Partial Differential Equations with Fourier Series and Boundary Value Problems [2 ed.] 0131480960, 9780131480964

Table of contents :
Front Cover
Title Page
Copyright Page
Table of Contents
Preface
1. A Preview of Applications and Techniques
1.1 What Is a Partial Differential Equation?
1.2 Solving and Interpreting a Partial Differential Equation
2. Fourier Series
2.1 Periodic Functions
2.2 Fourier Series
2.3 Fourier Series of Functions with Arbitrary Periods
2.4 Half-Range Expansions: The Cosine and Sine Series
2.5 Mean Square Approximation and Parseval's Identity
2.6 Complex Form of Fourier Series
2.7 Forced Oscillations
Supplement on Convergence
2.8 Proof of the Fourier Series Representation Theorem
2.9 Uniform Convergence and Fourier Series
2.10 Dirichlet Test and Convergence of Fourier Series
3. Partial Differential Equations in Rectangular Coordinates
3.1 Partial Differential Equations in Physics and Engineering
3.2 Modeling: Vibrating Strings and the Wave Equation
3.3 Solution of the One Dimensional Wave Equation: The Method of Separation of Variables
3.4 D'Alembert's Method
3.5 The One Dimensional Heat Equation
3.6 Heat Conduction in Bars: Varying the Boundary Conditions
3.7 The Two Dimensional Wave and Heat Equations
3.8 Laplace's Equation in Rectangular Coordinates
3.9 Poisson's Equation: The Method of Eigenfunction Expansions
3.10 Neumann and Robin Conditions
3.11 The Maximum Principle
4. Partial Differential Equations in Polar and Cylindrical Coordinates
4.1 The Laplacian in Various Coordinate Systems
4.2 Vibrations of a Circular Membrane: Symmetric Case
4.3 Vibrations of a Circular Membrane: General Case
4.4 Laplace's Equation in Circular Regions
4.5 Laplace's Equation in a Cylinder
4.6 The Helmholtz and Poisson Equations
Supplement on Bessel Functions
4.7 Bessel's Equation and Bessel Functions
4.8 Bessel Series Expansions
4.9 Integral Formulas and Asymptotics for Bessel Functions
5. Partial Differential Equations in Spherical Coordinates
5.1 Preview of Problems and Methods
5.2 Dirichlet Problems with Symmetry
5.3 Spherical Harmonics and the General Dirichlet Problem
5.4 The Helmholtz Equation with Applications to the Poisson, Heat,and Wave Equations
Supplement on Legendre Functions
5.5 Legendre's Differential Equation
5.6 Legendre Polynomials and Legendre Series Expansions
5.7 Associated Legendre Functions and Series Expansions
6. Sturm-Liouville Theory with Engineering Applications
6.1 Orthogonal Functions
6.2 Sturm-Liouville Theory
6.3 The Hanging Chain
6.4 Fourth Order Sturm-Liouville Theory
6.5 Elastic Vibrations and Buckling of Beams
6.6 The Biharmonic Operator
6.7 Vibrations of Circular Plates
7. The Fourier Transform and Its Applications
7.1 The Fourier Integral Representation
7.2 The Fourier Transform
7.3 The Fourier Transform Method
7.4 The Heat Equation and Gauss's Kernel
7.5 A Dirichlet Problem and the Poisson Integral Formula
7.6 The Fourier Cosine and Sine Transforms
7.7 Problems Involving Semi-Infinite Intervals
7.8 Generalized Functions
7.9 The Nonhomogeneous Heat Equation
7.10 Duhamel's Principle
8. The Laplace and Hankel Transforms with Applications
8.1 The Laplace Transform
8.2 Further Properties of the Laplace Transform
8.3 The Laplace Transform Method
8.4 The Hankel Transform with Applications
9. Finite Difference Numerical Methods
9.1 The Finite Difference Method for the Heat Equation
9.2 The Finite Difference Method for the Wave Equation
9.3 The Finite Difference Method for Laplace's Equation
9.4 Iteration Methods for Laplace's Equation
10. Sampling and Discrete Fourier Analysis with Applications to Partial Differential Equations
10.1 The Sampling Theorem
10.2 Partial Differential Equations and the Sampling Theorem
10.3 The Discrete and Fast Fourier Transforms
10.4 The Fourier and Discrete Fourier Transforms
11. An Introduction to Quantum Mechanics
11.1 Schrodinger's Equation
11.2 The Hydrogen Atom
11.3 Heisenberg's Uncertainty Principle
Supplement on Orthogonal Polynomials
11.4 Hermite and Laguerre Polynomials
12. Green's Functions and Conformal Mappings
12.1 Green's Theorem and Identities
12.2 Harmonic Functions and Green's Identities
12.3 Green's Functions
12.4 Green's Functions for the Disk and the Upper Half-Plane
12.5 Analytic Functions
12.6 Solving Dirichlet Problems with Conformal Mappings
12.7 Green's Functions and Conformal Mappings
12.8 Neumann Functions and the Solution of Neumann Problems
A. Ordinary Differential Equations:Review of Concepts and Methods
A.l Linear Ordinary Differential Equations
A.2 Linear Ordinary Differential Equations with Constant Coefficients
A.3 Linear Ordinary Differential Equations with Nonconstant Coefficients
A.4 The Power Series Method, Part I
A.5 The Power Series Method, Part II
A.6 The Method of Frobenius
B. Tables of Transforms
B.1 Fourier Transforms
B.2 Fourier Cosine Transforms
B.3 Fourier Sine Transforms
B.4 Laplace Transforms
References
Answers to Selected Exercises
Index
Solutions Manual
1. A Preview of Applications and Techniques
1.1 What Is a Partial Differential Equation?
1.2 Solving and Interpreting a Partial Differential Equation
2. Fourier Series
2.1 Periodic Functions
2.2 Fourier Series
2.3 Fourier Series of Functions with Arbitrary Periods
2.4 Half-Range Expansions: The Cosine and Sine Series
2.5 Mean Square Approximation and Parseval’s Identity
2.6 Complex Form of Fourier Series
2.7 Forced Oscillations
Supplement on Convergence
2.9 Uniform Convergence and Fourier Series
2.10 Dirichlet Test and Convergence of Fourier Series
3.Partial Differential Equations in Rectangular Coordinates
3.1 Partial Differential Equations in Physics and Engineering
3.3 Solution of the One Dimensional Wave Equation: The Method of Separation of Variables
3.4 D’Alembert’s Method
3.5 The One Dimensional Heat Equation
3.6 Heat Conduction in Bars: Varying the Boundary Conditions
3.7 The Two Dimensional Wave and Heat Equations
3.8 Laplace’s Equation in Rectangular Coordinates
3.9 Poisson’s Equation: The Method of Eigenfunction Expansions
3.10 Neumann and Robin Conditions
4. Partial Differential Equations in Polar and Cylindrical Coordinates
4.1 The Laplacian in Various Coordinate Systems
4.2 Vibrations of a Circular Membrane: Symmetric Case
4.3 Vibrations of a Circular Membrane: General Case
4.4 Laplace’s Equation in Circular Regions
4.5 Laplace’s Equation in a Cylinder
4.6 The Helmholtz and Poisson Equations
Supplement on Bessel Functions
4.7 Bessel’s Equation and Bessel Functions
4.8 Bessel Series Expansions
4.9 Integral Formulas and Asymptotics for Bessel Functions
5. Partial Differential Equations in Spherical Coordinates
5.1 Preview of Problems and Methods
5.2 Dirichlet Problems with Symmetry
5.3 Spherical Harmonics and the General Dirichlet Problem
5.4 The Helmholtz Equation with Applications to the Poisson, Heat,and Wave Equations
Supplement on Legendre Functions
5.5 Legendre’s Differential Equation
5.6 Legendre Polynomials and Legendre Series Expansions
6. Sturm–Liouville Theory with Engineering Applications
6.1 Orthogonal Functions
6.2 Sturm–Liouville Theory
6.3 The Hanging Chain
6.4 Fourth Order Sturm–Liouville Theory
6.6 The Biharmonic Operator
6.7 Vibrations of Circular Plates
7. The Fourier Transform and Its Applications
7.1 The Fourier Integral Representation
7.2 The Fourier Transform
7.3 The Fourier Transform Method
7.4 The Heat Equation and Gauss’s Kernel
7.5 A Dirichlet Problem and the Poisson Integral Formula
7.6 The Fourier Cosine and Sine Transforms
7.7 Problems Involving Semi-Infinite Intervals
7.8 Generalized Functions
7.9 The Nonhomogeneous Heat Equation
7.10 Duhamel’s Principle

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