Proceedings of the Sixth International Colloguium on Differential Equations: Plovdiv, Bulgaria, 18–23 August, 1995 [Reprint 2020 ed.] 9783112314050, 9783112302781

Table of contents :
CONTENTS
Preface
Generating Optimal Runge-Kutta Methods
Minimization Techniques in Neutral Network Supervised Training
An Extended Marquardt-Levenberg Method for Function Minimization
Approximate Iterative Method for a System of Nonlocal Parabolic Functional-Differential Problems
On a Delay Logistic Equation with Nonlinear Average Growth Rate
Griffiths' Formalism on the Calculus of Variations
A Nonlinear Elliptic Equation with Nonlinearity Crossing Eigenvalues
The Partial Differential Equation Associated to the Generalized Kac-Feynman Integral Equation
A primal-dual method for discrete lp approximation
Utilizing Curvature Information in Multi-step Quasi-Newton methods for Optimization
Utilizing Curvature Information in Multi-Step Quasi-Newton Methods for Optimization
On Numerical Results for the Generalized Curve Shortening Equations
Analytic continuation of holomorphic solutions of partial differential equations
The Condition (S) of Kawai, the Property of Completely Regular Growth and the Estimate of the Micro-Support of the Complex of a Convolution Operator
One Approach to Exact Solutions of a piecewise Linear Differential Equation
Numerical integration of irregularly oscillating functions by parameter-dependent compound quadrature formulae
Initial Value Problem for Abstract Differential Equations of Sobolev Type
A parallel approach to bivariate splines
Recent results on Liénard systems
Extension of Holomorphic Mappings in Infinite Dimension
Numerical approximations to some nonlinear diffusion equation with strong absorption
On the Initial Boundary Value Problem for the Linearized MHD Equation
Modified Gompertz Curve and it's Applications I
An Experimental Algorithm for Ending Eigenvalues
On a Degenerate Boundary Value Problem for Second Order Quasilinear Elliptic Operators
Some Improperly Posed Problems and their Nonlocal Regularization
Methods for the Computation of Periodic Solutions of Dynamical Systems
The Computation of Equilibrium Points of Up-to-Three Dimensional Dynamical Systems Using the Topological Degree Theory
The boundary behavior of a bounded function
Application of Statistical Moments on Exterior Dirichlet Problem
Discrete Trigonometric Methods for Boundary Integral Equations in Hòlder-Zygmund Spaces
On the conditioning and the solution, by means of multigrid methods, of symmetric (block) Toeplitz linear systems
Differential Game Theory at Control System design for a Manipulator
Two-Point Formulas in Hermite-Birkhoff Quadratures
Numerical Direct Methods for Initial Value Problems of Any r-th-Order Ordinary Differential Equations
A Stochastic Compartmental Model for Dynamic Biological Systems
Round-Off Error of Numerical Integral by a Finite Chebyshev Series in a Floating-Point Number System
Floating-Point Number Solutions in a Simple Linear Equation with Multiplication Algorithm
Multi-Peak System Identification with Gauss-Weierstrass Transform
Continuation of bounded holomorphic functions from one-codimensional subspace to pseudoconvex domains
Singularities of surfaces defined by Monge-Ampere equations of hyperbolic type
De Rham Cohomology of Toroidal Groups and Complex Line Bundles
Generalized Bisection Methods for Imprecise Problems
Periodic Orbits and Invariant Surfaces of Nonlinear Mappings
Expected Behavior of Bisection Based Methods for Counting and Computing All the Roots of a Function
Approximate Lie Symmetries for Diffusion Equations with Non-linear Perturbations
On the Stability of a Multistep Method
Numerical Treatment of Mathematical Models for Infectious Diseases
Applications of the Beltrami Function in the Physics of Plasmas
Necessary and sufficient conditions for Fredholmness of irregular singular partial differential operators
On the Computation of Zeros of Bessel and Bessel-Related Functions
Linearized Viscoelastic Wave Propagation

Citation preview

Proceedings of the Sixth International Colloquium on Differential Equations

Also available from V S P Proceedings of the Fifth International Colloquium on Differential Equations Plovdiv, Bulgaria, 1 8 - 2 3 August 1994 Proceedings of the Fourth International Colloquium on Differential Equations Plovdiv, Bulgaria, 1 8 - 2 2 August 1993 Proceedings of the Third International Colloquium on Differential Equations Plovdiv, Bulgaria, 1 8 - 2 3 August 1992

Proceedings of the Third International Colloquium on Numerical Analysis Plovdiv, Bulgaria, 13 - 17 August 1994 Proceedings of the Second International Colloquium on Numerical Analysis Plovdiv, Bulgaria, 13 - 17 August 1993 Proceedings of the First International Colloquium on Numerical Analysis Plovdiv, Bulgaria, 13 - 17 August 1992

Proceedings of the

Sixth International Colloquium on Differential Equations Plovdiv, Bulgaria, 1 8 - 2 3 August, 1995

Editor: D. Bainov

/f/VSP///

Utrecht, The Netherlands, 1996

VSP B V P.O. B o x 3 4 6 3 7 0 0 A H Zeist The Netherlands

© V S P B V 1996 First p u b l i s h e d in 1 9 9 6 ISBN 90-6764-203-7

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

Printed in The Netherlands

by Koninklijke

Wöhrmann bv,

Zutphen.

CONTENTS

Preface Generating Optimal Runge-Kutta Methods G.S. Androulakis,

T.N. Grapsa and M.N. Vrahatis

Minimization Techniques in Neural Network Supervised Training G.S. Androulakis, G.D. Magoulas and M.N. Vrahatis

1 9

An Extended Marquardt-Levenberg Method for Function Minimization C.A. Botsaris and J.N. Lambrinos

17

Approximate Iterative Method for a System of Nonlocal Parabolic Functional-Differential Problems L. Byszewski

25

On a Delay Logistic Equation with Nonlinear Average Growth Rate M.-P. Chen and J. S. Yu

35

Griffiths' Formalism on the Calculus of Variations W.-S. Cheung

45

A Nonlinear Elliptic Equation with Nonlinearity Crossing Eigenvalues Q.-H. Choi, S. Chun and T. Jung

53

The Partial Differential Equation Associated to the Generalized Kac-Feynman Integral Equation D.M. Chung and S.J. Chang

61

A Method for Discrete ( p Approximation A. Primal-Dual Dax

69

Utilizing Curvature Information in Multi-Step Quasi-Newton Methods for Optimization J.A. Ford and I.A. Moghrabi

79

Accelerating the Convergence of Newton's Approximation Scheme T.N. Grapsa, M.N. Vrahatis and F.A. Zafiropoulos

87

On Numerical Results for the Generalized Curve Shortening Equations T. Idogawa

95

Analytic Continuation of Holomorphic Solutions of Partial Differential Equations K. Igari

103

The Condition (S) of Kawai, the Property of Completely Regular Growth and the Estimate of the Micro-Support of the Complex of a Convolution Operator

vi

Contents

R. Ishimura

111

One Approach to Exact Solutions of a Piecewise Linear Differential Equation K. Katori, M. Otake, Y. Manome and T. Mishima

119

Numerical Integration of Irregularly Oscillating Functions by Parameter-Dependent Compound Quadrature Formulae P. Köhler

127

Initial Value Problem for Abstract Differential Equations of Sobolev Type T. Kojo and M. Tsutsumi

135

A Parallel Approach to Bivariate Splines P. Lamberti

143

Recent Results on Lienard Systems S. Lynch

151

Extension Y. Matsudaof Holomorphic Mappings in Infinite Dimension Numerical Approximations to some Nonlinear Diffusion Equations with Strong Absorption T. Nakaki and K. Tomoeda

159

165

On the Initial Boundary Value Problem for the Linearized MHD Equation M. Ohno and T. Shirota

173

Modified Gompertz Curve and It's Applications I K. Oshima and I. Hofuku

181

An Experimental Algorithm for Horimoto Finding Eigenvalues K. Oshima, S. Nomura and K.

189

On a Degenerate Boundary Value Problem for Second Order Quasilinear Elliptic Operators D.K. Palagachev and P.R. Popivanov

197

Some Improperly Posed Problems and their Nonlocal Regularization N.I. Popivanov

209

Methods for the Computation of Periodic Solutions of Dynamical Systems 0. Ragos, M.N. Vrahatis and G.S. Androulakis

213

The Computation of Equilibrium Points of Up-to-Three Dimensional Dynamical Systems Using the Topological Degree Theory 0. Ragos, M.N. Vrahatis and F.A. Zafiropoulos

221

The Boundary Behavior of a Bounded Function F. Sakanishi Application of Statistical Moments on Exterior Dirichlet Problem

229

Contents

vii

M.M. Saleh, A.E. Sedeek and I.L. El-Kalia

235

Discrete Trigonometric Methods for Boundary Integral Equations in Hòlder-Zygmund Spaces L. Schroderus

241

On the Conditioning and the Solution, by Means of Multigrid Methods, of Symmetric (Block) Toeplitz Linear Systems S. Serra

249

Differential Game Theory at Control System Design for a Manipulator Y. Shimomoto, H. Kisu and T. Ishimatsu

257

Two-Point Formulas in Hermite-Birkhoff Quadratures Ch. Suzuki

265

Numerical Direct Methods for Initial Value Problems of Any r-th-Order Ordinary Differential Equations Ch. Suzuki

273

A Stochastic Compartmental Model for Dynamic Biological Systems S. Tseng

281

Round-Off Error of Numerical Integral by a Finite Chebyshev Series in a Floating-Point Number System K. Tsuji

289

Floating-Point Number Solutions in a Simple Linear Equation with Multiplication Algorithm K. Tsuji

297

Multi-Peak System Identification with Gauss-Weierstrass Transform M. Tsuji

305

Continuation of Bounded Holomorphic Functions from One-Codimensional Subspace to Pseudoconvex Domains M. Tsuji

313

Singularities of Surfaces Defined by Monge-Ampere Equations of Hyperbolic Type Mikio Tsuji

321

De Rham Cohomology of Toroidal Groups and Complex Line Bundles T. Umeno

329

Generalized Bisection Methods for Imprecise Problems M.N. Vrahatis

337

Periodic Orbits and and T.C. Invariant Surfaces of Nonlinear Mappings M.N. Vrahatis Bountis

345

Expected Behavior of Bisection Based Methods for Counting and

viii

Contents

Computing All the Roots of a Function M.N. Vrahatis and D.J. Kavvadias

353

Approximate Lie Symmetries for Diffusion Equations with Nonlinear Perturbations R.J. Wiltshire and T.J. Zastawniak

361

On the Stability of a Multistep Method H. Yanagiwara

369

Numerical Treatment of Mathematical Models for Infectious Diseases A. Yanaghiya, J. Teramoto and S. Kikuchi

377

Applications of the Beltrami Function in the Physics of Plasmas Z. Yoshida

387

Necessary and Sufficient Conditions for Fredholmness of Irregular Singular Partial Differential Operators M. Yoshino and M. Miyake

401

On the Computation of Zeros of Bessel and Bessel-Related Functions F.A. Zafiropoulos, T.N. Grapsa, 0. Ragos and M.N. Vrahatis

409

Linearized Viscoelastic Wave Propagation F.A. Zafiropoulos, O. Ragos and M.N. Vrahatis

417

Preface The Sixth International Colloquium on Differential Equations was organized by the Institute for Basic Science of Inha University, the International Federation of Nonlinear Analysts, the Mathematical Society of Japan, Pharmaceutical Faculty of the Medical University of Sofia, University of Catania and UNESCO, with the cooperation of the Association Suisse d'lnformatique, the Canadian Mathematical Society, Kyushu University, the London Mathematical Society, Technical University of Plovdiv, and it was partially sponsored by the Bulgarian Ministry of Education, Science and Technologies under Grant MM-422. The Colloquim was held on August 18-23, 1995, in Plovdiv, Bulgaria. The subsequent colloquia will take place each year from 18 to 23 of August in Plovdiv, Bulgaria. The publishing of this volume is fully supported by the Bulgarian Ministry of Education, Science and Technologies under Grant MM-422. Organizing

Committee:

H. Adeli, A.S.A. Al-Hammadi, D. Bainov (Chairman), Q.-H. Choi, V. Covachev (Secretary), J. Diblik, J. Kajiwara (Vice Chairman). V. Lakshmikantham, M. Marino, A. Maugeri (Vice Chairman), E. Minchev (Secretary), N. Popivanov, P. Popivanov, H.M. Srivastava, Ch. Tsokos, M.N. Vrahatis, S. Zlatev (Secretary). Scientific

Committee:

D. Bainov (Bulgaria), Ph. Brener (Sweden), C. Dafermos (USA), H. Fujita (Japan), J. Kajiwara (Japan), V. Lakshmikantham (USA), L. Medeiros (Brazil), A. Maugeri (Italy), N. Pavel (USA), P. Popivanov (Bulgaria), P. Sobolevskii (Israel), H.M. Srivastava (Canada), Fr. Stenger (USA), M. Tsutsumi (Japan), M.N. Vrahatis (Greece). The address of the Organizing Committee is: Drumi Bainov, P.O.Box 45, 1504 Sofia, Bulgaria. The

Editor

6th Int. Coll. on Differential Equations, D Bainov © VSP 1996

pp. 1-7

GENERATING OPTIMAL RUNGE-KUTTA METHODS G . S . ANDROULAKIS, T . N .

G R A P S A AND M . N .

Department of Mathematics,

VRAHATIS

University of Patras, GR-261.10 Patras, Greece

Abstract. A procedure which generates optimal Runge-Kutta methods is presented. Using this, we were able to obtain optimal explicit Runge-Kutta methods as well as optimal ones for a special class of implicit Runge-Kutta schemes, named DIRK (Diagonally Implicit Runge-Kutta) used for the initial value problem in stiff ordinary differential equations. This procedure obtains minimum truncation errors by expanding the corresponding Runge-Kutta algebraic system of equations with the "truncation error equations". An efficient scheme for solving the above system in order to obtain the coefficients of the new Runge-Kutta methods, is applied. Numerical applications are presented. Keywords: Optimal Runge-Kutta methods, explicit Runge- Kutta methods, implicit Runge-Kutta methods, diagonally implicit Runge-Kutta methods, minimization of truncation error, derivation of coefficients of new Runge-Kutta methods, singly implicit method.

1.

Introduction

A well known class of methods for the numerical solution of the Initial Value Problem (IVP) : (1.1)

y{x 0 ) = yo, with x 0 € IR, yo € IR n , y : IR

IR", / : IR x IRn

IRn is Runge-Kutta methods.

Given a step-size h, an s-stage Runge-Kutta method is defined as follows : S

yn+i=yn + hY^biki,

71 = 1,2,..

(1.2)

where : Zn+l = xn + h, h =

f(xn,yn), (1.3)

2

G.S. Androuhkis,

T.N. Grapsa, M.N. Vrahatis

and it provides at each iteration ra, an approximation yn+i to y(xn+i)

where y(x) is the

exact solution of (1.1). T h e problem of constructing R u n g e - K u t t a methods has drawn a lot of attention and has evolved to a r a t h e r specialized branch of numerical mathematics ([1]—[12], etc.). It is convenient to display the coefficients occurring in (1.3) in the following form, known as a Butcher array : an a 21

Ci C2

c, a i s

h

au 022

•• ••

11» i, i = 1 , 2 , . . . , s t h e n each k, is individually defined by : ki = f(xn V

+ c; h, yn +

and t h a t m e t h o d s are called semi-implicit.

i=i

i

a t J kA, '

2,. . . , S,

(1.4)

A particularly efficient class of semi-implicit

R u n g e - K u t t a formula« was first suggested by N0rsett [13] who considered the case where the an are all equal a n d non-zero. These formula« were f u r t h e r studied by Crouzeix [14] and by Alexander [15]. An additional refinement introduced by Alexander was to derive formulae which are strongly S - s t a b l e [16], and on at least one of his test problems, strongly S - s t a b l e formula« performed b e t t e r t h a n ones which were only A-stable. T h e m a i n result proved by Alexander is t h a t an A - s t a b l e DIRK formula is strongly S-stable if it hits a Butcher m a t r i x of the form : ci C2

a a2i

0 a

c,~ 1 1

Gj-1,1

flj-1,2

a

0

bi

b,-i b,-i

a a

hi

•••

where a > 0 and where t h e usual row sum conditions hold. T h e n u m b e r s of fo's must be chosen large enough, a n d t h e parameters must be determined so t h a t t h e Taylor series of yn(xn + h) matches t h a t of y(xn + h) through t h e t e r m s involving hn.

These requirements generate a system of nonlinear algebraic

Generating Optimal Runge-Kutta

Methods

3

equations in the parameters. The number of equations grows rapidly with s and it is, in general, smaller than the number of the parameters. Since there are free parameters, there are infinitely many solutions of the above system. On the other hand there is only one which possesses the smallest local truncation error. To obtain such a method one must minimize an objective function that emanates from the equations expressing the truncation error which we call truncation error equations. To find such methods we enlarge the algebraic system by these truncation error equations and minimize the enlarged system. Specifically, we create an objective function from the sum of the squares of all the equations and we call it truncation error objective function. To obtain an s-stage DIRK method the minimum of this objective function has to fulfill the algebraic Runge-Kutta system. The global minimum of the objective function gives the parameters of the "optimal" s-stage DIRK method.

2. Minimizing the Truncation Error Objective Function For any s-stage Runge-Kutta method we are able, using the analysis described in [14,15,1-9], to formulate the algebraic system of equations produced by the conditions which must be satisfied. Thus, using Butcher trees and implementing the functions $ and 7 we are able to obtain any s-stage algebraic system. Testing Problem 2.1 Runge-Kutta s = 5 : This problem has Butcher array of the form : Ci C2 c3 C4 cs Ce

0 a2i «31 a41 asi 061 6i

0 0 0 0 0 0 0 0 0 0 0 0 0 o 32 0 0 0 ai2 «43 0 0 0 052 053 054 0 «62 063 064 065 ¿2 h ¿4 h be

Testing Problem 2.2 Embedded formula of order s = 2, [9] ; For this formula we have the following Butcher array : a c2

a 0 C2 — a a hi h

4

G.S.

Testing Problem 2.3

A n d r o u l a k i s ,

Third

o r d e r

Grapsa,

T.N.

M.N.

Vrahatis

[9] :

s t r o n g l y S-stable f o r m u l a ,

The Butcher array for this formula is : a

0

a c

C2

1

—

2

bi

a

0 0

a b2

a

b2

a

To find a method that possesses the smallest truncation error we have to find those coefficients which minimize, for each s, the upper bounds of all the truncation error expressions. Each one of these equations matches a coefficient which corresponds to a product of partial derivatives. Using Lotkin's bounds, i.e. : o

(

\

\ f ( x , y ( x ) ) \ < M ,

i + j < s ,

(2.1)

we axe able to bound the truncation error expressions with powers of M and L. Evidently, to each such bound corresponds a "weight" coefficient which we have to take into consideration for the minimization process. Of course, our goal is to find an "optimal" Runge-Kutta method which possesses the smallest local truncation error. Thus, we have to obtain a solution of the algebraic Runge-Kutta system which at the same time minimizes the truncation error. To do this we consider the following constrained optimization problem : « G(x) = y^ g\,

minimize

k=1

subject to

h(x) = 0 =

m

^/1

k=1

which, utilizing the penalty method, is equivalent to the following unconstrained optimization problem : minimize

G ( x ) +

p\H(x)\

and, consequently, to the minimization of the following objective function : Pi f ( b i , . . . , b „ c

u

. .

. , c

s

, a

n

, . . . , a „ )

=

A

+

Jt=i

(2.2)

it=i

where fk axe obtained from the fi\ s-stage Runge-Kutta order conditions and gk are the /12 equations for the truncation error. To minimize the truncation error objective function we utilize the algorithm OPTBIS: OPTimization using BISection [17]. This algorithm is based on a bisection method

Generating Optimal Runge-Kutta,

Methods

5

and always converges to a local minimum of an objective function / provided that the assumptions for the existence of such a minimum are fulfilled. Also, it converges in one iteration to quadratic functions of n variables and it rapidly minimizes general functions. OPTBIS avoids all information regarding the Hessian matrix and it does not perform matrix inversions. It requires only the algebraic signs of the function and gradient values to be correct, so that it can be applied to problems with imprecise function values. Moreover, OPTBIS is not affected when the objective function has more than one extrema in its domain and it does not require the starting estimate of an extremum to be close to it.

3.

Applications

The technique described in Section 2 has been implemented using the new FORTRAN program O P T R K . Our results have been obtained on the University of Patras H P 715 system as well as on an Express IBM PC. Our experience is that the algorithm behaves predictably and reliably and the results have been quite satisfactory. Some typical computational results are given below. For the following problems, the reported parameters indicate the Runge-Kutta coefficients. The "optimal" Runge-Kutta method for the Testing Problem 2.1 is exhibited in Table 1. For the Testing Problem 2.2 the analytical solution (see [9, 15, 14]) is given by: &! = (c2 - 1/2)/(C 2 - a ) b? = (a - l / 2 ) / ( a - c 2 ). Utilizing O P T R K for the extended system of Testing Problem 2.2 we have obtained the following optimal methods : Optimal methods for Problem 2.2 l/2±"l/2%/3 l / 2 ± l / 2 \ / 3 1/2 =F 1/2 \ / 3 Tl/v^ 1/2

0 1/2 ± 1/2 y/Z 1/2

For s = 3 and Testing Problem 2.3 the analytical solution (see [9, 15, 14]) is given by : c2 = (a 2 - 3 a / 2 + l / 3 ) / ( a 2 - 2 a + 1/2) bi = (c 2 /2 — 1/6)/((c 2 — a ) ( l — a ) )

6

G.S. Androulakis,

T.N. Grapsa, M.N.

Vrahatis

Table 1: The optimal Runge-Kutta method for the Testing Problem 2.1 111 = a 14 = a21 = a2 4 «31 = «34 = a41 =

a44

= -

«51

a 54

-

«61

—

064 =r

¿1 = ¿4 = Ci = c4 =

0.0000000000000000 0.0000000000000000 0.0441951335345095 0.0000000000000000 -2.3084219229081500 0.0000000000000000 5.8117739543108300 0.0000000000000000 -0.2135172891248340 0.8278584585862798 -0.2383774874724800 0.7409257504614182 -0.2271235477726650 0.1297952318387896 0.0000000000000000 0.5604493719471257

a 12 — «15 = «22 «25 -

«32

of the function E. First, we consider the sets B,, i = 1 , . . . , n to be those connected components of 2 = 1

(a)

a(pi + 2p2) = 1/2

(6)

(2.6)

L-stability is a desirable property for an one-step numerical integration scheme, so we force our integration scheme to be L-stable by introducing one more equation. Definition 2.1 A one-step from being A-stable,

numerical integration scheme is considered L-stable

when it is applied to the scalar initial value x(t) = A x(t),

if apart

problem

x(0) = 7]

(X being a complex constant with negative real part), the resultant numerical solution is given by xk+i = fJi(\t)xk with the characteristic

equation fJ-(Xt) having the property lim |„(A*)| = 0. 3?(A£)—oo

Applying previous definition to our integration scheme one more equation is obtained Pi=a. Solving the system of equations (2.6)-(2.7) yields

(2.7)

20

C.A. Botsaris

and J.N.

p2 = 1 — a,

Pi = a,

Lambrinos

where a = 1 — 0.5\/2.

(2.8)

Using the derived values of the parameters involved in the integration formula, a straight forward calculation leads to the space curve xk(t) = xk-

at[I + atHk]-lgk

- p2t[I + atHk]~2gk

(2.9)

xk(t) =xk-

t[I + 2atH k + a2t2Hk]~x [I + a2tHk]gk

(2.10)

or equivalently,

Notice that the iteration formula in its first form can be viewed as an extention to the well-known Marquardt-Levenberg formula. The search path in form (2.10) was also derived in ([3]) using a different approach to our problem of the numerical integration to the differential equation of the steepest descent curve. The properties of the curvilinear search path (2.9) axe discussed in the following section.

3.

PROPERTIES OF THE CURVILINEAR SEARCH PATH Let us set the curvilinear search path (2.9) in the form

where Mk(xk,t)

= 0,1,2,...

(3.1)

= t[I + 2atH k + a2t2Hl]~1 [I + a2tHk]

(3.2)

-

(3.3)

xk+1 = xk - Mk(xk,t)gk, is the direction matrix Mk(xk,t)

Then, ^

P

= - 9 l 9 k < 0,

and there exists an e > 0 such that f(xk(t)

\M*0

— f(xk)

< 0, Vi € (0, e). Hence (3.1) defines

a descent curve in x-space. Note now that for small t we have xk(t) =xk-

tgk

(3.4)

On the other hand, for t —> +oo and provided that the Hessian matrix is positive semidefinite, xk(t) — xk tends to the Newton step. Finally, if Hk ^ O, then as t —l/aXk, we obtain xk(t) - x

k

^ - [i(l + a2t\k)/{ 1 + a a * ) 2 ] {uTkgk)uk

(3.5)

where \k is the most negative eigenvalue of Hk and uk is the associated normalized eigenvector. Clearly, (3.5) is a vector of infinite length in the direction of the most negative curvature, provided that u k g k / 0.

An

Extended

Marquardt-Levenberg

Method

21

The previous argument proves that the path xk(t) is initially tangent to the direction of steepest descent, and as i

— l/a\k it becomes tangent to the eigenvector corre-

sponding to the most negative eigenvalue of matrix

Mk(xk,t)

Hk.

Moreover, for

t


) in E(n). By utilizing the theory and techniques of exterior differential systems, Griffiths [1] developed a powerful systematic formalism of the calculus of variations for functionals whose domain of definition consists of integral curves of an exterior differential system. Such a formalism is, while in greater generality than customary, particularly effective for geometrical problems, and it sheds new light on even the classical Lagrange problem. The formalism is subsequently generalized to the case of several independent variables by Cheung [2]. The purpose of this paper is to give a brief account on this formalism for the case of several independent variables, to work out some interesting and illustrative examples, and to discuss recent results in this discipline. 2. THE FORMALISM Let X be a manifold. On X a Pfaffian system [1,3,4] with independence condition (I,to) is given by an exterior differential system I c Si*(X) that is locally generated by some 1-forms {Ba : 1 < a < m}, or equivalently, generated by the smooth sections of a sub-bundle W* c T*(X), together with an n-form cj called the independence condition. An integral manifold of (l,u>) is a submanifold of X restricted to which 1 = 0 and u / 0 . The set of all integral manifolds of (I,w) is denoted by 1(1, u;). Observe that since as far as integral manifolds are concerned, the independence n-form u is only well-defined modulo I, what we shall be concerned with is thus the filtration of sub-bundles I f ' c i ' c T'X , where W' generates I and L'/W' is a rank n vector bundle such that induces a non-zero cross-section of A n ( L ' / W * ) .

u

e

f\nL')

Example 1. Let M be an m-manifold with local coordinates (ya : a = 1,... ,m). Consider r n X = J (R , M), the rth jet bundle of locally defined maps from R " t o M . If (x\ y j ) (1 :=ip + A i j " A u,i , := dip , and the Cartan

system C(4), we shall arrive at an involutive system (J, w), called the EulerLagrange

differential

system,

on an associated manifold y , called the momentum

space, with X(J,u>) =

u>). We shall not cariy out the detailed constructions of (J, u) and Y here, interested readers may consult [2] for the details. The important point here is that the construction of ( J , UJ) is algorithmic and hence is easily obtainable from (C(fl'), W). Furthermore, by studying the properties of (J, CJ) in Y, we know much details of the solutions of the Euler-Lagrange equations of the original variational problem.

5. N O E T H E R ' S T H E O R E M Analogous to the classical case, we shall call a vector field t> € T T X a ( i n f i n i t e s i m a l ) Noether of a variational problem (X\I,w;tp)

if f £„/ C I \ Cvip = 0

mod I .

symmetry

52

Wing-Sum Cheung

T h e o r e m 4.

then V-iip is a

conservation

law, that is, it is a closed (n — 1) form on each integral manifold N 6 X(I ,ui) which satisfies

the Euler-

Lagrange

(Noether's

equations

Proof. Let 6 =

Theorem)

If v is a Noether symmetry

of(X;I,u;i and v b e the vector field on Z = X x R n m induced by v on X by the product

structure. B y C a r t a n ' s formula and the Euler-Lagrange equations (3),

d(vj>!>) = d(v.i( Y in turn induces a vector field vy on Y. B y the natural one-to-one correspondence between I(J,u>)

and the solutions of the

Euler-Lagrange equations of (X; I,w, ; ip), then vy-npY is a conservation

a closed (n — 1) form on each integral manifold N 6

law, that is, it is

J(J,u>).

6. C O N C L U D I N G R E M A R K This new formalism of the calculus of variations by Griffiths and Cheung is very effective on geometrical problems, especially on those intrinsic ones. T h e only difficulty one may encounter in applying it is to set up the problem in our formalism correctly. Once the set up is done correctly, the rest of the computations are basically algorithmic. Other interesting applications including higher order Noether's theorem, higher order conservation laws, etc., and other recent results on the formalism can b e found in [1,2,9,10,11].

REFERENCES [1] P.A. Griffiths, Exterior differential systems and the calculus of variations. Birkhauser, Boston (1983). [2] W.S. Cheung, Variational problems and the exterior differential systems. 111. J . Math., vol.33, no.l (1987), 10-26. [3] R. Bryant, S.S. Chern and P.A. Griffiths, Essays on exterior differential systems. Proc. Beijing Symposium on Differential Geometry and Differential Equations (1980). [4] R. Bryant, S.S. Chern, R. Gardner, H. Goldschmidt and P. Griffiths, Exterior differential systems. Springer-Verlag, New York (1991). [5] I. Gelfand and G . Fomin, Calculus of variations. Prentice Hall, Englewood Cliffs, N . J . (1963). [6] R. Herman, Differential geometry and the calculus of variations. 2nd edition, Math. Sci. Press, Brookline, Mass. (1977). [7] W.S. Cheung, C°°-invariants on loop spaces. Proc. Amer. Math. Soc., vol.100, no.2 (1987), 322-328. [8] M. Kuranishi, On E. C a r t a n ' s prolongation theorem of exterior differential systems. Amer. J . Math., vol.79 (1957), 1-47. [9] W.S. Cheung, A simple proof to the complete integrabilitv of the free n-dimensional rigid body. Chinese J . Math, vol.15, no.l (1987), 17-30. [10] W . S . Cheung, Higher order conservation laws and a higher order Noether's theorem. Adv. Appl. Math., vol.8 (1987), 446-485. [11] L. Hsu, Calculus of variations via the Griffiths formalism. J . Diff. Geom., vol.36, no.3 (1992), 551589.

6tb Int. Coll. on Differential D. Bainov © VSP 1996

Equations,

p p . 53-60

A N O N L I N E A R ELLIPTIC E Q U A T I O N WITH N O N L I N E A R I T Y CROSSING EIGENVALUES

Q-HEUNG 1 2

CHOI1, SUNGKI CHUN2AND TACKSUN

JUNG3

Department of Mathematics, Inha University, Incheon 402-751, Korea

College of Science and Technology, Hongik University, Chochiwon 337-800, Korea

^Department of Mathematics, Kunsan National University, Kunsan 573-360, Korea

Abstract

Let Q b e a bounded domain in M n with smooth boundary and L b e an elliptic operator. We consider an nonlinear

elliptic boundary value problem under Dirichlet boundary condition Lu + 6 u + — a u " = / in fl. We assume t h a t a < Aj, A2 < &

A3 and / is generated by 4>\ and 2 - Then we reveal a relation between the multiplicity of solutions and source

terms / (cf. Theorem 2.1).

K e y w o r d s : Eigenvalue, eigenfunction, Lipschitz continuous, multiplicity of solutions, local degree.

0.

INTRODUCTION

L e t ft b e a b o u n d e d differential

w h e r e A,j =

d o m a i n in R n w i t h s m o o t h b o u n d a r y

dii

a n d let L

denote

the

operator

£ C°°(ft). W e consider a semilinear elliptic b o u n d a r y value problem under

Dirichlet b o u n d a r y

condition Lu

+

bu+

— au~ u

=

f(x)

=

0

in

ft,

1

on aft.

H e r e L i s a s e c o n d o r d e r l i n e a r e l l i p t i c d i f f e r e n t i a l o p e r a t o r a n d a m a p p i n g f r o m L2(Q) i t s e l f w i t h c o m p a c t i n v e r s e , w i t h e i g e n v a l u e s —A,, e a c h r e p e a t e d a s o f t e n a s 0 < Ai < A2 < In this paper, w e a s s u m e that a
0) can be explicitly calculated and they are cl{b-\l)4>1±kcl{b-\2)2 (d>0).

56

Choi, Chun and Jung

Therefore $ maps C\ onto the cone >„ I2 A. J 2\,(p2)- Then there exists a point q, of 7 such that q, = f)q for some P > 0. By Lemma 1.2, i " 1 ^ , ) and i f 1 (9) are on the same ray in C,-, starting from the origin and arg pi < arg i , " 1 (ft) < argp 2 , which is a contradiction. This completes the proof.

•

With Lemma 2.1, we have the following theorem, which is very important to study the multiplicity of solutions of a semilinear equation when the source term varies. THEOREM 2.1. For 1 < i < 4, the restriction maps Ci onto Ri. Therefore, maps V onto R3. In particular, and 3 are bijective.

$

The above theorem implies Theorem 1.1. Furthermore, we get the following theorem. THEOREM 2.2. solution.

If f does not belong to the cone R3, then equation

(2.1) has no

Nonlinear Elliptic

Equation

59

3 . A SHARP RESULT FOR THE MULTIPLICITY In this section, we continue the study of the multiplicity of solutions of equation (1.1). We shall make a sharp result for the multiplicity of solutions of this equation when the source term / belongs to the interior Int/j^ of the cone R\. Let ft be a bounded domain in R n with smooth boundary dfl and let L denote the differential operator E where a¡j = aJX 6 C°°(ft). eigenvalue problem

Given a function m € L°°(fl),

{— Lu = Amu

in

let us consider the linear ft,

/„ .

u= 0 on dfl. ^ ' ' An eigenvalue of (3.1) is a A such that (3.1) has a solution u ^ 0. Any ^ 0 satisfying (3.1) is an eigenfunction associated to the eigenvalue A. LEMMA 3 . 1 (COMPARISON P R O P E R T Y ) [ 5 ] .

Assume

L be an elliptic

operator.

If

m < M in ft, then > Ak{M); if m < M in a subset of positive measure, then A;t(m) > Ak(M). In particular, if m < At (resp. > A*), then A^(m) > 1 (resp. < I). Given u, we denote the characteristic function of the positive set of u, that is,

We set oc(u) = 6x(u) — ox(~u) DEFINITION 3.1 [5]. problem

when the measure of {x \ u(x) = 0} is zero. We say that u is a nondegenerate

{

— Lv = a(u)v v = 0

solution of (1.1) if the

in ii, on dfl

has only the trivial solution. In this section we have a concern only when L is the Laplacian operator. We denote by K the operator (—A) - 1 from H~l{fi) into //¿(ft) and we consider it as a compact operator. Given m 6 we consider the eigenvalue problem f — Av = vmv U = 0

in on 9ft.

ft, (

/„ ' '

It is well known (cf. [5]) that if m > 0 in a set of positive measure, then the positive numbers for which (3.2) has a nontrivial solution are the terms of a sequence i/i(m), i/ 2 (m), - • •, Vj{m), • • • diverging to +oo. Since each eigenvalue i/j has finite multiplicity, we can repeat it in the sequence as many times as its multiplicity.

60

Choi, Chun and Jung Now we consider the nonlinear equation Au + bu+ - au~ = f(x)

in

H.

(3.3)

LEMMA 3.2[5]. Assume f(x) = x + s24>2 6 Inti^. Let a < Aj and 6 < At for a given integer k > 2. Then if u is a solution of (3.3) which changes sign in ft, we have i>i(a(u)) < 1 < i/jfc_i(a(u)). Now we have a sharp result for the multiplicity of solutions of equation (1.1). THEOREM 3.1. Assume a < At < A2 < 6 < A3. Then, if f belongs to the interior IntJ?] of R\, then equation (3.3) has exactly four solutions and they are nondegenerate.

Proof. The statement follows from Lemma 3.2 which ensures that any solution which changes sign is nondegenerate and has local degree —1. We know that the solutions of constant sign are only up and un and they have local degree 1. Also we know [5] that deg(u - K(bu+ - au~), 5(0, r), -Kfa)

=0

for large positive r. By the homotopy invariance, if / £ Inti?, then deg(u - K(bu+ - au~), B{0, r), - K f ) = 0 for large positive r. This completes the proof.

•

REFERENCES

[1] A. Ambrosetti and G. Prodi, "A primer of nonlinear analysis", Cambridge, University Press, Cambridge Studies in Advanced Math., No. 34, 1993. [2] Q.H. Choi and T. Jung, An application of a variational reduction method to a nonlinear wave equation, J. Differential Equations, 117 (1995), 390-410. [3] A.C. Lazer and P.J. McKenna, Some multiplicity results for a class of semilinear elliptic and parabolic boundary value problems, J. Math. Anal. Appi., 107 (1985), 371-395. [4] P.J. McKenna, "Topological methods for asymmetric boundary value problems", Lecture Notes Series, No. 11, Research Institute of Math., Global Analysis Research Center, Seoul National University, 1993. [5] S. Solimini, Some remarks on the number of solutions of some nonlinear elliptic problems, Ann. Inst. Herni Poincaré, Vol.2, No.2 (1985), 143-156.

6th Int. Coll. on Differential Equations, pp. 61-68 D. Bainov © VSP 1996

T H E PARTIAL D I F F E R E N T I A L EQUATION ASSOCIATED TO T H E G E N E R A L I Z E D K A C - F E Y N M A N I N T E G R A L EQUATION

Dong Myung Chung* and Seung Jun Chang Department of Mathematics, Sogang University, Seoul 121-742, Korea Department of Mathematics, Dankook University, Cheonan 330-714, Korea Abstract Let ( R D , B D , n ) be the function space induced by a generalized Brownian motion processes determined by a R-valued function a( ) find a R-valued strictly increasing function &(•) with a(0) = 6(0) = 0. Let U be the function defined on [0, T] x R x R by u{t]LTi)=

~jèm

e x p {

~

( T ?

~ 2 b ( ! ) "

0 2

^

[ e x p {

/

+

°

d s

^

( t ) +

^

=

^

where •) is a C-valued bounded measurable function on [0, T] x R. In this paper, under appropriate regularity conditions on $(•, •), a(-) and i(-), we show that the function U satisfies the partial differential equation

with the initial condition lim ( _ 0 +

U{T\TJ)

=

6(T)

— ().

1. INTRODUCTION. Let (C 0 [0, T], £?(C0[0, T]), m w ) denote Wiener space where Co[0,T] is the space of all continuous functions x on [0, T] with x(0) = 0. Many physical problem can be formulated in terms of the conditional Wiener integral -E^F^Y] of the fimction of the form F{x) = exp{— f 6(s,x(s))ds}, Jo

x 6 C o [0,T],

where X(x) = x(t) and 0(-, •) is a sufficiently smooth function on [0, T] x R. It is indeed known from a theorem of Kac [6] that the function £/(•, •) defined on [0, T] x R by (1.1)

U(t,0

=

+ (0)\x(t)

is the solution of the partial differential equation ri

dU_ld*U

* Research supported by K O S E F and C A M at K A I S T

= i -

62

D.M. Chung

satisfying U((,0)

and S.J.

Chang

= S(( — £0). In [5], Donsker and Lions showed that the function

(1.3)

=-Eft.i-i.OOf'i*)]

is the solution of the partial differential equation (1.2) where Donsker's delta function formally defined by S,,((x)

= 1 - J ¿«'W-Udu,

x e

(i > 0, £ (E R) is the

Co[0,T}.

In [13], in order to provide a rigorous treatment of the fimction (1.3) involving the Donsker's delta function, Yeh introduced the concept of the conditional Wiener integral and derived a Fourier inversion formula for conditional Wiener integrals : (1.4)

E[F\x(t) = £]_I=exp{-g} =

jf e^E[e^F}du,

£GR

to obtain the explicit evaluation of (1.3). Using the inversion formula (1.4), Yeh [13] derived the Kac-Feynman equation for time independent continuous potential function #(£). In [3], Chung and Kang, using the Yeh's inversion formula, obtained similar results for time dependent bounded potential 0(s, £). In [9], Skoug and Park obtained a simple formula for expressing conditional Wiener integrals with vector-valued conditioning function in terms of ordinary Wiener integral, and then used the formula to derive the Kac-Feynman integral equation for time independent potential function 0(C)In this paper we extend the ideas of [3,8,9] from the Wiener processes to the more general stochastic processes. We note that the Wiener process is free of drift and stationary in time. However, the stochastic process considered in this paper is a process subject to drift and nonstationary in time. 2. THE GENERALIZED KAC-FEYNMAN INTEGRAL EQUATION. Let D = [0,T] and let ( f l , B , P ) be a probability measure space. A R-valued stochastic process X on (ii, B, P) and D is called a generalized Brownian motion process if X(0,tt>)=0 a.e. and for 0 < ¿o < i] < • • • < t n < T, the n-dimensional random vector ( X ( t i , c j ) , • • -,X(tn,u))) is normal distributed with the density function (2.1)

K ( t , i f ) = ((2*)» ¿ W i ) j=i rxn P

f l

1y

K*i-i)))~1/2

(fa- - °(*j)) - fri-1 - q(*j-i))) 2 1 ) J

where if = (TJI, • • •, r)n), TJQ — 0 and a(t) is a R-valued function with a(0) = 0 cind b(t) is a strictly increasing R-valued function with 6(0) = 0. As explained in [11, p. 18-20], X induces a probability measure /< on the measurable space ( R D , B D ) where R D is the space of all R-valued functions x(t), t £ D, and BD is the smallest cr-algebra of subsets of R D with respect to which all the coordinate

Generalized

Kac-Feynman

63

Equation

evaluation maps et(x) = x(t) defined on R ° are measurable. The triple ( R D , B D , f i ) is a probability measure space. This measure space is called the function space induced by the generalized Brownian motion process X determined by a(-) and &(•). Let X be an R"-valued measurable function and Y a C-valued /¿-integrable function on ( R D , B D , f i ) . Let ^(X) denote the