The first two editions of An Introduction to Partial Differential Equations with MATLAB® gained popularity among instructors and students at various universities throughout the world. Plain mathematical language is used in a friendly manner to provide a basic introduction to partial differential equations (PDEs).

Suitable for a one- or two-semester introduction to PDEs and Fourier series, the book strives to provide physical, mathematical, and historical motivation for each topic. Equations are studied based on method of solution, rather than on type of equation.

This third edition of this popular textbook updates the structure of the book by increasing the role of the computational portion, compared to previous editions. The redesigned content will be extremely useful for students of mathematics, physics, and engineering who would like to focus on the practical aspects of the study of PDEs, without sacrificing mathematical rigor. The authors have maintained flexibility in the order of topics.

In addition, students will be able to use what they have learned in some later courses (for example, courses in numerical analysis, optimization, and PDE-based programming). Included in this new edition is a substantial amount of material on reviewing computational methods for solving ODEs (symbolically and numerically), visualizing solutions of PDEs, using MATLAB®'s symbolic programming toolbox, and applying various schemes from numerical analysis, along with suggestions for topics of course projects.

Students will use sample MATLAB® or Python codes available online for their practical experiments and for completing computational lab assignments and course projects.

Table of Contents

Chapter 1. Introduction

What are Partial Differential Equations?

PDEs We Can Already Solve

Initial and Boundary Conditions

Linear PDEs – Definitions

Linear PDEs – The Principle of Superposition

The Method of Characteristics I

The Method of Characteristics II

Separation of Variables for Linear, Homogeneous PDEs

Eigenvalue Problems

Chapter 2. 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

D'Alembert's Solution for the Infinite String Problem

General Second-Order Linear PDEs and Characteristics

Using Separation of Variables to Solve the Big Three PDEs

Chapter 3. Using MATLAB for Solving Differential Equations and Visualizing Solutions

Visualizing Solutions of ODEs

Symbolic Math Toolbox for Solving ODEs

Solving BVPs Numerically Using bvp4(5)c

Solving PDEs Numerically Using pdepe

Exercises for Chapter 3

Lab Assignment #1: Review Chapters 1-3

Chapter 4. Fourier Series

Introduction

Properties of Sine and Cosine

The Fourier Series

The Fourier Series, Continued

Fourier Sine and Cosine Series

Chapter 5. Solving the Big Three PDEs on Finite Domains

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

Chapter 6. Review of Numerical Methods for Solving ODEs

Approaches to Solving First-Order IVPs

Numerical Solutions Using Euler's Method

Numerical Solutions Using Runge–Kutta Methods

Solving Higher-Order ODEs Numerically

Implicit Approximations for BVPs

Exercises for Chapter 6

Chapter 7. Solving PDEs Using Finite Difference Approximations

Numerical Solutions for the Heat Equation

Explicit Scheme for the Wave Equation

Numerical Schemes for Laplace's Equation

Numerical Solution of First-Order PDEs

Exercises for Chapter 7

Lab Assignment #2: Review Chapters 6-7

Lab Assignment #3: Review Chapters 4-7

Chapter 8. Integral Transforms

The Laplace Transform for PDEs

Fourier Sine and Cosine Transforms

The Fourier Transform

The Infinite and Semi-Infinite Heat Equations

Other Integral Transforms and Integral Equations

Chapter 9. Using MATLAB's Symbolic Math Toolbox with Integral Transforms

Integral Transforms via Symbolic Programming

Solving ODEs Using the Laplace Transform in MATLAB

Symbolic Solution of PDEs Using the Laplace Transform

Symbolic Solution of PDEs Using the Fourier Transform

Exercises for Chapter 9

Lab Assignment #4: Review Chapters 8-9

Chapter 10. 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

Interlude 1: Bessel Functions

Interlude 2: The Legendre Polynomials

The Wave and Heat Equations in Polar Coordinates

Problems in Spherical Coordinates

The Infinite Wave Equation and Multiple Fourier Transforms

MATLAB Exercises for Chapter 10

Lab Assignment #5: Review Chapters 7 & 10

Chapter 11. Overview of Spectral, Finite Element, and Finite Volume Methods

Spectral Methods

Finite Element Methods

Finite Volume Methods

Exercises for Chapter 11

Appendix A: Important Definitions and Theorems

Appendix B: Bessel's Equation and the Method of Frobenius

Appendix C: A Menagerie of PDEs

Appendix D: Review of Math with MATLAB

Appendix E: Answers to Selected Exercises

References

Index

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