An Introduction to Partial Differential Equations with MATLAB

An Introduction to Partial Differential Equations with MATLAB exposes the basic ideas critical to the study of PDEs-- characteristics, integral transforms, Green’s functions, and, most importantly, Fourier series and related topics. The author approaches the subject from a motivational perspective, detailing equations only after a need for them has been established. He uses MATLAB® software to solve exercises and to generate tables and figures. This volume includes examples of many important PDEs and their applications.

The first chapter introduces PDEs and makes analogies to familiar ODE concepts, then strengthens the connection by exploring the method of separation of variables. Chapter 2 examines the “Big Three” PDEs-- the heat, wave, and Laplace equations, and is followed by chapters explaining how these and other PDEs on finite intervals can be solved using the Fourier series for arbitrary initial and boundary conditions.

Chapter 5 investigates characteristics for both first- and second-order linear PDEs, the latter revealing how the Big Three equations are important far beyond their original application to physical problems. The book extends the Fourier method to functions on unbounded domains, gives a brief introduction to distributions, then applies separation of variables to PDEs in higher dimensions, leading to the special funtions, including the orthogonal polynomials.

Other topics include Sturm-Liouville problems, adjoint and self-adjoint problems, the application of Green’s functions to solving nonhomogeneous PDEs, and an examination of practical numerical methods used by engineers, including the finite difference, finite element, and spectral methods.

Introduction

What are Partial Differential Equations?
PDEs We Can Already Solve
Initial and Boundary Conditions
Linear PDEs--Definitions
Linear PDEs--The Principle of Superposition
Separation of Variables for Linear, Homogeneous PDEs
Eigenvalue Problems

The Big Three PDEs

Second-Order,  Linear, Homogeneous PDEs with Constant Coefficients
The Heat Equation and Diffusion
The Wave Equation and the Vibrating String
Initial and Boundary Conditions for the Heat and Wave Equations
Laplace's Equation--The Potential Equation
Using Separation of Variables to Solve the Big Three PDEs

Fourier Series

Introduction
Properties of Sine and Cosine
The Fourier Series
The Fourier Series, Continued
The Fourier Series---Proof of Pointwise Convergence
Fourier Sine and Cosine Series
Completeness
Solving the Big Three PDE s
Solving the Homogeneous Heat Equation for a Finite Rod
Solving the Homogeneous Wave Equation for a Finite String
Solving the Homogeneous Laplace's Equation on a Rectangular
Domain
Nonhomogeneous Problems
Characteristicsfor Linear PDE s
First-Order PDEs with Constant Coefficients
First-Order PDEs with Variable Coefficients
D'Alembert's Solution for the Wave Equation--The Infinite
String
Characteristics for Semi-Infinite and Finite String Problems
General Second-Order Linear PDEs and Characteristics

Integral Transforms

The Laplace Transform for PDEs
Fourier Sine and Cosine Transforms
The Fourier Transform
The Infinite and Semi-Infinite Heat Equations
Distributions, the Dirac Delta Function and Generalized Fourier
Transforms
Proof of the Fourier Integral Formula
Bessel Functions and Orthogonal Polynomials
The Special Functions and Their Differential Equations
Ordinary Points and Power Series Solutions; Chebyshev, Hermite
and Legendre Polynomials
The Method of Frobenius; Laguerre Polynomials
Interlude: The Gamma Function
Bessel Functions
Recap: A List of Properties of Bessel Functions and Orthogonal
Polynomials
Sturm-Liouville Theory and Generalized Fourier Series
Sturm-Liouville Problems
Regular and Periodic Sturm-Liouville Problems
Singular Sturm-Liouville Problems; Self-Adjoint Problems
The Mean-Square or L 2 Norm and Convergence in the Mean
Generalized Fourier Series; Parseval's Equality and Completeness
PDEs in Higher Dimensions
PDEs in Higher Dimensions: Examples and Derivations
The Heat and Wave Equations on a Rectangle; Multiple Fourier
Series
Laplace's Equation in Polar Coordinates; Poisson's Integral
Formula
The Wave and Heat Equations in Polar Coordinates
Problems in Spherical Coordinates
The Infinite Wave Equation and Multiple Fourier Transforms
Postlude: Eigenvalues and Eigenfunctions of the Laplace Operator;
Green's Identities for the Laplacian
Nonhomogeneous Problems and Green's Functions
Green's Functions for ODEs
Green's Function and the Dirac Delta Function
Green's Functions for Elliptic PDEs (I): Poisson's Equation in
Two Dimensions
Green's Functions for Elliptic PDEs (II): Poisson's Equation in
Three Dimensions; the Helmholtz Equation
Green's Function's for Equations of Evolution

Numerical Methods

Finite Difference Approximations for ODEs
Finite Difference Approximations for PDEs
Spectral Methods and the Finite Element Method
References
Uniform Convergence; Differentiation and Integration of Fourier Series
Important Theorems: Limits, Derivatives, Integrals, Series, and Interchange of Operations
Existence and Uniqueness Theorems
A Menagerie of PDEs
MATLAB Code for Figures and Exercises
Answers to Selected Exercises