2α +  IαRL hðtÞ  I 2α ¼ x0 ⊙ RL hðtÞ, Γðα + 1Þ Γð2α + 1Þ If it is proceeding to more n ! ∞ : xn ðtÞ  gH xðtÞ ¼ ¼ x0 ⊙

∞ i X λ ðt  t0 Þiα i¼1

¼ x0 ⊙

ðt X ∞ i1 λ ðt  sÞiα1 ⊙ hðsÞds ΓðiαÞ t0 i¼1

ðt X ∞ i λ ðt  sÞiα + ðα1Þ  ⊙ hðsÞds Γðiα + 1Þ Γðiα + αÞ t0 i¼0

∞ i X λ ðt  t0 Þiα i¼1

¼ x0 ⊙

Γðiα + 1Þ



∞ i X

λ ðt  t0 Þiα  Γðiα + 1Þ i¼1

ðt t0

ðt  sÞα1

∞ i X λ ðt  sÞiα i¼0

Γðiα + αÞ

⊙ hðsÞds

Let us consider λ 0 and x is i  gH differentiable (increasing length): ðt xðtÞ ¼ x0 Eα,1 ðλðt  t0 Þα Þ  ðt  sÞα1 Eα,α ðλðt  sÞα Þ ⊙ hðsÞds t0

Let us consider λ < 0 and x is ii  gH differentiable (decreasing length): ðt xðtÞ ¼ x0 Eα,1 ðλðt  t0 Þα Þ  H ð1Þ ðt  sÞα1 Eα, α ðλðt  sÞα Þ ⊙ hðsÞds t0

such that the following function is called the Mittag-Leffler function: Eα,β ðtÞ ¼

∞ X

ti , α > 0, β > 0 Γðiα + βÞ i¼0

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Soft Numerical Computing in Uncertain Dynamic Systems

6.4.1.5 Some Properties of the Mittag-Leffler Function The basic Mittag-Leffler function is defined as follows: Eα ðtÞ ¼

∞ X

ti , α>0 Γðiα + 1Þ i¼0

If α ¼ 1: E1 ðtÞ ¼

∞ i X ti t ¼ ¼ et Γ ð i + 1 Þ i! i¼0 i¼1

∞ X

Hence Eα(t) is a generalization of exponential series. One generalization of Eα(t) as a two-parameter generalization of is Eα(t) is: Eα,β ðtÞ ¼

∞ X

ti , α > 0, β > 0 Γðiα + βÞ i¼0

A three-parameter generalization of Eα(t) is denoted by Eγα, β(t) and is defined as: Eγα,β ðtÞ ¼

∞ X

ðγ Þi , α > 0, β > 0 i!Γ ð iα + βÞ i¼0

where (γ)i is the Pochhammer symbol standing for: ðγ Þi ¼ γ ðγ + 1Þðγ + 2Þ…ðγ + i  1Þ, ðγ Þ0 ¼ 1, γ 6¼ 0 Here, there is no other condition on (γ)i, and γ could be a negative integer also. In that case, the series is going to terminate into a polynomial. However, if (γ)i is to be written in terms of a gamma function as: ðγ Þi ¼

Γ ðγ + iÞ , γ>0 Γ ðγ Þ

if more parameters are to be incorporated, then we can consider:   ∞ ðγ Þ ⋯ γ X xi γ 1 ,γ 2 ,…, γ p 1 i  p i , α > 0, β > 0 Eα, β, δ1 ,…, δq ðxÞ ¼ Γðiα + βÞ i¼0 i!ðδ1 Þi ⋯ δq i where: δj 6¼ 0,  1,  2, …, j ¼ 1, 2, …,q: No other restrictions on γ 1, …, γ p and δ1, …, δq are there other than the conditions for the convergence of the series. A δj can be a negative integer provided there is a γ r, a negative integer such that (γ r)k ¼ 0 first before (δr)k ¼ 0, such as γ 2 ¼  3 and δ1 ¼  5 so that (γ 2)4 ¼ 0 and (δ1)4 6¼ 0 (Mathai & Haubold, 2017).

Numerical Solutions of Uncertain Fractional Differential Equations 317 Example. Consider the following fuzzy fractional initial value problem with: 1 λ ¼ 1, α ¼ , hðtÞ ¼ 0, tð0, 1 2 1

D2CKgH xðtÞ ¼ xðtÞ, xðt0 , r Þ ¼ x0 ðr Þ ¼ ð2, 2  r Þ 1

1

1

such that xðtÞCððt0 , T , R Þ: Since I 2RL D2CKgH xðtÞ ¼ xðtÞ  gH x0 ¼ I 2RL xðtÞ and it has increasing length then we can use the successive approximation method: xn ðtÞ  gH x0 ¼ I αRL xn1 ðtÞ, n ¼ 1,2, …, The i  gH differentiable solution (with increasing length) is:

ð 1 2 t s2 2 2 xðtÞ ¼ x0 ⊙ E1,1 t ¼ x0 ⊙ 1 + pffiffiffi e ds et π 2 0 Example. Consider the following fuzzy fractional initial value problem with: 1 λf1, 1g, α ¼ , hðtÞ ¼ c ⊙ t2 , tð0, 1 2 1

D2CKgH xðtÞ ¼ λ ⊙ xðtÞ  hðtÞ, c, x0 R Let us consider that λ ¼ 1 and x is i  gH differentiable: ðt

1 1 1 2 2 2 xigH ðtÞ ¼ x0 Eα,1 t  ðt  sÞ E1, 1 ðt  sÞ ⊙ c ⊙ s2 ds 0

2 2

where: i X ∞ 1 t2

, E1,1 t2 ¼ i 2 i¼0 Γ +1 2 i

X

ð ∞ pffi 1 ðt  sÞ2 2 t s2 1

¼ t 1 + pffiffiffi e ds + E1 , 1 ð t  s Þ 2 ¼ i 1 1 π 0 2 2 i¼0 Γ + Γ 2 2 2

The solution is obtained as follows:



ð ðt 1 1 2 t s2 2 t2 2 p ffiffiffi e ds e  ðt  sÞ E1, 1 ðt  sÞ ⊙ c ⊙ s2 ds xigH ðtÞ ¼ x0 ⊙ 1 + π 0 2 2 0

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Soft Numerical Computing in Uncertain Dynamic Systems

Now let us consider that λ ¼  1 and x is ii  gH differentiable:



ð ðt 1 1 2 t s2 2 t2 xiigH ðtÞ ¼ x0 ⊙ 1 + pffiffiffi e ds e  H ð1Þ ðt  sÞ E1, 1 ðt  sÞ2 ⊙ c ⊙ s2 ds π 0 2 2 0

6.5

Fuzzy Generalized Taylor’S Expansion

Let us consider the fuzzy continuous function Dkα CgH xðtÞR in (a, b) for k ¼ 0, 1, …, n, such that 0 < α  1. Then we have the following items (Armand et al., 2019). Case 1. If x(t) is i  CgH differentiable of order kα, then there exists ξ  (a, b) such that: xðtÞ ¼ xðaÞ 

ðn + 1Þα n X DCgH xðξÞ ðt  aÞkα ⊙ Dkα ⊙ ðt  aÞðn + 1Þα x ð a Þ  CgH Γ ð kα + 1 Þ Γ ð ð n + 1 Þα + 1 Þ k¼1

Case 2. If x(t) is ii  CgH differentiable of order kα, then there exists ξ  (a, b) such that: ðn + 1Þα n X DCgH xðξÞ ðt  aÞkα kα ⊙ DCgH xðaÞ  H ð1Þ xðtÞ ¼ xðaÞ  H ð1Þ ⊙ ðt  aÞðn + 1Þα Γ ð kα + 1 Þ Γ ð ð n + 1 Þα + 1 Þ k¼1

n Case 3. If x(t) is i  CgH differentiable of order 2kα,k n ¼ 0, 1,…, 2 and also it is ii  CgH differentiable of order ð2k  1Þα,k ¼ 0, 1,…, 2 , then there exists ξ  (a, b) such that: xðtÞ ¼ xðaÞ  H ð1Þ

n X ðt  aÞkα ⊙ Dkα CgH xðaÞ  Γ ð kα + 1 Þ k¼1, odd ðn + 1Þα

 H ð1Þ

DCgH

xðξÞ

Γððn + 1Þα + 1Þ

n X

ðt  aÞkα ⊙ Dkα CgH xðaÞ Γ ð kα + 1 Þ k¼1, even

⊙ ðt  aÞðn + 1Þα

Case 4. If ζ  [a, b], and x(t) is ii  CgH differentiable before ζ and i  CgH after it, and types of differentiability for DCgHkαx(t), k ¼ 1, 2, …, n are i  CgH differentiable, then there exists ξ  (a, b) such that: 8 n X ð t  aÞ α ðt  aÞkα > α > > x ð a Þ  ⊙ D ⊙ Dkα ð 1 Þ x ð a Þ  H > CgH CgH xðaÞ > Γ ð α + 1 Þ Γ ð kα + 1 Þ > k¼2 > > > ðn + 1Þα < DCgH xðξÞ xðtÞ ¼ ⊙ ðt  aÞðn + 1Þα , atζ  > Γ ð ð n + 1 Þα + 1 Þ > > > ðn + 1Þα > n > X DCgH xðξÞ ðt  aÞkα > kα > > ⊙ D ⊙ ðt  aÞðn + 1Þα , ζ  t  b x ð a Þ  x ð a Þ  : CgH Γðkα + 1Þ Γððn + 1Þα + 1Þ k¼1

Numerical Solutions of Uncertain Fractional Differential Equations 319 Case 5. If ζ  [a, b], and x(t) is i  CgH differentiable before ζ and ii  CgH after it, and types of differentiability for DCgHkαx(t), k ¼ 1, 2, …, n are ii  CgH differentiable, then there exists ξ  (a, b) such that: 8 n X > ð t  aÞ α ðt  aÞkα > α > ⊙ D ⊙ Dkα x ð a Þ  ð 1 Þ x ð a Þ  > H CgH CgH xðaÞ > > Γ ð α + 1Þ Γ ð kα + 1 Þ > k¼2 > > ðn + 1Þα > > DCgH xðξÞ > > > ⊙ ðt  aÞðn + 1Þα , a  t  ζ  H ð1Þ < Γððn + 1Þα + 1Þ xðtÞ ¼ n X > ðt  aÞkα > > ⊙ Dkα ð 1 Þ x ð a Þ  > H CgH xðaÞ  H > > Γðkα + 1Þ > k¼1 > > ðn + 1Þα > > DCgH xðξÞ > > > ⊙ ðt  aÞðn + 1Þα , ð1Þ ζtb : Γððn + 1Þα + 1Þ In general, in accordance with gH-difference, we have the following relations. Since: ðn + 1Þα

nα Inα RL DCgH xðtÞ  gH I RL n

X k¼0

ðk + 1Þα

kα I kα RL DCgH xðtÞ  gH I RL

ðn + 1Þα

DCgH

ðk + 1Þα

DCgH

xðtÞ ¼

ðt  aÞnα ⊙ Dnα CgH xðaÞ Γðnα + 1Þ

 ðn + 1Þα ðn + 1Þα xðtÞ ¼ xðtÞ  gH I RL DCgH xðtÞ

n

n  X X ðt  aÞkα ðk + 1Þα ðk + 1Þα kα ⊙ Dkα I kα DCgH xðtÞ ¼ RL DCgH xðtÞ  gH I RL CgH xðaÞ Γ ð kα + 1 Þ k¼0 k¼0 ðn + 1Þα

xðtÞ  gH I RL xðtÞ ¼

ðn + 1Þα

DCgH

xðtÞ ¼

n X ðt  aÞkα ⊙ Dkα CgH xðaÞ Γ ð kα + 1 Þ k¼0

n X ðt  aÞkα ðn + 1Þα ðn + 1Þα ⊙ Dkα DCgH xðtÞ CgH xðaÞ  I RL Γ ð kα + 1 Þ k¼0

xðtÞ ¼ xðaÞ 

n X ðt  aÞkα ðn + 1Þα ðn + 1Þα ⊙ Dkα DCgH xðtÞ CgH xðaÞ  I RL Γ ð kα + 1 Þ k¼1

using the fractional mean value theorem: ðn + 1Þα ðn + 1Þα IRL DCgH xðtÞ ¼

1 ⊙ Γððn + 1ÞαÞ ðn + 1Þα

¼

DCgH

ðt

xðξÞ

Γððn + 1Þα + 1Þ

a

ðn + 1Þα

ðt  τÞðn + 1Þα1 ⊙ DCgH

⊙ ðt  aÞðn + 1Þα

xðτÞdτ

320

Soft Numerical Computing in Uncertain Dynamic Systems

Proof of Case 1. Since x(t) is i  CgH differentiable of order kα, then all gH-differences are defined in the sense of i  gH difference and we have: xðtÞ ¼ xðaÞ 

n X ðt  aÞkα ðn + 1Þα ðn + 1Þα ⊙ Dkα DCgH xðζ Þ CgH xðaÞ  I RL Γ ð kα + 1 Þ k¼1

Proof of Case 2. Since x(t) is ii  CgH differentiable of order kα, then all gH-differences are defined in the sense of ii  gH difference and also fractional mean value theorem we have: ðn + 1Þα

nα I nα RL DCgH xðtÞ  ð1ÞI RL

ðn + 1Þα

DCgH

xðtÞ ¼

ðt  aÞnα ⊙ Dnα CgH xðaÞ Γðnα + 1Þ

ðn + 1Þα n X DCgH xðξÞ ðt  aÞkα kα ⊙ DCgH xðaÞ  H ð1Þ ⊙ ðt  aÞðn + 1Þα xðtÞ ¼ xðaÞ  H ð1Þ Γ ð kα + 1 Þ Γ ð ð n + 1 Þα + 1 Þ k¼1

Proof of Case 3. n If x(t) is i  CgH differentiable of order 2kα,k n ¼ 0,1, …, 2 and also it is ii  CgH differentiable of order ð2k  1Þα, k ¼ 0, 1,…, 2 , then there exists ξ  (a, b): n

X k¼0

ðk + 1Þα

kα I kα RL DCgH xðtÞ  gH I RL

ðk + 1Þα

DCgH

 xðtÞ ¼ xðtÞ  ð1ÞIαRL DαCgH xðtÞ

2α  H ð1ÞI αRL DαCgH xðtÞ  H I 2α RL DCgH xðtÞ ðn + 1Þα

nα  ⋯  I nα RL DCgH xðtÞ  H ð1ÞI RL

ðn + 1Þα

DCgH

xðtÞ

By similar reasoning in Case 1: ðn + 1Þα

xðtÞ  ð1ÞIRL

ðn + 1Þα

DCgH

¼ x ðaÞ  H

xðtÞ ¼ xðaÞ  H

n X k¼1, odd

kα I kα RL DCgH xðaÞ 

n X ðt  aÞkα ⊙ Dkα CgH xðaÞ  Γðkα + 1Þ k¼1, odd

n X k¼1, even

kα I kα RL DCgH xðaÞ

n X

ðt  aÞkα ⊙ Dkα CgH xðaÞ Γðkα + 1Þ k¼1, even

By using the definition of H-difference: xðtÞ ¼ xðaÞ  H ð1Þ

n X ðt  aÞkα ⊙ Dkα CgH xðaÞ  Γ ð kα + 1 Þ k¼1, odd

n X

ðt  aÞkα ⊙ Dkα CgH xðaÞ Γ ð kα + 1 Þ k¼1, even

Numerical Solutions of Uncertain Fractional Differential Equations 321 ðn + 1Þα

 H ð1Þ

DCgH

xðξÞ

Γððn + 1Þα + 1Þ

⊙ ðt  aÞðn + 1Þα

The proof of Cases 4 and 5 are very similar to Cases 1–3. One of the applications of fuzzy fractional Taylor is the fuzzy Euler fractional method, which is the immediate consequence of the Taylor expansion. We shall now cover the Fuzzy fractional Euler method.

6.6

Fuzzy Fractional Euler Method

Consider the following fuzzy fractional differential equation: DαCgH xðtÞ ¼ f ðt, xðtÞÞ, xðt0 Þ ¼ x0 , t½t0 , T , 0 < α  1 where f : ½t0 , T   R ! R is a fuzzy number valued function, x(t) is a continuous fuzzy set valued solution, and DCgHα is the fractional Caputo fractional operator. In this method, a sequence of approximations to the solution x(t) will be obtained at several points, called grid points. To derive Euler’s method, the interval [t0, T] is divided to into N equal subintervals, each of length h, by the grid points ti ¼ t0 + ih, 0 i ¼ 0, 1, 2, …, N. The distance between points, h ¼ Tt N , is called the grid size. To explain the method, we should discuss the type of differentiability of the solution and to this end we will have several cases, as with fuzzy Taylor expansion (Armand et al., 2019). Case 1. Suppose that the fuzzy solution x(t) is i  CgH differentiable on [t0, T]. The Taylor expansion about tk on any subinterval [tk, tk+1] for k ¼ 0, 1, …, N  1 can be expressed as the following form, where h ¼ tk+1  tk, 9 ζ k  [tk, tk+1]: xðtk + 1 Þ ¼ xðtk Þ 

ðtk + 1  tk Þα ðtk + 1  tk Þ2α ⊙ DαCgH xðtk Þ  ⊙ D2α CgH xðζ k Þ Γðα + 1Þ Γð2α + 1Þ

xðtk + 1 Þ ¼ xðtk Þ 

hα h2α ⊙ Dkα ⊙ D2α CgH xðtk Þ  CgH xðζ k Þ Γðα + 1Þ Γð2α + 1Þ

Since DCgHαx(tk) ¼ f(t, x(tk)), then by substituting, we have: xðtk + 1 Þ ¼ xðtk Þ  where the term

h2α Γð2α + 1Þ

hα h2α ⊙ f ðtk , xðtk ÞÞ  ⊙ D2α CgH xðζ k Þ Γ ð α + 1Þ Γð2α + 1Þ

⊙ D2α CgH xðζ k Þ can be denoted as the error for the numerical

method. As a result, the Fuzzy fractional Euler method on [tk, tk+1] can be introduced as: xðtk + 1 Þ ¼ xðtk Þ 

hα ⊙ f ðtk , xðtk ÞÞ, h ¼ tk + 1  tk , k ¼ 0, 1,…,N  1 Γðα + 1Þ

322

Soft Numerical Computing in Uncertain Dynamic Systems

Case 2. Suppose that the fuzzy solution x(t) is ii  CgH differentiable on [t0, T]. The Taylor expansion about tk on any subinterval [tk, tk+1] for k ¼ 0, 1, …, N  1 can be expressed as the following form, where h ¼ tk+1  tk, 9 ζ k  [tk, tk+1]: xðtk + 1 Þ ¼ xðtk Þ  H ð1Þ

hα hα ⊙ DαCgH xðtk Þ  H ð1Þ ⊙ D2α CgH xðζ k Þ Γ ð α + 1Þ Γð2α + 1Þ

Since DCgHαx(tk) ¼ f(t, x(tk)), then by substituting, we have: xðtk + 1 Þ ¼ xðtk Þ  H ð1Þ

hα h2α ⊙ f ðtk , xðtk ÞÞ  H ð1Þ ⊙ D2α CgH xðζ k Þ Γðα + 1Þ Γð2α + 1Þ

2α h where the term Γð2α + 1Þ ⊙ DCgH xðζ k Þ can be denoted as the error for the numerical method. As a result, the Fuzzy fractional Euler method on [tk, tk+1] can be introduced as follows: 2α

xðtk + 1 Þ ¼ xðtk Þ  H ð1Þ

hα ⊙ f ðtk , xðtk ÞÞ, h ¼ tk + 1  tk , k ¼ 0, 1,…, N  1 Γ ð α + 1Þ

Case 3. Suppose that the fuzzy solution x(t) has a switching point at ξ  [t0, T] and it is i  CgH differentiable at the points t0, t1, …, tj and ii  CgH differentiable at the points tj+1, tj+2, …, tN on [t0, T]. The Taylor expansion about tk on any subinterval [tk, tk+1] for k ¼ 0, 1, …, N  1 can be expressed as the following form: 8 hα > > ⊙ f ðtk , xðtk ÞÞ, k ¼ 1, 2, …, j < xðtk + 1 Þ ¼ xðtk Þ  Γ ð α + 1Þ α h > > ⊙ f ðtk , xðtk ÞÞ, k ¼ j + 1, j + 2, …,N : xðtk + 1 Þ ¼ xðtk Þ  H ð1Þ Γ ð α + 1Þ Case 4. Suppose that the fuzzy solution x(t) has a switching point at ξ  [t0, T] and it is ii  CgH differentiable at the points t0, t1, …, tj and i  CgH differentiable at the points tj+1, tj+2, …, tN on [t0, T]. The Taylor expansion about tk on any subinterval [tk, tk+1] for k ¼ 0, 1, …, N  1 can be expressed as the following form: 8 hα > > ⊙ f ðtk , xðtk ÞÞ, k ¼ 1, 2, …, j < xðtk + 1 Þ ¼ xðtk Þ  H ð1Þ Γðα + 1Þ hα > > ⊙ f ðtk , xðtk ÞÞ, k ¼ j + 1, j + 2, …, N : xðtk + 1 Þ ¼ xðtk Þ  Γ ð α + 1Þ Example. Consider the following fuzzy fractional differential equation: DαCgH xðtÞ ¼ ð0, 1, 1:5Þ ⊙ Γðα + 1Þ, xð0Þ ¼ 0, t½0, 1, 0 < α  1 The exact solution is: xðtÞ ¼ ð0, 1, 1:5Þ ⊙ tα ≔c ⊙ tα

Numerical Solutions of Uncertain Fractional Differential Equations 323 This function is i  gH differentiable, because the length of x(t) is increasing. Note. The Caputo gH-derivative of x(t) ¼ (0, 1, 1.5) ⊙ tα is: ðt 1 ðð0, 1, 1:5Þ ⊙ τα Þ0 ⊙ dτ DαCgH ðð0, 1, 1:5Þ ⊙ tα Þ ¼ ðt  τ Þα Γ ð1  α Þ 0 ð t α1 ð0, 1, 1:5Þα τ ¼ ⊙ α dτ Γð1  αÞ 0 ðt  τ Þ Since: pffiffiffi τα1 4α π ΓðαÞt2α

dτ ¼ α 1 0 ðt  τ Þ Γ α+ 2

ðt

then: DαCgH ðð0, 1, 1:5Þ ⊙ tα Þ ¼ ð0, 1, 1:5Þ ⊙

pffiffiffi α4α π ΓðαÞt2α

1 Γ α + Γ ð1  α Þ 2

In the level-wise form: pffiffiffi α4α π ΓðαÞt2α

DαC ðrtα Þ ¼ r 1 Γ α + Γð1  αÞ 2 pffiffiffi α4α π ΓðαÞt2α

DαC ðð1:5  1:5r Þtα Þ ¼ ð1:5  1:5r Þ 1 Γð1  αÞ Γ α+ 2 so the Euler method is in the form of Case 1: xðtk + 1 Þ ¼ xðtk Þ 

hα ⊙ ð0, 1, 1:5Þ ⊙ Γðα + 1Þ, k ¼ 0, 1,…, N  1 Γðα + 1Þ

For instance, consider the step size h ¼ 0.1 and α ¼ 0.5. The level-wise form of the Euler method is: xðtk + 1 , r Þ ¼ xðtk , r Þ  ð0:1Þ0:5 ⊙ ðr, 1:5  1:5r Þ, k ¼ 0,1, …, N  1

 xðtk + 1 , r Þ ¼ xðtk , r Þ  ð0:1Þ0:5 r, ð0:1Þ0:5 ð1:5  1:5r Þ , k ¼ 0,1, …, N  1 In the level-wise form: xl ðtk + 1 , r Þ ¼ xl ðtk , r Þ + ð0:1Þ0:5 r, k ¼ 0, 1,…, N  1, 0  r  1

324

Soft Numerical Computing in Uncertain Dynamic Systems xu ðtk + 1 , r Þ ¼ xu ðtk , r Þ + ð0:1Þ0:5 ð1:5  1:5r Þ, k ¼ 0, 1, …,N  1, 0  r  1

Using the recursive method and finding the values for x(tk), k ¼ 0, 1, …, 9, finally we get: xl ðt10 , r Þ ¼ 10ð0:1Þ0:5 r, xu ðt10 , r Þ ¼ 10ð0:1Þ0:5 ð1:5  1:5r Þ, 0  r  1 In Table 6.1, the numerical solutions with order α ¼ 0.5 and h ¼ 0.1 are expressed. Example. Consider the following fuzzy fractional differential equation: DαCgH xðtÞ ¼ ð1Þ ⊙ xðtÞ, xð0Þ ¼ ð0, 1, 2Þ, t½0, 1, 0 < α  1 The exact solution is: xðtÞ ¼ ð0, 1, 2Þ ⊙ Eα ðtα Þ≔c ⊙ Eα ðtα Þ where: Eα ðtα Þ ¼

∞ X ðtα Þi Γðiα + 1Þ i¼0

This function is ii  gH differentiable, because the length of x(t) is decreasing. Therefore, the Euler method is in the form of Case 2: xðtk + 1 Þ ¼ xðtk Þ  H ð1Þ

  hα ⊙ ð0, 1, 2Þ ⊙ Eα tαk , k ¼ 0, 1,…,N  1 Γðα + 1Þ

TABLE 6.1 Numerical results for step size h ¼ 0.1. t

h = 0.01, level = 0.5

t

h = 0.01, level = 0.75

0.1

0, 0.1, 0.15

0.1

0, 0.03, 0.04

0.2

0, 0.2, 0.30

0.2

0, 0.06, 0.09

0.3

0, 0.3, 0.45

0.3

0, 0.09, 0.14

0.4

0, 0.4, 0.60

0.4

0, 0.12, 0.18

0.5

0, 0.5, 0.75

0.5

0, 0.15, 0.23

0.6

0, 0.6, 0.90

0.6

0, 0.18, 0.28

0.7

0, 0.7, 1.05

0.7

0, 0.22, 0.33

0.8

0, 0.8, 1.20

0.8

0, 0.25, 0.37

0.9

0, 0.9, 1.35

0.9

0, 0.28, 0.42

1.0

0, 1.0, 1.50

1.0

0, 0.31, 0.47

Numerical Solutions of Uncertain Fractional Differential Equations 325 For instance, consider the step size h ¼ 0.1 and α ¼ 0.5. The level-wise form of the Euler method is:   0:5α E0:5 t0:5 k xðtk + 1 , r Þ ¼ xðtk , r Þ  H ð1Þ ⊙ ðr  1, 2  r Þ, Γð0:5 + 1Þ for k ¼ 0, 1, …, N  1: 0:687498 0:775758 pffiffiffiffi ¼ pffiffiffiffi ð tk + 1ÞΓð0:5 + 1Þ ð tk + 1Þ Since the value

0:5α E0:5 ðt0:5 Þ Γð0:5 + 1Þ > 0,

then, in the level-wise form:

0:775758 xl ðtk + 1 , r Þ ¼ xl ðtk , r Þ + pffiffiffiffi ð2  r Þ, k ¼ 0,1, …, N  1, 0  r  1 ð t k + 1Þ 0:775758 ðr  1Þ, k ¼ 0,1, …, N  1, 0  r  1 xu ðtk + 1 , r Þ ¼ xu ðtk , r Þ + pffiffiffiffi ð t k + 1Þ In Table 6.2, the numerical solutions with order α ¼ 0.5 and h ¼ 0.1 are expressed. Example. Consider another fuzzy fractional differential equation: 



 πt1α α 1 α 3 α 1 DαCgH xðtÞ ¼ ⊙ 0, , 1 ⊙ 1 F1 1; 1  ,  ;  π 2 t2 α2 , 1  t  2 Γð2  αÞ 2 2 2 2 4

TABLE 6.2 Numerical results for step size h ¼ 0.01. t

h = 0.01, level = 0.5

t

h = 0.01, level = 0.75

0.1

0, 0.3020, 0.6040

0.1

0, 0.70, 1.40

0.2

0, 0.0912, 0.1824

0.2

0, 0.49, 0.99

0.3

0, 0.0275, 0.0551

0.3

0, 0.34, 0.69

0.4

0, 0.0083, 0.0166

0.4

0, 0.24, 0.49

0.5

0, 0.0025, 0.0050

0.5

0, 0.17, 0.34

0.6

0, 0.0009, 0.0017

0.6

0, 0.12, 0.24

0.7

0, 0.0003, 0.0006

0.7

0, 0.08, 0.17

0.8

0, 0.0001, 0.0002

0.8

0, 0.05, 0.11

0.9

0, 0.0000, 0.0001

0.9

0, 0.04, 0.08

1.0

0, 0.0000, 0.0000

1.0

0, 0.03, 0.06

326

Soft Numerical Computing in Uncertain Dynamic Systems

with fuzzy initial value x(1) ¼ (0,0.5,1) ⊙ sin απ, where 1F1(a; b; z) is a generalized hypergeometric function and defined as the following form:   ∞   X ð a 1 Þ n ⋯ ap n z n  

p Fq a1 , a2 , …, ap ; b1 , b2 , …, bq ; z ¼ n! n¼0 ðb1 Þn ⋯ bq n Here: F ða; b; zÞ ¼ 1 1

∞ X ð aÞ

n

n¼0

ðbÞn



∞ zn X Γ ð a + nÞ ΓðbÞ zn ¼



n! n¼0 ΓðaÞ Γðb + nÞ n!

where: ð Þn ¼

Γ ð + nÞ Γð Þ

Note that by the ratio test, the series is convergent. In this example, using α ¼ 0.5, the exact solution is: xðtÞ ¼ ð0; 0:5; 1Þ ⊙ sin ðαπtÞ The solution has a switching point at the point t ¼ 1.463 and the switching point is type I; before the point it is i  gH differentiable and after it is ii  gH differentiable. In the subintervals [tk, tk+1], k ¼ 0, 1, …, N  1 with assumption that switching point is in the interval [tj, tj+1], the fuzzy Euler’s method is denoted as: 8 hα > > ⊙ f ðtk , xðtk ÞÞ, xðtk + 1 Þ ¼ xðtk Þ  k ¼ 1, 2, …, j > < Γ ð α + 1Þ > > > : xðtk + 1 Þ ¼ xðtk Þ  H ð1Þ

hα ⊙ f ðtk , xðtk ÞÞ, k ¼ j + 1, j + 2, …,N Γ ð α + 1Þ

where: 



 πt1α 1 α 3 α 1 22 2 k α f ðtk , xðtk ÞÞ ¼ ⊙ 0, , 1 ⊙ 1 F1 1; 1  ,  ;  π tk α Γ ð2  α Þ 2 2 2 2 4 Suppose h ¼ 0.1, k ¼ 0, α ¼ 0.5, t0 ¼ 1, x(1) ¼ (0,0.5,1) ⊙ sin απ. Then: xðt1 Þ ¼ ð0; 0:5; 1Þ 

h0:5 ⊙ f ð1, ð0; 0:5; 1ÞÞ Γð0:5 + 1Þ

where: 



 π0:5 1 3 5 1 f ð1, ð0;0:5; 1ÞÞ ¼ ⊙ 0, , 1 ⊙ 1 F1 1; , ;  π 2 3 2 4 4 16 Γ 2

Numerical Solutions of Uncertain Fractional Differential Equations 327

  3 5 1 2 n 

X  Γ  , π ∞ 3 5 1 Γð1 + nÞ 4 4 16 

¼ , ;  π2

 1 F1 1; 3 5 n! 4 4 16 Γ ð 1 Þ n¼0 Γ + n, + n 4 4

n

3 2 3 1 2 5 1 2 n π Γ  Γ  π ∞ ∞ X 6X 7 Γ ð 1 + nÞ Γ ð 1 + nÞ 4 16 4 16 7¼

,



¼6

4 5 3 5 n! n! Γ ð 1 Þ Γ ð 1 Þ n¼0 n¼0 Γ Γ 4 4   16 16 16 ¼ , ¼ 16 + π 2 16 + π 2 16 + π 2 thus:

π0:5 1 16 f ðt0 , xðt0 ÞÞ ¼ f ð1, ð0; 0:5; 1ÞÞ ¼ ⊙ 0, , 1 ⊙ 3 2 16 + π 2 Γ 2 π0:5 16 Since  1:1 and positive, then f(t0, x(t0)) ¼ 1.1 ⊙ (0,0.5,1) and in the 3 16 + π2 Γ 2 level-wise form:

fl ðt0 , xðt0 Þ, r Þ ¼

1:1 1:1 r, fu ðt0 , xðt0 Þ, r Þ ¼ 1:1  r 2 2

and the method for approximate the solution at the second point x(t1) is:

1 h0:5 1:1 1 h0:5 1:1

1:1  r, xu ðt1 , r Þ ¼ 1  r + r x l ðt 1 , r Þ ¼ r + Γð0:5 + 1Þ 2 Γð0:5 + 1Þ 2 2 2 Putting h ¼ 0.1 and in the level, for instance, r ¼ 0.5, we have: 1 0:50:5 1:1 3 0:50:5 3

1:1  1:41  0:47, xu ðt1 , r Þ ¼ + xl ðt1 , 0:5Þ ¼ + 4 Γð1:5Þ 4 4 Γð1:5Þ 4 In general, for α ¼ 0.5 we have:



πto:5 1 1 22 k 0:5 f ðtk , xðtk ÞÞ ¼ ⊙ 0, , 1 ⊙ 1 F1 1; ½0:5; 1;  π tk Γð1:5Þ 2 16 and: 8 h0:5 > > ⊙ f ðtk , xðtk ÞÞ, k ¼ 1, 2, …, j < xðtk + 1 Þ ¼ xðtk Þ  Γð1:5Þ > h0:5 > : xðtk + 1 Þ ¼ xðtk Þ  H ð1Þ ⊙ f ðtk , xðtk ÞÞ, k ¼ j + 1, j + 2, …,N Γð1:5Þ The results for h ¼ 0.01 are listed in Table 6.3.

328

Soft Numerical Computing in Uncertain Dynamic Systems

TABLE 6.3 Numerical results with order α ¼ 0.5 for step size h ¼ 0.01, 0.001. t

h = 0.01

h = 0.01

1.1

0, 0.4936, 0.9875

0, 0.4942, 0.9823

1.2

0, 0.4745, 0.9514

0, 0.4747, 0.9512

1.3

0, 0.4468, 0.8923

0, 0.4452, 0.8920

1.4

0, 0.4024, 0.8092

0, 0.4023, 0.8031

1.5

0, 0.3526, 0.7072

0, 0.3523, 0.7062

1.6

0, 0.2945, 0.5835

0, 0.2923, 0.5832

1.7

0, 0.2248, 0.4583

0, 0.2246, 0.4545

1.8

0, 0.1536, 0.3148

0, 0.1536, 0.3903

1.9

0, 0.0264, 0.1265

0, 0.0011, 0.0234

1.0

0, −0.018, −0.012

0, 0.0000, 0.0000

References Armand, A., Allahviranloo, T., Abbasbandy, S., Gouyandeh, Z., 2019. The fuzzy generalized Taylor’s expansion with application in fractional differential equations. Iran. J. Fuzzy Syst. 16 (2), 57–72. Chehlabi, M., Allahviranloo, T., 2016. Concreted solutions to fuzzy linear fractional differential equations. Appl. Soft Comput. 44, 108–116. Garrappa, R., Kaslik, E., Popolizio, M., 2019. Evaluation of fractional integrals and derivatives of elementary functions: overview and tutorial. Mathematics 7, 407. Mathai, A.M., Haubold, H.J., 2017. An Introduction to Fractional Calculus. Nova Science Publishers, Inc., Hauppauge, New York Van Hoa, N., Ho, V., Duc, T.M., 2019. Fuzzy fractional differential equations under Caputo–Katugampola fractional derivative approach. Fuzzy Sets Syst. 375, 70–99.

7.1

Chapter 7

Numerical solutions of uncertain partial differential equations Introduction

In this chapter, first fuzzy heat and Poisson equations with fuzzy initial values are considered. The concept of generalized Hukuhara differentiation is interpreted thoroughly in the univariate and multivariate cases. The first objective of this chapter is to prove the uniqueness of a solution for a fuzzy heat equation and show that a fuzzy heat equation can be modeled as two systems of fuzzy differential equations by considering the type of differentiability of solutions. Then the fuzzy Fourier transform is applied for solving the fuzzy heat equation. As the second object, the fuzzy Poisson’s equation and the fuzzy finite difference method are introduced. Then, the fuzzy Poisson’s equation is discretized by the fuzzy finite difference method and it is solved as a linear system of equations. In addition, we discuss the fuzzy Laplace equation as a special case of the fuzzy Poisson’s equation. In addition, the convergence of method is taken into account.

329 Soft Numerical Computing in Uncertain Dynamic Systems. https://doi.org/10.1016/B978-0-12-822855-5.00007-0 © 2020 Elsevier Inc. All rights reserved.

330

7.1.1

Soft Numerical Computing in Uncertain Dynamic Systems

PARTIAL

ORDERING

For two fuzzy numbers A,B  R , we call ≼ a partial order notation and: A ≼ ð
ÞB if any only if Al ðr Þ  ð 0 subject to DH(x(t), x(t0)) < E whenever x is an arbitrary value from j x  x0 j < δ.

7.1.3

MINIMUM

AND MAXIMUM

For any gH-differentiable fuzzy number function x(t) at the inner point, such as c  (a, b), if the function has local minimum or maximum, then xgH0 (c) ¼ 0 and: 8t½a, b 9tMin 9tMax ½a, b s:t: xðtMin Þ ≼ x ðtÞ ≼ xðtMax Þ where max atb xðtÞ ¼ xðtMax Þ and min atb xðtÞ ¼ xðtMin Þ. For more information, see Allahviranloo et al. (2015) and Gouyandeha et al. (2017).

7.1.4

PRODUCTION

IN PARTIAL GH-DIFFERENTIABILITY

Suppose that uðx, tÞ : R  R0 ! R is a fuzzy number valued function and it is gH-partial differentiable at (x, t) on D (Bede & Gal, 2005a, b; Stefanini & Bede, 2009). Also let us suppose the function v(x, t) : R  R0 ! R is a differentiable real function in the same region. Then: ∂tgH ðu ⊙ vÞðx, tÞ ¼ ∂tgH uðx, tÞ ⊙ vðx, tÞ v0 ðx, tÞ ⊙ uðx, tÞ

7.1.5

FUZZY

INTEGRATING FACTOR

Consider the following two-point fuzzy boundary value problem for t  [a, b]: y00gH ðtÞ pðtÞ ⊙ y0gH ðtÞ ¼ 0, yðaÞ ¼ c0 , yðbÞ ¼ c1 Ð

where pðtÞR, c0 , c1 R . Multiplying both sides by the integrating factor e we get:

p(t)dt

,

Numerical solutions of uncertain partial differential equations 331 ð

ð pðtÞdt

e

pðtÞdt

⊙ y00gH ðtÞ pðtÞe

⊙ y0gH ðtÞ ¼ 0

Using the production in gH-derivative, we have: ð 0  pðtÞdt ¼0 e ⊙ y0gH ðtÞ gH

So, by taking the integral: ð pðtÞdt e

⊙ y0gH ðtÞ ¼ λ0 R

Again, integrating both sides: ðt a

y0gH ðsÞds ¼ λ0 ⊙

ð

ðt

pðsÞds



e

ds

a

Hence: yðtÞ  gH c0 ¼ λ0 ⊙

ð

ðt



pðsÞds

e

ds

a

Ð Ð Therefore, if y(t) is i  gH differentiable, we obtain y(t) ¼ c0 λ0 ⊙ Ðtae p(s)dsds Ð and if y(t) is ii  gH differentiable, we obtain y(t) ¼ c0 H(1)λ0 ⊙ tae p(s)dsds:

λ0 ¼

c1  gH c0 ð ð b  pðsÞds e ds a

7.2

The fuzzy heat equation

Assume that we have a rod of some material of constant cross-section that is surrounded by insulation so that heat can only flow along the rod and not out of the cylindrical surface. We assume that the rod is infinitely long (∞ < x < ∞) and that no boundary conditions are required. Moreover, the temperature is uniform across each cross-section. A fuzzy heat equation models the flow of heat in this rod that is insulated everywhere except at the two ends. We start with an initial temperature distribution u(x, 0) ¼ f (x), such that f ðxÞR , then the fuzzy heat equation is as follows:

332

Soft Numerical Computing in Uncertain Dynamic Systems ∂tgH uðx, tÞ  gH κ ⊙ ∂xxgH uðx, tÞ ¼ Fðx, tÞ, κR0

where κ is the constant heat conductivity coefficient and the fuzzy function uðx, tÞ : R  R0 ! R that models heat flow should satisfy the fuzzy heat equation. First, we shall discuss the existence and uniqueness of the solution, and to this end we need the maximum principle. This principle has a simple interpretation: if we have an insulated rod of length L that can only be heated at the ends of the rod when x¼ 0 or x ¼ L, then the temperature cannot rise above the maximum of the initial temperatures and the temperatures at the ends of the rod.

7.2.1

THEOREM—FUZZY

MAXIMUM PRINCIPLE

Let us consider that the fuzzy number valued function u(x, t) is the solution of the fuzzy heat equation: ∂tgH uðx, tÞ  gH κ ⊙ ∂xxgH uðx, tÞ ≼ 0 in D ¼ {(x, t)j0 < x < L, 0 < t  T} and is a fuzzy continuous in D ¼ fðx, tÞ j 0 < x < L, 0 < t  Tg. Hence u(x, t) attains its maximal values on the set: Γ ¼ fðx, tÞj x ¼ 0, 0  t  T; x ¼ L, 0  t  T; 0 < x < L, t ¼ 0 g Proof. We first prove that if v(x, t) is a fuzzy continuous function in D and satisfies the inequality in D: ∂tgH vðx, tÞ  gH ∂xxgH vðx, tÞ
0 and it assumes its maximum on Γ. Suppose that v has a local maximum at P ¼ (x0, t0) in D. According to the maximum and minimum theorem, we have: ∂tgH vðx0 , t0 Þ ¼ 0, ∂xxgH vðx0 , t0 Þ ≼ 0 Since κ > 0, and based on the properties of the partial ordering, we have: ∂tgH vðx, tÞ  gH κ ⊙ ∂xxgH vðx, tÞ≽0 and indeed, it contradicts the fact that v(x, t) satisfies: ∂tgH vðx, tÞ  gH ∂xxgH vðx, tÞ
0 Moreover, since v(x, t) is continuous in D, we suppose that M ¼ max Γ vðx, tÞ and MR . Now if we define: vðx, tÞ ¼ uðx, tÞ + Ex2 , E > 0 then:   ∂tgH vðx, tÞ  gH κ ⊙ ∂xxgH vðx, tÞ ¼ ∂tgH uðx, tÞ  gH κ ⊙ ∂xxgH uðx, tÞ + Ex2

Numerical solutions of uncertain partial differential equations 333   ¼ ∂tgH uðx, tÞ  gH κ ⊙ ∂xxgH uðx, tÞ + Ex2  2κE ¼ 2κE
0 then: ∂tgH vðx, tÞ  gH κ ⊙ ∂xxgH vðx, tÞ
0 and it assumes its maximum on Γ. Therefore: 8ðx, tÞR,vðx, tÞ ¼ uðx, tÞ + Ex2 ≼ M + EL2 By removing the arbitrary small E we have: uðx, tÞ ≼ vðx, tÞ ≼ M The proof is completed (Allahviranloo et al., 2015; Armand et al., 2019; Gouyandeha et al., 2017).

7.2.2

THEOREM—EXISTENCE

There exists at most one solution of the fuzzy heat equation:  ∂tgH uðx, tÞ  gH ∂xxgH uðx, tÞ ¼ Fðx, tÞ,  ∞ < x < ∞, t > 0 uðx, 0Þ ¼ f ðxÞ, ∞ 0 0, x 0 Then we define:   wðξ, τÞ ¼ Ec u Ea ξ, Eb τ , ER, c > 0 Note that ξ(x) and τ(t) are strictly increasing, so this changing variable gives: ∂tgH uðx, tÞ ¼ Ec ⊙ ∂τgH wðξ, τÞ ⊙ ∂tgH τ ¼ Ebc ⊙ ∂τgH wðξ, τÞ ∂xgH uðx, tÞ ¼ Ec ⊙ ∂ξgH wðξ, τÞ ⊙ ∂xgH ξ ¼ Eac ⊙ ∂ξgH wðξ, τÞ   ∂xxgH uðx, tÞ ¼ Eac ⊙ ∂ξgH ∂ξgH wðξ, τÞ ⊙ ∂xgH ξ ¼ E2ac ⊙ ∂ξξgH wðξ, τÞ

Numerical solutions of uncertain partial differential equations 335 So, the fuzzy heat equation transforms into: Ebc ⊙ ∂τgH wðξ, τÞ  gH κE2ac ⊙ ∂ξξgH wðξ, τÞ ¼ 0 and is invariant under the dilatation transformation if b ¼ 2a for all ε. Thus, if u solves the equation at x, t then w ¼ E cu solves the equation at x ¼ ε aξ, t ¼ ε bτ. We can build some groupings of independent variables, such as: ξ a τb

¼

x a tb

¼

Ea x a

ðEb tÞb

x which define the variable ηðx, tÞ ¼ pffiffiffiffiffi , since b ¼ 2a. This implies that: 2κτ

w c τb

¼

u c tb

¼

Ec c

ðEb t Þb

¼ vðη Þ c

We look for the solution of fuzzy heat equation of the form uðx, tÞ ¼ t2a ⊙ vðηÞ, where vðηÞR . Under the assumptions that all gH  p  derivatives of u(x, t) exist, we examine the solution of the fuzzy heat equation: •

η xffiffiffiffiffi If x > 0 then ∂tgH ηðx, tÞ ¼ 2tp ¼ 1 2t ηðx, tÞ≔ 2t < 0, then: 2κt

∂t, igH uðx, tÞ ¼

c c c 1 t2a ⊙ vðηÞ t2a ⊙ ∂t, iigH ηv0iigH ðηÞ ¼ 2a

 1 c c ¼ t2a1 ⊙ vðηÞ  gH η ⊙ v0iigH ðηÞ 2 a c c c ∂t, iigH uðx, tÞ ¼ t2a1 ⊙ vðηÞ t2a ⊙ ∂t, igH ηv0iigH ðηÞ ¼ 2a  1 c c ¼ t2a1 ⊙ vðηÞ  gH η ⊙ v0igH ðηÞ 2 a

•

xffiffiffiffiffi If x < 0 then ∂tgH ηðx, tÞ ¼ 2tp ¼ 2tη > 0, then: 2κt

∂t, igH uðx, tÞ ¼

c c c 1 t2a ⊙ vðηÞ t2a ⊙ ∂t, igH ηv0igH ðηÞ ¼ 2a

 1 c c ¼ t2a1 ⊙ vðηÞ  gH η ⊙ v0igH ðηÞ 2 a c c c 1 ∂t, iigH uðx, tÞ ¼ t2a ⊙ vðηÞ t2a ⊙ ∂t, igH ηv0iigH ðηÞ ¼ 2a

336

Soft Numerical Computing in Uncertain Dynamic Systems  1 c c ¼ t2a1 ⊙ vðηÞ  gH η ⊙ v0iigH ðηÞ 2 a

ffi > 0, we get: Also, considering that ∂x η ¼ p1ffiffiffiffi 2κt c

1

c t2a 2 ∂xgH uðx, tÞ ¼ t2a ⊙ v0gH ðηÞ ⊙ ∂x η ¼ pffiffiffiffiffi ⊙ v0gH ðηÞ 2κ

Therefore, we observe that: c

1

t2a 2 ∂xxgH uðx, tÞ ¼ pffiffiffiffiffi ⊙ v00gH ðηÞ 2κ This means that ∂xη does not change the type of differentiability of u(x, t) with respect to x. We then obtain the following expressions. Case 1. Let assume that the type of gH  p-differentiability for u(x, t) and ∂xu(x, t) do not change with respect to x, so we obtain the following: 1.1. If u(x, t) is i  gH  p differentiable with respect to t then using the mentioned partial differentiability in two cases x > 0 and x < 0, and: c

1

t2a 2 ∂xxgH uðx, tÞ ¼ pffiffiffiffiffi ⊙ v00gH ðηÞ 2κ The fuzzy heat equation reduces to the following fuzzy heat equations:  8 γ  < t21 v00i, gH ðηÞ η ⊙ v0ii, gH ðηÞ  gH γ ⊙ vðηÞ ¼ 0, if x > 0  γ  : t21 v00 ðηÞ η ⊙ v0 ðηÞ  γ ⊙ vðηÞ ¼ 0, if x < 0 gH i, gH i, gH γ

γ

ffi. Since u ¼ t2 vðηÞ ! u0 as η ! + ∞ with γ ¼ ac such that uðx, tÞ ¼ t2 vðηÞ and ¼ pxffiffiffiffi 2κt where u0 does not depend on t, γ must be zero. Hence, v(η) is the solution of the fuzzy boundary value differential equations: 8  00 v ðηÞ η ⊙ v0ii, gH ðηÞ ¼ 0 > > if x > 0 > i, gH < vð +∞Þ ¼ u0 R  00 > vi, gH ðηÞ η ⊙ v0i, gH ðηÞ ¼ 0 > > if x < 0 : vð∞Þ ¼ 0R In the case where x > 0, we use the fuzzy integrating factoring method to get the following relations: η2 e2

η2 ⊙ v00i, gH ðηÞ ηe 2

⊙ v0ii, gH ðηÞ ¼

 2 0 η 0 e 2 ⊙ vii, gH ðηÞ i, gH

¼0

Numerical solutions of uncertain partial differential equations 337 then: η2

e 2 ⊙ v0ii, gH ðηÞ ¼ λ0 ¼)v0ii, gH ðηÞ ¼ λ0 ⊙ e hence:

ð +∞ η

ð +∞

v0ii, gH ðyÞdy ¼ λ0 ⊙

e

η2 2

y2 2 dy

η

since: ð +∞ η

so:

v0ii, gH ðyÞdy ¼ vð +∞Þ  gH vðηÞ,

ð +∞ η

y2 pffiffiffi e 2 dy ¼ 2λ0

⊙

ð +∞ ffi

η p 2

es ds 2

ð +∞ pffiffiffi 2 vð +∞Þ  gH vðηÞ ¼ 2λ0 ⊙ es ds η pffi 2 ð +∞ pffiffiffi 2 vðηÞ ¼ u0 ð1Þ 2λ0 ⊙ es ds η pffi 2

It is concluded that in this case, v(η) is ii  gH differentiable. In the case where x < 0, we use the fuzzy integrating factoring method to get the following relations:  2 0 η2 η2 η e 2 ⊙ v00i, gH ðηÞ ηe 2 ⊙ v0i, gH ðηÞ ¼ e 2 ⊙ v0i, gH ðηÞ

¼0

i, gH

then: η2

e 2 ⊙ v0i, gH ðηÞ ¼ λ0 ¼)v0i, gH ðηÞ ¼ λ0 ⊙ e hence:

ðη ∞

v0i, gH ðyÞdy ¼ λ0

⊙

ðη

η2 2

y2 2 dy

e ∞

since: ðη ∞

v0i, gH ðyÞdy ¼ vðηÞ  gH vð∞Þ,

ðη ∞

y2 pffiffiffi e 2 dy ¼ 2λ0

so: ð pffiffi pffiffiffi 2 s2 e ds vðηÞ  gH vð∞Þ ¼ 2λ0 ⊙ η

∞

⊙

ð pηffiffi 2

∞

es ds 2

338

Soft Numerical Computing in Uncertain Dynamic Systems ð pffiffi pffiffiffi 2 s2 vðηÞ ¼ vð∞Þ 2λ0 ⊙ e ds η

∞

Finally, by adding both sides of the following relations, we get: ð +∞ ð pffiffi pffiffiffi pffiffiffi 2 2 s2 es ds,vðηÞ  gH vð∞Þ ¼ 2λ0 ⊙ e ds vð +∞Þ  gH vðηÞ ¼ 2λ0 ⊙ η pffi ∞ 2 η

We have: ð +∞ ð pffiffi pffiffiffi pffiffiffi 2 s2 s2 e ds 2λ0 ⊙ e ds vð +∞Þ  gH vð∞Þ ¼ 2λ0 ⊙ η pffi ∞ 2 η

! ð pηffiffi ð +∞ ð∞ pffiffiffi pffiffiffi 2 2 s2 s2 ¼ 2λ 0 ⊙ e ds + e ds ¼ 2λ0 ⊙ es ds η p ffi ∞ ∞ 2 Indeed: ð +∞ pffiffiffi 2 es ds u0  gH 0 ¼ 2λ0 ⊙ Since,

Ð +∞ ∞

es

2

∞

pffiffiffi ds ¼ π then: u0 λ0 ¼ pffiffiffiffiffi 2π

Case x > 0 and v(η) is ii  gH differentiable: u0 vðηÞ ¼ u0 ð1Þ pffiffiffi ⊙ π

ð +∞ ffi

η p 2

es ds 2

Case x < 0 and v(η) is i  gH differentiable: u0 vðηÞ ¼ pffiffiffi ⊙ π

ð pηffiffi 2

es ds 2

∞

since: γ c x γ ¼ , uðx, tÞ ¼ t2 vðηÞ, η ¼ pffiffiffiffiffiffi a 2κt

Hence the analytical solution of the fuzzy heat equations can be obtained as follows:

Numerical solutions of uncertain partial differential equations 339 0 1 8 ð +∞ > > γ > u0 2 B C > > uðx, tÞ ¼ t2 @u0 ð1Þ pffiffiffi ⊙ x es dsA, x > 0 > > π < pffiffiffi 2 κt 0 1 x > ð pffiffiffi > > γ u 2 > 0 > > uðx, tÞ ¼ t2 ⊙ @pffiffiffi ⊙ 2 κt es dsA, x < 0 > : π ∞

1.2. If u(x, t) is ii  gH  p differentiable with respect to t, then according to the process described above, we obtain the following equations: 8 ( 00 vi, gH ðηÞ η ⊙ v0i, gH ðηÞ ¼ 0 > > > > > < vð +∞Þ ¼ u0 R ( > > v00i,gH ðηÞ η ⊙ v0ii, gH ðηÞ ¼ 0 > > > : vð∞Þ ¼ 0R

if x > 0 if x < 0

Using the same process, the analytical solution of the fuzzy heat equations can be obtained as follows: 0 1 8 > ð > +∞ γ > u0 2 B C > > uðx, tÞ ¼ t2 @u0  gH pffiffiffi ⊙ x es dsA, x > 0 > > > π < pffiffiffi > > > > > > > > :

0 γ uðx, tÞ ¼ t2

u0 ⊙ @pffiffiffi ⊙ π

2 κt

ð px ffiffiffi 2 κt s2 e

∞

1 dsA, x < 0

Case 2. Let assume that the type of gH  p-differentiability for u(x, t) and ∂xu(x, t) changes with respect to x, so we obtain the following:

2.1. If u(x, t) is i  gH  p differentiable with respect to t then, according to the process described above, we obtain the following equations: 8  00 vii, gH ðηÞ η ⊙ v0ii, gH ðηÞ ¼ 0 > > > < vð +∞Þ ¼ u  0 R  00 0 > ð η Þ η ⊙ v v > ii, gH i, gH ðηÞ ¼ 0 > : vð∞Þ ¼ 0R

if x > 0 if x < 0

340

Soft Numerical Computing in Uncertain Dynamic Systems

Hence the analytical solution of the fuzzy heat equations can be obtained as follows: 0 1 8 > ð > +∞ γ > u0 2 B C > > uðx, tÞ ¼ t2 @u0 ð1Þ pffiffiffi ⊙ x es dsA, x > 0 > > > π < pffiffiffi > > > > > > > > :

0 γ uðx, tÞ ¼ t2

u0 ⊙ @pffiffiffi ⊙ π

2 κt

ð pxffiffiffi 2 κt s2 e

∞

1 dsA, x < 0

2.2. If u(x, t) is ii  gH  p differentiable with respect to t then, according to the process described above, we obtain the following equations: 8( > v00ii, gH ðηÞ η ⊙ v0i, gH ðηÞ ¼ 0 > > > > < vð +∞Þ ¼ u0 R ( > > v00ii, gH ðηÞ η ⊙ v0ii, gH ðηÞ ¼ 0 > > > : vð∞Þ ¼ 0R

if x > 0 if x < 0

0 1 8 > ð +∞ > γ > u0 2 B C > > uðx, tÞ ¼ t2 @u0  gH pffiffiffi ⊙ x es dsA, x > 0 > > > π < pffiffiffi 2 κt

0 1 > > ð pxffiffiffi > γ > u 2 > 0 > > uðx, tÞ ¼ t2 ⊙ @pffiffiffi ⊙ 2 κt es dsA, > : π ∞

x ð +∞ > γ > u0 2 C B > > uðx, tÞ ¼ t2 @u0 ð1Þ pffiffiffi ⊙ x es dsA, x > 0 > > > π < pffiffiffi 2 κt

0 1 > > ð pxffiffiffi > γ > u 2 > 0 > > uðx, tÞ ¼ t2 ⊙ @pffiffiffi ⊙ 2 κt es dsA, > : π ∞

x > γ > u0 2 B C > > uðx, tÞ ¼ t2 @u0  gH pffiffiffi ⊙ x es dsA, x > 0 > > π < pffiffiffi 2 κt 0 1 x > ð pffiffiffi > > γ u 2 > 0 > > x : π ∞

Example. Consider the following heat equation:  ∂t, igH uðx, tÞ  gH ∂xx, iigH uðx, tÞ ¼ 0, uðx, 0Þ ¼

ð1, 3, 5Þ, x > 0 0, x 0 we have:

    1 x p ffiffiffi 1 , uðx, t, r Þ ¼ ð2r + 1Þ + ð2r + 5Þ erf 2 2 π     1 x ð2r + 5Þ + ð2r + 1Þ erf pffiffiffi  1 2 2 π If x < 0: uðx, t, rÞ ¼

        1 x 1 x ð2r + 1Þ erf pffiffiffi + 1 , ð2r + 5Þ erf pffiffiffi + 1 2 2 π 2 2 π

For r ¼ 12 and t  [1, 4], Fig. 7.1 shows the fuzzy solutions in two cases x > 0 and x < 0.

4 3 u(x,t)

u(x,t)

1.6 1.0 0.5 0.0

−4

4 −3

3 −2 x

2

−1 1

t

2 1 4

0 3

1 2 x

2

3

t

41

Fig. 7.1 The fuzzy solutions in two cases x > 0 (left) and x < 0 (right) for r ¼ 2:1

342

Soft Numerical Computing in Uncertain Dynamic Systems

Example. Consider the following heat equation:  ∂t,iigH uðx, tÞ  gH 2 ⊙ ∂xx, igH uðx, tÞ ¼ 0,

uðx, 0Þ ¼

ð3:5,  2,  1Þ, x < 0 0, x>0

Since u(x, t) is ii  p  gH differentiable with respect to t, then for x > 0 we have:

      pffiffiffi  2x 3 1 3 uðx, t, r Þ ¼ r  3:5 + r  3:5 erf pffiffiffi  1 , 4 π 2 2 2 p ffiffi ffi     2x 1 ðr  1Þ + ðr  1Þ erf pffiffiffi  1 4 π 2 If x < 0:

   pffiffiffi     pffiffiffi   2x 2x 1 3 1 uðx, t, r Þ ¼ r  3:5 erf pffiffiffi + 1 , ðr  1Þ erf pffiffiffi + 1 4 π 4 π 2 2 2

For r ¼ 13, Fig. 7.2 shows the fuzzy solutions in two cases x > 0 and x < 0.and for any x  R the solution is demonstrated as shown in Fig. 7.3. Example. Consider the following heat equation:  ∂t,igH uðx, tÞ  gH 0:76 ⊙ ∂xx, igH uðx, tÞ ¼ 0, uðx, 0Þ ¼

ð0:6;2:2;4:7Þ, 0,

x0

Since u(x, t) is ii  p  gH differentiable with respect to t, then for x > 0 we have: 0 0 1 2 1 6 B B x C C 1 B B C  1C, uðx, t, r Þ ¼ 6 4ð1:6r + 0:6Þ + 2 ð2:5r + 4:7Þ@ erf @rffiffiffiffiffi A A 19 t 25 0 0 1 13

−1.0 −1.5 −2.0 −2.5 −3.0

u(x,t)

u(x,t)

B B x C C7 1 7 rffiffiffiffiffi C ð2:5r + 4:7Þ + ð1:6r + 0:6ÞB erf B  1C @ @ A A5 2 19 t 25

4

0

3

1 2 x

1

3 40

Fig. 7.2

2 t

−1.0 −1.5 −2.0 −2.5 −3.0 −4

4 −3

3 −2 x

−1

1

2 t

00

The fuzzy solutions in two cases x > 0 (left) and x < 0 (right) for r ¼ 13.

Numerical solutions of uncertain partial differential equations 343

u(x,t)

−1.0 −1.5 −2.0 −2.5 −3.0

4

−4

3

−2 0 x

1

2

2 t

40

Fig. 7.3 The fuzzy solution for all x  R.

If x < 0: 2

0

0

1

1

0

0

1

13

61 B B x C C 1 B B C C7 B B C + 1C, ð2:5r + 4:7ÞB erf Brxffiffiffiffiffi C + 1C7 uðx, t, r Þ ¼ 6 42 ð1:6r + 0:6Þ@ erf @rffiffiffiffiffi A A @ @ 2 19 19 A A5 t t 25 25

For r ¼ 13, Fig. 7.4 shows the fuzzy solutions in two cases x > 0 and x < 0.

7.3.1

THE

FUNDAMENTAL SOLUTION OF THE FUZZY HEAT EQUATION

3

1.5

2

1.0

u(x,t)

u(x,t)

In this section, we are going to find the solution of a fuzzy heat equation based on the fuzzy Fourier transforms. These transforms have been discussed with more illustration in Chapter 3. Here are some concepts that we need in this section (Gouyandeha et al., 2017).

1 0

4 3 2 x

2 1

4 0

t

0.5 0.0

0

0 −1

−2 x

2 −3

4

t

−4

Fig. 7.4 The fuzzy solutions in two cases x > 0 (left) and x < 0 (right) for r ¼ 13.

344

Soft Numerical Computing in Uncertain Dynamic Systems

7.3.2

FUZZY FOURIER

TRANSFORM

Consider the function f : R ! R is a fuzzy number valued function. The fuzzy Fourier transform of f(x) denoted by ðF ff ðxÞg : R ! C Þ is given by the following integral: ð 1 ∞ f ðxÞ ⊙ eiwx dx ¼ FðwÞ F ff ðxÞg ¼ pffiffiffiffiffi 2π ∞ Here C is the set of all fuzzy numbers on complex numbers.

7.3.3

FUZZY

INVERSE

FOURIER

TRANSFORM

If F(w) is the fuzzy Fourier transform of f(x), then the fuzzy inverse Fourier transform of F(w) is defined as follows: ð 1 ∞ F 1 fFðwÞg ¼ pffiffiffiffiffi f ðwÞ ⊙ eiwx dw ¼ f ðxÞ 2π ∞ Also: F fa ⊙ f ðxÞ b ⊙ gðxÞg ¼ a ⊙ F ff ðxÞg b ⊙ F fgðxÞg

F a ⊙ f ðxÞ  gH b ⊙ gðxÞ ¼ a ⊙ F ff ðxÞg  gH b ⊙ F fgðxÞg

7.3.4

FOURIER

TRANSFORM OF GH-DERIVATIVE

Let f(x) be fuzzy continuous, fuzzy absolutely integrable, and converge to zero as jx j ! ∞. Furthermore, if fgH0 (x) is fuzzy absolutely integrable on (∞, ∞). Then: n o 0 ðxÞ ¼ iwF ff ðxÞg F fgH 0 (x) be fuzzy continuous on (∞, +∞) and f(k) Let f(x) and f gH gH ! 0 as j x j ! ∞ , ( j) k ¼ 0, 1 and fgH is absolutely integrable in (∞, +∞), j ¼ 0, 1, 2, then: n o 00 ðxÞ ¼  gH w2 F ff ðxÞg F fgH

Consider the following fuzzy homogenous heat equation: 8  ∞ < x < ∞, t  0 < ∂tgH uðx, tÞ  gHðκ ⊙ ∂xxgH uðx, tÞ ¼ 0, : uðx, 0Þ ¼ f ðxÞ,

∞

∞

DH ðf ðxÞ, 0Þdx < ∞,

∞ > > > < u2, 0,l  4u1,0,l + u0,0,l + u1,1,l ¼ h2 f1,0,l > ⋮ > > > : 4uN,0,l + uN1,0,l + uN,1,l ¼ h2 fN,0, l , uN + 1,0,l ¼ 0 8 u1,0,u  4u0,0,u + u0,1,u ¼ h2 f0,0,u , u1, 0,u ¼ 0 > > > < u 2 2,0,u  4u1,0,u + u0,0,u + u1,1,u ¼ h f1,0, u > ⋮ > > : 4uN,0,u + uN1,0,u + uN,1,u ¼ h2 fN,0,u , uN + 1,0,u ¼ 0 If j ¼ 1 is fixed and i ¼ 0, 1, …, N, then: ui + 1,1  H 4 ⊙ ui,1 ui1,1 ui,2 ¼ h2 fi,1 , i, j ¼ 0, 1,…, N: In level-wise form: ui + 1,1,l  4ui,1,l + ui1,1,l + ui,2,l ¼ h2 fi, 1,l , ui + 1,1,u  4ui,1,u + ui1,1,u + ui,2, u ¼ h2 fi,1,u ,

i, j ¼ 0, 1, …,N: i, j ¼ 0,1, …, N:

For i ¼ 0, 1, …, N, we have: 8  4u0,1, l + u0,2,l ¼ h2 f0, 1,l ,u1,1,l ¼ 0 u > > < 1,1,l u2, 1,l  4u1,1,l + u0,1,l + u1,2,l ¼ h2 f1,1,l ⋮ > > : 4uN,1,l + uN1,1,l + uN,2,l ¼ h2 fN,1, l , uN + 1,1,l ¼ 0 8 u  4u0,1,u + u0,2,u ¼ h2 f0,1,u , u1, 1,u ¼ 0 > > < 1,1,u u2,1,u  4u1,1,u + u0,1,u + u1,2,u ¼ h2 f1,1, u ⋮ > > : 4uN,1,u + uN1,1,u + uN,2,u ¼ h2 fN,1,u , uN + 1,1,u ¼ 0 Finally, if j ¼ N is fixed and i ¼ 0, 1, …, N, then: ui + 1,N  H 4 ⊙ ui, N ui1,N ui,N1 ¼ h2 fi, N , ui, N + 1 ¼ 0, i, j ¼ 0,1, …,N: In level-wise form:

Numerical solutions of uncertain partial differential equations 355 ui + 1,N, l  4ui,N,l + ui1, N, l ¼ h2 fi,1, l , ui, N + 1,l ¼ 0, i, j ¼ 0, 1,…, N: ui + 1,N, u  4ui, N,u + ui1, N, u ¼ h2 fi,1,u , ui, N + 1,u ¼ 0, i, j ¼ 0, 1, …,N: For i ¼ 0, 1, …, N, we have: 8  4u0,N, l ¼ h2 f0,1,l , u0, N + 1,l ¼ 0 u > > < 1,N, l u2,N, l  4u1,N, l + u0,N, l + u1, N, l ¼ h2 f1, N, l ⋮ > > : 4uN, N,l + uN1, N, l ¼ h2 fN, N, l ,uN + 1,N, l ¼ uN, N + 1,l ¼ 0 8 u  4u0, N, u ¼ h2 f0,1,u ,u0, N + 1,u ¼ 0 > > < 1, N, u u2, N, u  4u1, N, u + u0, N, u + u1, N, u ¼ h2 f1, N, u ⋮ > > : 4uN, N, u + uN1, N, u ¼ h2 fN, N, u , uN + 1,N, u ¼ uN, N + 1,u ¼ 0 In the compact form: 

A1 O O A1 0



A BO A1 ¼ @ ⋮ O

Ul Uu



O O O O ⋮ ⋮ O …

 ¼h O O ⋮ O

2

Fl Fu

… … ⋮ O



1 O OC ⋮A A

where I is a specific and O is a zero matrix and: 0 4 1 0 0 0 … 0 B 1 4 1 0 0 … 0 B A ¼ B 0 1 4 1 0 … 0 @ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 0 … 0 1 4 0 B B B B B B B B B B Ul ¼ B B B B B B B B B @

u0,0,l u1,0,l ⋮ uN,0,l u0,1,l u1,1,l ⋮ uN,1,l ⋮ u0, N,l u1, N,l ⋮ uN,N, l

1

0

C B B C B C C B B C B C B C B C B C B C C , Uu ¼ B C B C B C B C B C B C B C B B C A @

u0,0,u u1,0,u ⋮ uN,0,u u0,1,u u1,1,u ⋮ uN,1,u ⋮ u0,N, u u1,N, u ⋮ uN, N,u

1

0

C B B C B C C B B C B C B C B C B C B C C, F l ¼ B C B C B C B C B C B C B C B B C A @

f0,0,l f1,0,l ⋮ fN,0, l f0,1,l f1,1,l ⋮ fN,1, l ⋮ f0,N, l f1,N, l ⋮ fN, N, l

1 C C C A 1

0

C B B C B C C B B C B C B C B C B C B C C, Fu ¼ B C B C B C B C B C B C B C B B C A @

f0,0,u f1,0,u ⋮ fN,0, u f0,1,u f1,1,u ⋮ fN,1, u ⋮ f0,N, u f1,N, u ⋮ fN, N, u

1 C C C C C C C C C C C C C C C C C C C A

356

Soft Numerical Computing in Uncertain Dynamic Systems

Cases 1 and 6. The derivatives with respect to x are ∂xigH, xigH, differentiable and with respect to y are ∂yigH, yiigH differentiable: ∂xigH , xigH uji, j ¼ ∂yigH ,yiigH uji, j ¼

 1 ui + 1, j  H 2 ⊙ uij ui1, j 2 h

 1 ui, j + 1  H uij  H ð1Þui, j1 ð1Þuij 2 h

By substituting in:   ∂xigH , xigH uji, j ∂yigH , yiigH uji, j ¼ f xi , yj  1  1 ui + 1, j  H 2 ⊙ uij ui1, j 2 ui, j + 1  H uij  H ð1Þui, j1 ð1Þuij ¼ h2 h   ¼ f xi , yj ui + 1, j  H 3 ⊙ uij ð1Þuij ui1, j ui, j + 1  H ð1Þui, j1 ¼ h2 fij , i, j ¼ 0, 1,…, N: The level-wise form is: ui + 1, j, l  3ui, j, l  ui, j, u + ui1, j,l + ui, j + 1,l + ui, j1, u ¼ h2 fi, j, l ui + 1, j,u  3ui, j, u  ui, j, l + ui1, j, u + ui, j + 1, u + ui, j1, l ¼ h2 fi, j, u By a similar procedure to that for Cases 1 and 5, the compact matrix form can be described as follows:  0

B B I B1 ¼ @ ⋮ O

B1 B2 B2 B1



Ul Uu



 ¼h

2

1 0 O O O … O B O O … OC B , B ¼ ⋮ ⋮ ⋮ ⋮ ⋮A 2 @ O … O I B

Fl Fu



1 I I O O … O O I I O … O C ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ A O O … O O I

where I is a specific and O is a zero matrix and: 0

3 B 1 B B¼B 0 @ ⋮ 0

1 0 0 0 … 0 3 1 0 0 … 0 1 3 1 0 … 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 … 0 1 3

1 C C C A

Numerical solutions of uncertain partial differential equations 357 Cases 1 and 7. The derivatives with respect to x are ∂xigH, xigH differentiable and with respect to y are ∂yiigH, yigH differentiable: ∂xigH , xigH uji, j ¼ ∂yiigH ,yigH uji, j ¼

 1 ui + 1, j  H 2 ⊙ uij ui1, j 2 h

 1 ð1Þuij  H ð1Þui, j + 1  H ui, j ui, j1 2 h

By substituting in:   ∂xigH , xigH uji, j ∂yiigH , yigH uji, j ¼ f xi , yj  1  1 ui + 1, j  H 2 ⊙ uij ui1, j 2 ð1Þuij  H ð1Þui, j + 1  H ui, j ui, j1 ¼ 2 h h   ¼ f xi , yj ui + 1, j  H 3 ⊙ uij ð1Þuij ui1, j  H ð1Þui, j + 1 ui, j1 ¼ h2 fij , i, j ¼ 0,1, …,N: The level-wise form is: ui + 1, j, l  3ui, j, l  ui, j, u + ui1, j, l + ui, j + 1, u + ui, j1, l ¼ h2 fi, j, l ui + 1, j, u  3ui, j, u  ui, j, l + ui1, j, u + ui, j + 1,l + ui, j1, u ¼ h2 fi, j, u By a similar procedure to that for Cases 1 and 5, the compact matrix form can be described as follows: 

0

B O B B1 ¼ @ ⋮ O

B1 B2 B2 B1

I O O … B I O … ⋮ ⋮ ⋮ ⋮ O O … O

with the same matrices and: 0

3 B 1 B B¼B 0 @ ⋮ 0



Ul Uu



 ¼ h2

Fl Fu



1 0 O I O OC B I I , B ¼ ⋮ A 2 @ ⋮ ⋮ B O …

1 0 0 0 … 0 3 1 0 0 … 0 1 3 1 0 … 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 … 0 1 3

1 O … O O O … O OC ⋮ ⋮ ⋮ ⋮ A O 0 I I

1 C C C A

358

Soft Numerical Computing in Uncertain Dynamic Systems

Cases 1 and 8. The derivatives with respect to x are ∂xigH, xigH differentiable and with respect to y are ∂yiigH, yiigH differentiable: ∂xigH , xigH uji, j ¼ ∂yiigH ,yigH uji, j ¼

 1 ui + 1, j  H 2 ⊙ uij ui1, j 2 h

 1 ð2Þuij  H ð1Þui, j + 1  H ð1Þui, j1 2 h

By substituting in:   ∂xigH , xigH uji, j ∂yiigH , yiigH uji, j ¼ f xi , yj  1  1 ui + 1, j  H 2 ⊙ uij ui1, j 2 ð2Þuij  H ð1Þui, j + 1  H ð1Þui, j1 ¼ 2 h h   ¼ f x i , yj ui + 1, j  H 2 ⊙ uij ð2Þuij ui1, j  H ð1Þui, j + 1  H ð1Þui, j1 ¼ h2 fij , i, j ¼ 0, 1,…, N: The level-wise form is: ui + 1, j, l  2ui, j, l  2ui, j, u + ui1, j, l + ui, j + 1, u + ui, j1, u ¼ h2 fi, j, l ui + 1, j, u  2ui, j, u  2ui, j, l + ui1, j, u + ui, j + 1,l + ui, j1, l ¼ h2 fi, j, u By a similar procedure to that for Cases 1 and 5, the compact matrix form can be described as follows:  0

C BO C1 ¼ @ ⋮ O

O O C O ⋮ ⋮ O …

C1 C2 C2 C1

O … ⋮ O

… O ⋮ O



Ul Uu



 ¼ h2

Fl Fu



1 0 O 2I I OC B I 2I ,C ¼ ⋮A 2 @ ⋮ ⋮ B O …

O I ⋮ O

… O ⋮ O

1 O O … O C ⋮ ⋮ A I 2I

where I is a specific and O is a zero matrix and the matrix C is as follows: 0

2 B 1 B C¼B 0 @ ⋮ 0

1 0 0 0 … 0 2 1 0 0 … 0 1 2 1 0 … 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 … 0 1 2

1 C C C A

Numerical solutions of uncertain partial differential equations 359 Cases 2 and 5. The derivatives with respect to x are ∂xigH, xiigH differentiable and with respect to y are ∂yigH, yigH differentiable:  1 ui + 1, j  H uij  H ð1Þui1, j ð1Þuij 2 h  1 ∂yigH , yigH uji, j ¼ 2 ui, j + 1  H 2 ⊙ uij ui, j1 h

∂xigH ,xiigH uji, j ¼

By substituting in:   ∂xigH , xiigH uji, j ∂yigH , yigH uji, j ¼ f xi , yj  1  1 ui + 1, j  H uij  H ð1Þui1, j ð1Þuij 2 ui, j + 1  H 2 ⊙ uij ui, j1 2 h h   ¼ f x i , yj ui + 1, j  H 3 ⊙ uij ð1Þuij  H ð1Þui1, j ui, j + 1 ui, j1 ¼ h2 fij i, j ¼ 0,1, …,N: In level-wise form: ui + 1, j, l  3ui, j, l  ui, j, u + ui1, j, u + ui, j + 1, l + ui, j1, l ¼ h2 fi, j, l ui + 1, j, u  3ui, j, u  ui, j, l + ui1, j, l + ui, j + 1,u + ui, j1, u ¼ h2 fi, j, u In the compact form: 

D1 D2 D2 D1



Ul Uu



 ¼h

2

0

1 0 D I O O … O B I D I … O OC B D1 ¼ @ , D2 ¼ @ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮A O O … O I D

Fl Fu

E O ⋮ O



O O E O ⋮ ⋮ O …

O … ⋮ O

… O ⋮ O

1 O OC ⋮A E

where I is a specific and O is a zero matrix and: 0

3 B 0 B D¼B 0 @ ⋮ 0

1 0 0 0 … 0 3 1 0 0 … 0 0 3 1 0 … 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 … 0 0 3

1

0 1 C C B 1 C, E ¼ @ ⋮ A 0

0 1 ⋮ …

1 0 … 0 0 … 0 C ⋮ ⋮ ⋮A 0 1 1

360

Soft Numerical Computing in Uncertain Dynamic Systems

Cases 2 and 6. The derivatives with respect to x are ∂xigH, xiigH differentiable and with respect to y are ∂yigH, yiigH differentiable:  1 ui + 1, j  H uij  H ð1Þui1, j ð1Þuij 2 h  1 ∂yigH ,yiigH uji, j ¼ 2 ui, j + 1  H uij  H ð1Þui, j1 ð1Þuij h ∂xigH ,xiigH uji, j ¼

By substituting in:   ∂xigH , xiigH uji, j ∂yigH , yiigH uji, j ¼ f xi , yj  1 ui + 1, j  H ui, j  H ð1Þui1, j ð1Þui, j 2 h    1 2 ui, j + 1  H ui, j  H ð1Þui, j1 ð1Þui, j ¼ f xi , yj h ui + 1, j  H 2 ⊙ ui, j ð2Þui, j  H ð1Þui1, j ui, j + 1  H ð1Þui, j1 ¼ h2 fij , i, j ¼ 0, 1,…, N: The level-wise form is: ui + 1, j, l  2ui, j, l  2ui, j, u + ui1, j, u + ui, j + 1, l + ui, j1, u ¼ h2 fi, j, l ui + 1, j, u  2ui, j, u  2ui, j, l + ui1, j, l + ui, j + 1,u + ui, j1, l ¼ h2 fi, j, u By a similar procedure to that for Cases 1 and 5, the compact matrix form can be described as follows:  0

F BO F1 ¼ @ ⋮ O

F1 F2 F2 F1

I O O … F I O … ⋮ ⋮ ⋮ ⋮ O O … O



Ul Uu



 ¼ h2

Fl Fu



0 T 1 O F O B I FT OC , F ¼B ⋮ A 2 @ ⋮ ⋮ F O …

where I is a specific and O is a zero matrix and: 0 2 1 0 0 0 … 0 B 0 2 1 0 0 … 0 B F ¼ B 0 0 2 1 0 … 0 @ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 0 … 0 0 2

1 O … O O O … O OC C ⋮ ⋮ ⋮ ⋮ A O 0 I FT

1 C C C A

Numerical solutions of uncertain partial differential equations 361 Cases 2 and 7. The derivatives with respect to x is ∂xigH, xiigH differentiable and with respect to y is ∂yiigH, yigH differentiable.  1 ui + 1, j  H uij  H ð1Þui1, j ð1Þuij 2 h  1 ∂yiigH , yigH uji, j ¼ 2 ði1Þuij  H ð1Þui, j + 1  H ui, j ui, j1 h ∂xigH ,xiigH uji, j ¼

By substituting in:   ∂xigH , xiigH uji, j ∂yigH , yigH uji, j ¼ f xi , yj  1 ui + 1, j  H uij  H ð1Þui1, j ð1Þuij 2 h    1 2 ð1Þuij  H ð1Þui, j + 1  H ui, j ui, j1 ¼ f xi , yj h ui + 1, j  H 2 ⊙ uij ð2Þuij  H ð1Þui1, j  H ð1Þui, j + 1 ui, j1 ¼ h2 fij , i, j ¼ 0,1, …,N: The level-wise form is: ui + 1, j, l  2ui, j, l  2ui, j, u + ui1, j, l + ui, j + 1, u + ui, j1, l ¼ h2 fi, j, l ui + 1, j, u  2ui, j, u  2ui, j, l + ui1, j, u + ui, j + 1,l + ui, j1, u ¼ h2 fi, j, u By a similar procedure to that for Cases 1 and 5, the compact matrix form can be described as follows: 

0

F B I F1 ¼ @ ⋮ O

F1 F2 F2 F1



Ul Uu



 ¼ h2

Fl Fu



0 T 1 1 O O … O O F I O O … O B O FT I O … O C F O … O OC C , F ¼B ⋮ ⋮ ⋮ ⋮ ⋮ A 2 @ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ A … O 0 I F O O O … O FT

where I is a specific and O is a zero matrix and: 0

2 B 0 B F¼B 0 @ ⋮ 0

1 0 0 0 … 0 2 1 0 0 … 0 0 2 1 0 … 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 … 0 0 2

1 C C C A

362

Soft Numerical Computing in Uncertain Dynamic Systems

Cases 2 and 8. The derivatives with respect to x are ∂xigH, xiigH differentiable and with respect to y are ∂yiigH, yiigH differentiable:  1 ui + 1, j  H uij  H ð1Þui1, j ð1Þuij 2 h  1 ∂yiigH ,yigH uji, j ¼ 2 ð2Þuij  H ð1Þui, j + 1  H ð1Þui, j1 h

∂xigH ,xiigH uji, j ¼

By substituting in:   ∂xigH , xiigH uji, j ∂yigH , yigH uji, j ¼ f xi , yj  1 ui + 1, j  H uij  H ð1Þui1, j ð1Þuij 2 h    1 2 ð2Þuij  H ð1Þui, j + 1  H ð1Þui, j1 ¼ f xi , yj h ui + 1, j  H uij ð3Þuij  H ð1Þui1, j  H ð1Þui, j + 1  H ð1Þui, j1 ¼ h2 fij , i, j ¼ 0, 1,…, N: The level-wise form is: ui + 1, j, l  ui, j, l  3ui, j, u + ui1, j, u + ui, j + 1,u + ui, j1, u ¼ h2 fi, j, l ui + 1, j, u  ui, j, u  3ui, j, l + ui1, j, l + ui, j + 1, l + ui, j1, l ¼ h2 fi, j, u By a similar procedure to that for Cases 1 and 5, the compact matrix form can be described as follows:  0

G BO G1 ¼ @ ⋮ O

O O G O ⋮ ⋮ O …

G1 G2 G2 G1

O … ⋮ O

… O ⋮ O



Ul Uu



 ¼h

2

Fl Fu



0 T 1 O D I B I DT OC , G2 ¼ B @ ⋮ ⋮ ⋮A G O …

O I ⋮ O

… O ⋮ O

1 O O … O C C ⋮ ⋮ A I DT

where I is a specific and O is a zero matrix and the matrix G is as follows: 0

1 B 0 B G¼B 0 @ ⋮ 0

1 0 0 0 … 0 1 1 0 0 … 0 0 1 1 0 … 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 … 0 0 1

1 C C C, A

0

3 B 0 B D¼B 0 @ ⋮ 0

1 0 0 0 … 0 3 1 0 0 … 0 0 3 1 0 … 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 … 0 0 3

1 C C C A

Numerical solutions of uncertain partial differential equations 363 Cases 3 and 5. The derivatives with respect to x are ∂xiigH, xigH differentiable and with respect to y are ∂yigH, yigH differentiable:  1 ð1Þui, j  H ð1Þui + 1, j  H ui, j ui1, j 2 h  1 ∂yigH , yigH uji, j ¼ 2 ui, j + 1  H 2 ⊙ ui, j ui, j1 h

∂xiigH ,xigH uji, j ¼

By substituting in:   ∂xiigH , xigH uji, j ∂yigH , yigH uji, j ¼ f xi , yj  1  1 ð1Þui, j  H ð1Þui + 1, j  H ui, j ui1, j 2 ui, j + 1  H 2 ⊙ ui, j ui, j1 h2 h   ¼ f xi , yj  H ð1Þui + 1, j  H 3 ⊙ uij ð1Þui, j ui1, j ui, j + 1 ui, j1 ¼ h2 fij i, j ¼ 0,1, …,N: In level-wise form: ui + 1, j, u  3ui, j, l  ui, j, u + ui1, j, l + ui, j + 1, l + ui, j1, l ¼ h2 fi, j, l ui + 1, j, l  3ui, j, u  ui, j, l + ui1, j, u + ui, j + 1,u + ui, j1, u ¼ h2 fi, j, u In the compact form:  0

DT I B I DT G1 ¼ B @ ⋮ ⋮ O …

O I ⋮ O

G1 G2 G2 G1 … O ⋮ O



Ul Uu 1



 ¼ h2

0 O O C … O C B , G2 ¼ @ ⋮ ⋮ A I DT

Fl Fu G O ⋮ O



O O G O ⋮ ⋮ O …

O … ⋮ O

… O ⋮ O

1 O OC ⋮A G

where I is a specific and O is a zero matrix and the matrix G is as follows: 0

1 B 0 B G¼B 0 @ ⋮ 0

1 0 0 0 … 0 1 1 0 0 … 0 0 1 1 0 … 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 … 0 0 1

1 C C C, A

0

3 B 0 B D¼B 0 @ ⋮ 0

1 0 0 0 … 0 3 1 0 0 … 0 0 3 1 0 … 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 … 0 0 3

1 C C C A

364

Soft Numerical Computing in Uncertain Dynamic Systems

Cases 3 and 6. The derivatives with respect to x are ∂xiigH, xigH differentiable and with respect to y are ∂yigH, yiigH differentiable:  1 ð1Þui, j  H ð1Þui + 1, j  H ui, j ui1, j 2 h  1 ∂yigH ,yiigH uji, j ¼ 2 ui, j + 1  H ui, j  H ð1Þui, j1 ð1Þui, j h ∂xiigH ,xigH uji, j ¼

By substituting in:   ∂xiigH ,xigH uji, j ∂yigH , yiigH uji, j ¼ f xi , yj  1 ð1Þui, j  H ð1Þui + 1, j  H ui, j ui1, j 2 h    1 2 ui, j + 1  H ui, j  H ð1Þui, j1 ð1Þui, j ¼ f xi , yj h  H ð1Þui + 1, j  H 2 ⊙ ui, j ð2Þui, j ui1, j ui, j + 1  H ð1Þui, j1 ¼ h2 fij , i, j ¼ 0, 1,…, N: The level-wise form is: ui + 1, j, u  2ui, j, l  2ui, j, u + ui1, j, l + ui, j + 1, l + ui, j1, u ¼ h2 fi, j, l ui + 1, j, l  2ui, j, u  2ui, j, l + ui1, j, u + ui, j + 1,u + ui, j1, l ¼ h2 fi, j, u By a similar procedure to that for Cases 1 and 5, the compact matrix form can be described as follows:  0

2I I B I 2I C1 ¼ @ ⋮ ⋮ O …

O I ⋮ O

C1 C2 C2 C1 … O ⋮ O



Ul Uu



 ¼h

2

Fl Fu

1 0 O O … O C B , C ¼ ⋮ ⋮ A 2 @ I 2I



C O ⋮ O

O O C O ⋮ ⋮ O …

O … ⋮ O

… O ⋮ O

1 O OC ⋮A B

where I is a specific and O is a zero matrix and the matrix C is as follows: 0 1 2 1 0 0 0 … 0 B 1 2 1 0 0 … 0 C B C C ¼ B 0 1 2 1 0 … 0 C @ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ A 0 0 0 … 0 1 2

Numerical solutions of uncertain partial differential equations 365 Cases 3 and 7. The derivatives with respect to x are ∂xiigH, xigH differentiable and with respect to y are ∂yiigH, yigH differentiable:  1 ð1Þui, j  H ð1Þui + 1, j  H ui, j ui1, j 2 h  1 ∂yiigH ,yigH uji, j ¼ 2 ð1Þui, j  H ð1Þui, j + 1  H ui, j ui, j1 h ∂xiigH ,xigH uji, j ¼

By substituting in:   ∂xigH , xiigH uji, j ∂yigH , yigH uji, j ¼ f xi , yj  1 ð1Þui, j  H ð1Þui + 1, j  H ui, j ui1, j 2 h    1 2 ð1Þui, j  H ð1Þui, j + 1  H ui, j ui, j1 ¼ f xi , yj h  H ð1Þui + 1, j  H 2 ⊙ uij ð2Þuij ui1, j  H ð1Þui, j + 1 ui, j1 ¼ h2 fij , i, j ¼ 0,1, …,N: The level-wise form is: ui + 1, j, u  2ui, j, l  2ui, j, u + ui1, j,l + ui, j + 1,u + ui, j1, l ¼ h2 fi, j, l ui + 1, j,l  2ui, j, u  2ui, j, l + ui1, j, u + ui, j + 1, l + ui, j1, u ¼ h2 fi, j, u By a similar procedure to that for Cases 1 and 5, the compact matrix form can be described as follows:  0

I O B I I B1 ¼ @ ⋮ ⋮ O …

B1 B2 B2 B1



Ul Uu



 ¼h

2

1 0 O … O O O … O OC B , B ¼ ⋮ ⋮ ⋮ ⋮ A 2 @ O 0 I I

with the same matrices and: 0

3 B 1 B B¼B 0 @ ⋮ 0

Fl Fu

B O ⋮ O



I O O … B I O … ⋮ ⋮ ⋮ ⋮ O O … O

1 0 0 0 … 0 3 1 0 0 … 0 1 3 1 0 … 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 … 0 1 3

1 C C C A

1 O OC ⋮ A B

366

Soft Numerical Computing in Uncertain Dynamic Systems

Cases 3 and 8. The derivatives with respect to x are ∂xiigH, xigH differentiable and with respect to y are ∂yiigH, yiigH differentiable:  1 ð1Þui, j  H ð1Þui + 1, j  H ui, j ui1, j 2 h  1 ∂yiigH ,yigH uji, j ¼ 2 ð2Þui, j  H ð1Þui, j + 1  H ð1Þui, j1 h

∂xiigH ,xigH uji, j ¼

By substituting in:   ∂xigH , xiigH uji, j ∂yigH , yigH uji, j ¼ f xi , yj  1 ð1Þui, j  H ð1Þui + 1, j  H ui, j ui1, j 2 h    1 2 ð2Þui, j  H ð1Þui, j + 1  H ð1Þui, j1 ¼ f xi , yj h  H ð1Þui + 1, j  H uij ð3Þuij ui1, j  H ð1Þui, j + 1  H ð1Þui, j1 ¼ h2 fij , i, j ¼ 0, 1,…, N: The level-wise form is: ui + 1, j, u  ui, j, l  3ui, j, u + ui1, j,l + ui, j + 1,u + ui, j1, u ¼ h2 fi, j, l ui + 1, j, l  ui, j, u  3ui, j, l + ui1, j, u + ui, j + 1, l + ui, j1, l ¼ h2 fi, j, u By a similar procedure to that for Cases 1 and 5, the compact matrix form can be described as follows:      G1 G2 Ul 2 Fl ¼h G2 G1 Uu Fu 0 1 0 1 G O O O … O DT I O … O O B I DT I O … O C B O G O … O OC C G1 ¼ @ , G2 ¼ B @ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮A ⋮ ⋮ ⋮ A O O … O O G O … O O I DT where I is a specific and O is a zero matrix and the matrix G is as follows: 0

1 B 0 B G¼B 0 @ ⋮ 0

1 0 0 0 … 0 1 1 0 0 … 0 0 1 1 0 … 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 … 0 0 1

1 C C C, A

0

3 B 0 B D¼B 0 @ ⋮ 0

1 0 0 0 … 0 3 1 0 0 … 0 0 3 1 0 … 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 … 0 0 3

1 C C C A

Numerical solutions of uncertain partial differential equations 367 Note. The rest of the cases have a similar structure and the only difference is in the permutation of submatrices. Cases 4 and 8 are the symmetric cases of Cases 1 and 5:      O A1 Ul Fl ¼ h2 A1 O Uu Fu 0

A BO A1 ¼ @ ⋮ O

O O O O ⋮ ⋮ O …

where I is a specific matrix and O is a 0 4 1 B 1 4 B A¼B 0 1 @ ⋮ ⋮ 0 0

O O ⋮ O

… … ⋮ O

1 O OC ⋮A A

zero matrix and: 0 0 1 0 4 1 ⋮ ⋮ 0 …

0 … 0 0 … 0 0 … 0 ⋮ ⋮ ⋮ 0 1 4

1 C C C A

As we know, the finite difference method needs error analysis or convergence investigation. In the following, we shall discuss the error analysis of the method (see Abdi & Allahviranloo, 2019).

7.4.1.1 Error analysis Suppose that Ui, j is the exact value and ui, j  Ui, j is the approximate value at the point (xi, yj), which is obtained by the difference method. Consider the error ei, j ¼ ui, j HUi, j at the same point and ui, j ¼ Ui, j ei, j: ∂2xx, gH uðx, yÞji, j ∂2yy, gH uðx, yÞji, j ¼ fi, j Since u(x, y) ¼ U(x, y) e(x, y), then by substituting we have: ∂2xx, gH ðU ðx, yÞ eðx, yÞÞ ∂2yy, gH ðU ðx, yÞ eðx, yÞÞ ¼ ¼ ∂2xx, gH U ðx, yÞ ∂2yy, gH Uðx, yÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} f ðx, yÞ

∂2xx, gH eðx, yÞ ∂2yy, gH eðx, yÞ ¼ f ðx, yÞ 7ðx, yÞ

We assume that: ∂2xx, gH eðx, yÞ ∂2yy, gH eðx, yÞ ¼ h2 7ðx, yÞ where 7ðx, yÞ is called a reminder function and it is proportional to the fourth order derivative of U(x, y). Indeed, our problem is now:

368

Soft Numerical Computing in Uncertain Dynamic Systems ∂2xx, gH eðx, yÞ ∂2yy, gH eðx, yÞ ¼ h2 7ðx, yÞ

It should be noted that e(x, y) has the same properties of the solution function u(x, t). Here we should show that the error function intends zero when the step size h intends zero. To this end, we show that the norm of error function at discretized points (as a vector) intends zero. In fact, we have: ∂2xx, gH eðx, yÞji, j ∂2yy, gH eðx, yÞji, j ¼ h2 7i, j   h2 ∂ 4 ∂4 7i, j ∝ U ðx, yÞ 4 Uðx, yÞ 12 ∂x4 ∂y i, j and considering: (   4  )  ∂4  ∂  M ¼ max  4 U ðx, yÞ ,  4 U ðx, yÞ ∂x i, j ∂y i, j then: 7i, j ∝

Mh2 12

As with the previously mentioned cases, eight cases happen here too. In each case, we have a system of linear equations, which are denoted in the following form, generally:      M1 M2 El 2 Hl ¼h M2 M1 Eu Hu where M1 and M2 are three-diagonal sparse and block matrices. If the matrix     M1 M2 El is nonsingular, then the vector solution can be found easily. M2 M1 Eu On the other hand: 

M1 M2 det M2 M1

 ¼ det ðM1  M2 Þ det ðM1 + M2 Þ

and: 

M1 M2 det M2 M1





El 6 0⟺ ¼ Eu



 ¼h

2

M1 M2 M2 M1

1 

Hl Hu



To show the convergence, it is enough that we show it under the infinity norm of the vectors and matrices:

Numerical solutions of uncertain partial differential equations 369     1    1       Mh2  El   Hl  M M M M     1 2 1 2 2 2   h    ¼h     Eu   Hu  M M M M     12 2 1 2 1 ∞ ∞ ∞

∞

 1     M1 M2  So, if   ¼ M, then we have:  M2 M1  ∞   4  El    M Mh 0 > u1,0, l  u0,0,l  3u0,0,u + u0,1,u ¼ y0 sin x0 > > 4 > > > > 1 > > > u1,0, u  u0,0,u  3u0,0, l + u0,1,l ¼ y0 sin x0 > > 4 > > > > 1 > > < u2,0, l  u1,0,l  3u1,0, u + u0,0,u + u1,1,u ¼ y0 sin x1 4 i ¼ 0,1, 2, j ¼ 0 > 1 > > u2,0,u  u1,0,u  3u1,0,l + u0,0, l + u1,1,l ¼ y0 sin x1 > > 4 > > > > 1 > > > u3,0, l  u2,0,l  3u2,0, u + u1,0,u + u2,1,u ¼ y0 sin x2 > > 4 > > > > 1 > : u3,0,u  u2,0,u  3u2,0,l + u1,0, l + u2,1,l ¼ y0 sin x2 4

370

Soft Numerical Computing in Uncertain Dynamic Systems

8 1 > > u1,1,l  u0, 1,l  3u0,1,u + u0,2,u + u0,0,u ¼ y1 sin x0 > > 4 > > > > 1 > > > u1,1,u  u0,1,u  3u0, 1,l + u0,2,l + u0,0,l ¼ y1 sin x0 > > 4 > > > > 1 > > < u2,1,l  u1,1,l  3u1,1,u + u0,1,u + u1,2,u + u1,0, u ¼ y1 sin x1 4 i ¼ 0, 1,2, j ¼ 1 > 1 > > u2,1,u  u1,1,u  3u1,1,l + u0,1,l + u1,2, l + u1,0,l ¼ y1 sin x1 > > 4 > > > > 1 > > > u3,1,l  u2,1,l  3u2,1,u + u1,1,u + u2,2,u + u2,0, u ¼ y1 sin x2 > > 4 > > > > 1 > : u3,1,u  u2,1,u  3u2,1,l + u1,1,l + u2,2,l + u2, 0,l ¼ y1 sinx2 4 8 1 > > u1,2, l  u0,2,l  3u0,2, u + u0,1,u ¼ y2 sin x0 > > 4 > > > > 1 > > > u  u0,2,u  3u0,2, l + u0,1,l ¼ y2 sin x0 > > 1,2, u 4 > > > > 1 > > < u2,2,l  u1,2,l  3u1,2, u + u0,2,u + u1,1, u ¼ y2 sin x1 4 i ¼ 0,1, 2, j ¼ 2 > 1 > > u2,2,u  u1,2,u  3u1,2,l + u0,2,l + u1,1,l ¼ y2 sin x1 > > 4 > > > > 1 > > > u3,2, l  u2,2,l  3u2,2,u + u1,2,u + u2,1,u ¼ y2 sin x2 > > 4 > > > > 1 > : u3,2,u  u2,2,u  3u2,2,l + u1,2,l + u2,1,l ¼ y2 sin x2 4 where u1, ¼ u , 3 ¼ 0, and u0, j, ui, 0, i, j ¼ 0, 1, 2 are known. After solving the linear systems, the results are obtained as the following triangular fuzzy numbers at several points. For instance, assume that:    37r 35r 5r , , u1,0 ¼ , 128 128 64    25r 23r 7r , , u1,1 ¼ , u0,1 ¼ 64 64 64    37r 35r 5r , , u1,2 ¼ , u0,2 ¼ 128 128 64

u0,0 ¼

  3r 61r , u2,0 ¼ , 64 128   r 41r , u2,1 ¼ , 64 64   3r 61r , u2,2 ¼ , 64 128

 59r 128  39r 64  59r 128

Now, in Cases 3 and 8, the derivatives with respect to x are ∂xiigH, xigH differentiable and with respect to y are ∂yiigH, yiigH differentiable:       61r 59r 5r 3r 37r 35r u0,0 ¼ , , u1,0 ¼ , , u2,0 ¼ , 128 128 64 64 128 128

Numerical solutions of uncertain partial differential equations 371    41r 39r 7r , , u1,1 ¼ , 64 64 64    61r 59r 5r , , u , u0,2 ¼ 1,2 ¼ 128 128 64 u0,1 ¼

   r 25r 23r , u2,1 ¼ , 64 64 64    3r 37r 35r , u2,2 ¼ , 64 128 128

Now, the derivatives with respect to x are ∂xiigH, xiigH differentiable and with respect to y are ∂yiigH, yigH differentiable:       61r 59r 41r 39r 61r 59r u0,0 ¼ , , u1,0 ¼ , , u2,0 ¼ , 128 128 64 64 128 128       5r 3r 7r r 5r 3r , , u1,1 ¼ , , u2,1 ¼ , u0,1 ¼ 64 64 64 64 64 64       37r 35r 25r 23r 37r 35r , , u1,2 ¼ , , u2, 2 ¼ , u0,2 ¼ 128 128 64 64 128 128 Now, the derivatives with respect to x are ∂xiigH, xiigH differentiable and with respect to y are ∂yiigH, yiigH differentiable:     11r 11 11 11r 7r 7 7 7r u0,0 ¼  ,  , u1,0 ¼  ,  , 64 64 64 64 32 32 32 32   11r 11 11 11r  ,  u2,0 ¼ 64 64 64 64     7r 7 7 7r 9r 9 9 9r  ,  , u1, 0 ¼  ,  , u0,0 ¼ 32 32 32 32 32 32 32 32   7r 7 7 7r u2, 0 ¼  ,  32 32 32 32     11r 11 11 11r 7r 7 7 7r  ,  , u1, 0 ¼  ,  , u0, 0 ¼ 64 64 64 64 32 32 32 32   11r 11 11 11r u2,0 ¼  ,  64 64 64 64

References Abdi, M., Allahviranloo, T., 2019. Fuzzy finite difference method for solving Poisson’s equation. J. Intell. Fuzzy Syst. 37 (3), 1–16. Allahviranloo, T., 2020. Uncertain information and linear systems. In: Studies in Systems, Decision and Control. vol. 254. Springer. Allahviranloo, T., Gouyandeh, Z., Armand, A., Hasanoglu, A., 2015. On fuzzy solutions for heat equation based on generalized Hukuhara differentiability. Fuzzy Set. Syst. 265, 1–23.

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Armand, A., Allahviranloo, T., Abbasbandy, S., Gouyandeh, Z., 2019. The fuzzy generalized Taylor’s expansion with application in fractional differential equations. Iran. J. Fuzzy Syst. 16 (2), 57–72. Bede, B., Gal, S.G., 2005a. Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Set. Syst. 151, 581–599. Bede, B., Gal, S.G., 2005b. Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Set. Syst. 151, 581–599. Gouyandeha, Z., Allahviranloob, T., Abbasbandyb, S., Armand, A., 2017. A fuzzy solution of heat equation under generalized Hukuhara differentiability by fuzzy Fourier transform. Fuzzy Set. Syst. 309, 81–97. Stefanini, L., Bede, B., 2009. Generalized Hukuhara differentiability of interval-valued functions and interval differential equations. Nonlinear Anal. 71, 1311–1328.

Index

Note: Page numbers followed by f indicate figures, and t indicate tables. A Absolutely convergence, 115–117, 116–117f Addition, 48 Ambiguity, 3 Analytical solution, of fuzzy heat equation, 334–351 B Bimathematic two-compartment model homogeneous fuzzy linear differential systems, 190–191, 190f nonhomogeneous fuzzy linear differential systems, 193–195, 193–194f Biomathematic three-compartment model, 191–192, 191f C Canonical Liu process, uncertain process, 128–129 Caputo gH-differentiability, 293–296 Caputo-Katugampola derivative, 301–318 Caputo-Katugampola gH-fractional derivative, 296–301 Cauchy’s fuzzy mean value theorem, 91–92 Chain rule, 130–131 Characteristic theorem, 168 fuzzy differential equations, 138–139 Z-differential equations, 200–201 Cognitive models, 7 Combination property, of fuzzy RL derivative, 290–291 Compact matrix, 356–358, 360–362, 364–366 Constraint, 52 Convergence, 177–178, 182 theorem, 214–217, 231–236, 262 Convolution, 56 D Derivative theorem, 118 Difference, 48–49 of fuzzy numbers, 30–32, 31f

of two Z-numbers, 56–57 Differentiability, 153–154, 156, 182, 193, 347, 349 Differential operator, 80–113, 140 Dilatation transformation, 334–335 Division, of two Z-numbers, 57–58 Duffing’s differential equation, 178 Dynamic system modeling, 4 E Economics, 204–205 Electrical circuit, 283, 285f, 286t Error analysis, 248–254, 367–371 Error propagation, 220 Euler’s method, 9 Existence and uniqueness, 310–315 fuzzy differential equations, 142–145 in fuzzy fractional differential equation, 309 fuzzy heat equation, 331–334 in real fractional differential equation, 309–310 Existence of solution, fuzzy differential equations, 147–151 Existence theorem, 124 Expected value concept, 3, 68–70 F First order differential equation, 135, 135f First order Taylor expansion, 130 First translation theorem, 117 Fourier transform of gH-derivative, 125–126, 344–351 operator, 122–126 Fractional derivative operator, 304–306 Fractional mean value theorem, 319–320 Fractional operators, 10 Fuzzy calculated operations, 183 difference, 184 production, 183–184 summation, 183 Fuzzy Caputo fractional derivative, 293 Caputo gH-differentiability, 293–296

374

Index

Fuzzy Caputo fractional derivative (Continued) Caputo-Katugampola gH-fractional derivative, 296–301 Fuzzy comparison test, 182–183 Fuzzy differential equations, 8, 134–197 characteristic theorem, 138–139 existence and uniqueness, 142–145 existence of solution, 147–151 laplace transform, 162–166 Legendre differential equation, 179–188 fuzzy calculated operations, 183–185 fuzzy power series method, 185–188 power series with fuzzy coefficients, 181–182 properties, 182–183 length function, 151, 156–162 definition, 151 differentiability and length, 153–154 fuzzy functions, nonlinear property of, 152, 154–155 integral equation, derivative of, 155–156 nonlinear property, 151–152 remark, 153 linear systems of, 188–197 bimathematic two-compartment model, 190–191, 190f biomathematic three-compartment model, 191–192, 191f fuzzy forced harmonic oscillator problem, 195–197 homogeneous fuzzy linear differential systems, 189–192 nonhomogeneous fuzzy linear differential systems, 192–195 second order fuzzy differential equations to system of first order equations, reduction of, 195–197 second order, 166–173 variational iteration method, 173–179 existence and convergence, 177–178 variation of constants, 145–147 Fuzzy Euler method, 208, 210–226 analysis of, 212–226, 229–241 for fuzzy hybrid differential equations, 242–246 for fuzzy impulsive differential equations, 246–254 instability of, 219f Fuzzy explicit method definition of, 255 three steps method, 256–259 Fuzzy finite difference method, 329 for solving fuzzy Poisson’s equation, 351–371

Fuzzy forced harmonic oscillator problem, 195–197 Fuzzy Fourier transform, 329, 343–344 definition of, 122 examples, 122–123 Fuzzy fractional differential equations, 301–318, 325–326 definition, 302–308 existence and uniqueness, 310–315 in real fractional differential equation, 309–310 of solution, 309 Mittag-Leffler function, properties of, 316–318 Fuzzy fractional Euler method, 321–327 fuzzy fractional differential equation, 321, 325–326 generalized hypergeometric function, 325–326 level-wise form of, 325 numerical solutions, 324–325, 324–325t, 328t Fuzzy function, 9 Fuzzy generalized power series method, 179 Fuzzy generalized Taylor’s expansion, 318–321 Fuzzy heat equation, 331–334 analytical solution of, 334–351 fuzzy fourier transform, 344 fuzzy inverse fourier transform, 344 GH-derivative, fourier transform of, 344–351 existence, 333–334 fundamental solution of, 343 fuzzy maximum principle, 332–333 fuzzy solutions, 341f, 342 Fuzzy hybrid differential equations, 242–246 Fuzzy implicit method definition of, 255–256 two steps method, 259–260 Fuzzy improper integral, 119–122 integral and derivative theorem, 121–122 interchanging integrals theorem, 120 uniform convergence, 120 Fuzzy impulsive differential equations, 246–254 Fuzzy initial value problem, 136, 146, 164, 265, 304–305, 310 Fuzzy integrating factor, 330–331 method, 336–338 Fuzzy inverse Fourier transform, 344 definition of, 123–124 Fuzzy k-step explicit method, 255 Fuzzy Laplace transform operator, 113–119 Fuzzy mean value theorem, 92 Fuzzy measure, 15 Fuzzy modified Euler method, 226–241 Fuzzy nth-order differential equations

Index numerical solution of, 267–286, 272–274t, 278–280t, 284t Fuzzy numbers absolute value of, 38 definition, 20 general form of, 25f level-wise form of, 20–23, 21–22f nonlinear, 24–25, 25f operations, 28–44 in parametric form, 23–24 and properties, 20–45 properties of addition and scalar product on, 45 singleton, 23 trapezoidal, 26 in triangular, 26–28, 28f Fuzzy number valued function, 90 definition of, 75 limit of, 74–78 Fuzzy Poisson’s equation, 329 fuzzy finite difference method for, 351–371 error analysis, 367–371 uniqueness, 351–371 Fuzzy predictor and corrector methods, 255–266 and three steps methods, 260–266, 266f Fuzzy Riemann integral operator, 78–80 properties of, 80 Fuzzy Riemann-Liouville derivative, 289–292 combination property, 290–291 integral operators, level-wise form of, 291 operators, 292 RL fractional integral operator, 291 theorem, 292 Fuzzy sequence, 182 Fuzzy set, 13 Fuzzy set valued function definition of, 75 Fuzzy Taylor method, 207–208, 289 G Gaussian membership function, 16, 16f Generalized difference, level-wise form of, 36 Generalized division, 41–44 examples, 43 properties of, 43 Generalized Hukuhara difference, 35–41 in level-wise form, 62–64 properties of, 64 Generalized hypergeometric function, 325–326 gH-derivative, Fourier transform of, 344–351 gH-difference, properties of, 36–37

375

gH-differentiability, 153, 198 Cauchy’s fuzzy mean value theorem, 91–92 composition of, 88–89 continuous fuzzy number valued function, 90 definition of, 81–113 difference in, 86–87 example, 81–83 Fuzzy mean value theorem, 92 high order differentiability, 94 integral of, 92–93 in level-wise form, 83 minimum and maximum, 89–90 part-by-part integration, 97 production in, 87–88 summation in, 85–86 switching points of, 83–85, 84–86f Taylor expansion, 97–106 gH-partial differentiability, 107–108 example, 107–108 higher order of, 110–111 integral relation in, 111 level-wise form of, 108 multivariate fuzzy chain rule in, 111–113 switching point in, 109, 110f Global errors, 239–240t Global truncation error, and convergence, 213–214, 231 H Hausdorff distance, 73–74, 261, 300 properties of, 144 H-difference, 147, 320–321 High order derivation theorem, 118–119 High order differentiability, 94 High order differential equation, 135, 136f Homogeneous fuzzy linear differential systems, 189–192 Hukuhara difference, 32–35, 33f in level-wise form, 61–62 Hukuhara differentiation, 329 I i-differentiability, 164–165 ii-differentiability, 165–166 Integral equation, derivative of, 155–156 Integral operator, Riemann-Liouville, 299 Integrating factor method, 157–158 Integration by parts method, 131 Interpolation problem, 255 Inverse Laplace, 164, 169–170 transform, 116 Iterative method, 228

376

Index

L Lagrange multiplier, 174 Language computations, 51 Laplace transform, 118–119, 162–166, 169, 171–173 Legendre differential equation, 179–188 Length function fuzzy differential equations, 151, 156–162 definition, 151 differentiability and length, 153–154 fuzzy functions, nonlinear property of, 152, 154–155 integral equation, derivative of, 155–156 nonlinear property, 151–152 remark, 153 Level-wise form, 136–137, 140, 153–154, 156, 165–166, 179, 199, 357–365, 369–370 Limit of difference of functions theorem, 76–77 Limit of multiplication theorem, 77–78 Limit of summation of functions theorem, 75–76 Linearity property theorem, 125 Linear systems, of fuzzy differential equations, 188–197 Lipschitz property, 139, 201 Liu integral, of uncertain process, 129 Local truncation error, and consistency, 212–213, 229–231 M Malthusian model, 8 Mathematical models, 5 Measurable sets, 14 Measurable space, 14–15 Measures, 14–15 Medicine, 203–204 Membership degree, 15 Membership function, 14, 16, 19–20 Mittag-Leffler function, 315–318 Multiplication, 49–50 of fuzzy numbers, 30, 30f of two Z-numbers, 57 Multivariate fuzzy chain rule, in gH-partial differentiability, 111–113 N Nonhomogeneous fuzzy linear differential systems, 192–195 Nonlinear fuzzy differential equation, 173–174 Nonlinear fuzzy number, 24–25, 25f Nonlinear property of fuzzy functions, 152 length function, 151–152

Numerical solution, of fuzzy nth-order differential equations, 267–286, 272–274t, 278–280t, 284t P Part-by-part integration, 175 Partial GH-differentiability, production in, 330 Partial ordering, 37–38, 330, 332 properties of, 37–38 P-distance, 72–73 Piece-wise membership function, 44, 44f Population biology, 202–203 Possibility restriction, 52 Power series method with fuzzy coefficients, 181–182 for solving Legendre’s equation, 185–188 Pseudo-octagonal sets, 45–51, 47f R Real fractional differential equation, existence and uniqueness in, 309–310 Residual values, 213, 230 S Scalar multiplication, in level-wise form, 61, 61f Second order fuzzy differential equations, 166–173 Second-order fuzzy linear differential equation, 276 Second order linear differential equation, 195 Second order operator, 140 Second translation theorem, 117 Singleton fuzzy number, 23 definition, 45, 45f Singleton fuzzy zero number, 38 Stability, 217–226, 221f, 223–225f error analysis, 248–254 global error results, 221t, 223t, 226t of modified fuzzy Euler method, 236–241, 241t Stacking theorem, 22 State of the system, in uncertain environment, 6 State-transition function, 6 Statistical models, 5–6 Summation of fuzzy numbers, 29–30, 29f in level-wise form, 60 of two Z-numbers, 56 Switching points of gH-differentiability, 83–85, 84–86f in gH-partial differentiability, 109, 110f in integration, 93–94

Index T Tangent line, of fuzzy number valued solution function, 219 Taylor expansion, 97–106 Taylor method, 9 Theory of probability, 2 Time-series data, 4 Trapezoidal fuzzy number, 26 Trapezoidal membership function, 16, 17f Triangular fuzzy number, 26–28, 28f Triangular membership function, 16, 17f Truncation error in first interval, 252–253t in second interval, 252t, 254t in third interval, 253–254t Two fuzzy numbers, distances of, 67, 70–74 Two-point fuzzy boundary value problem, 330–331 Two-stage predictor–corrector algorithm, 226 U Uncertain differential equations, 127–128 chain rule, 130–131 definition, 131–133 integration by parts, 131 remark, 133–134 uncertain process as canonical Liu process, 128–129 Liu integral of, 129 Uncertain dynamic systems, 4–7 history of, 7–8 Uncertain sets, 15–20

377

Uncertainty, 1–2 state-space of the system, 6 Uncertain variables, 15–20, 18–19f Uniform convergencek, definition of, 120 V Variational iteration, 9 method, 173–179 W Word computations, 51 Z Zadeh’s extension principle, 56 Z-differential equations, 197–205 characteristic theorem, 200–201 continuity, 197 economics, 204–205 medicine, 203–204 population biology, 202–203 Z-number initial value problem, 199–200 Z-process, 197 gH-differentiability of, 198 Zero interval, 32 Zigzag uncertain variables, 17 Z-number initial value problem, 199–200 Z-numbers, 54–64 level-wise form of, 58–60, 59f Z-process, 51–54, 197–198 definition, 52 examples, 53–54, 53–54f Z-valuation, 52