●Preface
Acknowledgments
1 Introduction
1.1 Model Equations
1.1.1 Advection Equation
1.1.2 Diffusion Equation
1.1.3 Laplace's Equation
1.1.4 Wave Equation
1.2 PDEs Are Everywhere
1.3 Exercise
2 First-Order PDEs
2.1 Constant Coefficient Equations
2.2 Linear Equations
2.3 Method of Characteristics
2.4 Quasilinear Equations
2.5 Higher-Dimensional Equations
2.6 Fully Nonlinear First-Order Equations
2.6.1 Method of Characteristics
2.6.2 Charpit's Method
2.7 Exercises
3 Second-Order Linear PDEs
3.1 Introduction
3.2 Standard Forms
3.2.1 Parabolic Standard Form
3.2.2 Hyperbolic Standard Form
3.2.3 Modified Hyperbolic Form
3.2.4 Regular Hyperbolic Form
3.2.5 Elliptic Standard Form
3.3 The Wave Equation
3.4 Exercises
4 Fourier Series
4.1 Fourier Series
4.2 Fourier Series on [-π,π]
4.3 Fourier Series on [-L,L]
4.4 Odd and Even Extensions
4.4.1 Sine Series
4.4.2 Cosine Series
4.5 Exercises
5 Separation of Variables
5.1 The Heat Equation
5.1.1 Nonhomogeneous Boundary Conditions
5.1.2 Nonhomogeneous Equations
5.1.3 Equations with a Solution-Dependent Source Term
5.1.4 Equations with a Solution-Dependent Convective Term
5.2 Laplace's Equation
5.2.1 Laplace's Equation on an Arbitrary Rectangular Domain
5.3 The Wave Equation
5.4 Exercises
6 Fourier Transform
6.1 Fourier Transform
6.2 Fourier Sine and Cosine Transforms
6.3 Exercises
7 Solutions
Author's Biography
編輯手記

這本書是對解偏微分方程方法的介紹。在介紹了可以被認為是應用數學基石的主要四種偏微分方程之後，作者還介紹了其他的多種偏微分方程，這些方程來自自然科學和工程的各個領域，是進入這一奇妙學科的跳板。具體包括：一階偏微分方程、二階偏微分方程、傅裡葉級數、分離變量和傅裡葉變換。在這些章節中，作者將介紹解決一階和二階偏微分方程的方法。每一章的結尾都有一繫列的練習來說明每一章的內容。本書可以作為任何介紹偏微分方程課程的教科書，這些課程通常在科學和工程項目中都有，並且已經在中阿肯色大學開設了十多年。