偏微分方程导论Introduction to partial differential equations : a computational approach

1　Setting the Scene
1.1 What Is a Differential Equation？
1.1.1 Concepts
1.2 The Solution and Its Properties
1.2.1 An Ordinary Differential Equation
1.3 A Numerical Method
1.4 Cauchy Problems
1.4.1 First Order Homogeneous Equations
1.4.2 First Order Nonhomogeneous Equations
1.4.3 The Wave Equation
1.4.4 The Heat Equation
1.5 Exercises
1.6 Projects
2 TWO-Point Boundary Value Problems
2.1 Poisson’S Equation in OBe Dimension
2.1.1 Green’S Function
2.1.2 Smoothness of the Solution
2.1.3 A Maximum Principle
2.2 A Finite Difference Approximation
2.2.1 Taylor Series
2.2.2 A System of Algebraic Equations.
2.2.3 Gaussian Elimination for Tridiagonal Linear Systems
2.2.4 Diagonal Dominant Matrices
　　2.2.5 Positive Definite Matrices
　2.3 Continuous and Discrete Solutions
2.3.1 Difference and Differential Equations.
2.3.2 Symmetry
2.3.3 Uniqueness
2.3.4 A Maximum Principle for the Discrete Problem.
2.3.5 Convergence of the Discrete Solutions
　2.4 Eigenvalue Problems
2.4.1 The COntinuous Eigenvalue Problem
2.4.2 The Discrete Eigenvalue Problem
　2.5 Exercises
　2.6 Projects
3 The Heat Equation
3.1 A Brief Overview
3.2 Separation of Variables
3.3 The Principle of Superposition
3.4 F0urier Coefficients
3.5 Other Boundary Conditions
3.6 The Neumann Problem
3.6.1 The Eigenvalue Problem
3.6.2 Particular Solutions
3.6.3 A F0rmal Solution
3.7 Energy Arguments
3.8 Differentiation of Integrals
3.9 Exercises
3.10 Projects
4 Finite Difference Schemes for the Heat Equation
4.1 An Explicit Scheme
4.2 F0urier Analysis of the Numerical Solution
4.2.1 Particular Solutions
4.2.2 Comparison of the Analytical and Discrete Solution
4.2.3 Stability Considerations
4.2.4 The Accuracy oftim Approximation
4.2.5 Summary of the Comparison
4.3 Von Neumann’S Stability Analysis
4.3.1 Particular Solutions：Continuous and Discrete
4.3.2 Examples
4.3.3 A Nonlinear Problem
4.4 An Implicit Scheme
4.4.1 Stability Anysis
4.5 Numerical Stability by Energy Arguments
4.6 Exercises
5 The Wave Equation
6 Maximum Principles
7 Poisson’s Equation in Two Space Dimensions
8 Orthogonality and General Fourier Series
9 Convergence of Fourier Series
10 The Heat Equation Revisited
11 Reaction-Diffusion Equations
12 Application of the Fourier Transform
References
Index