偏微分方程与数值方法 / 国外数学名著系列
￥118.00定价
作者： [瑞典]拉松
出版时间：2006年09月
出版社：科学出版社
- 科学出版社
- 9787030166760
- 356509
- 2006年09月
内容简介
《偏微分方程与数值方法》的作者Stig Larsson现任瑞典Chalmers大学数学系教授、瑞典科学院院士。本书将微分方程的数学分析及有限差分理论和有限元方法结合起来，讲述线性偏微分方程的基本理论及其常用的数值解法。分别用三章阐述椭圆型、抛物型及双曲型偏微分方程，一章关于其数学理论，一章关于其有限差分方法，一章关于其有限元方法。在论述椭圆型方程之前，讲述常微分方程的两点边值问题；类似地，在论述抛物型和双曲型发展问题之前，讲述常微分方程的初值问题。另有一章研究椭圆型特征值问题和特征函数的展开。附录提供了阅读本书所要求的线性泛函分析及索伯列夫空间的背景知识。阅读本书不需要高深的数学分析和泛函分析知识。本书适用于应用数学专业和工程专业的高年级本科生和低年级研究生。
目录
1 Introduction
1.1 Background
1.2 Notation and Mathematical Preliminaries
1.3 Physical Derivation of the Heat Equation
1.4 Problems
2 A Two-Point Boundary Value Problem
2.1 The Maximum Principle
2.2 Green's Function
2.3 Variational Formulation
2.4 Problems
3 Elliptic Equations
3.1 Preliminaries
3.2 A Maximum Principle
3.3 Dirichlet's Problem for a Disc. Poisson's Integral
3.4 Fundamental Solutions. Green's Function
3.5 Variational Formulation of the Dirichlet Problem
3.6 A Neumann Problem
3.7 Regularity
3.8 Problems
4 Finite Difference Methods for Elliptic Equations
4.1 A Two-Point Boundary Value Problem
4.2 Poisson's Equation
4.3 Problems
5 Finite Element Methods for Elliptic Equations
5.1 A Two-Point boundary Value Problem
5.2 A Model Problem in the Plane
5.3 Some Facts from Approximation Theory
5.4 Error Estimates
5.5 An A Posteriori Error Estimate
5.6 Numerical Integration
5.7 A Mixed Finite Element Method
5.8 Problems
6 The Elliptic Eigenvalue Problem
6.1 Eigenfunction Expansions
6.2 Numerical Solution of the Eigenvalue Problem
6.3 Problems
7 Initial-Value Problems for ODEs
7.1 The Initial Value Problem for a Linear System
7.2 Numerical Solution of ODEs
7.3 Problems
8 Parabolic Equations
8.1 The Pure Initial Value Problem
8.2 Solution by Eigenfunction Expansion
8.3 Variational Formulation. Energy Estimates
8.4 A Maximum Principle
8.5 Problems
9 Finite Difference Methods for Parabolic Problems
9.1 The Pure Initial Value Problem
9.2 The Mixed Initial-Boundary Value Problem
9.3 Problems
10 The Finite Element Method for a Parabolic Problem
10.1 The Semidiscrete Galerkin Finite Element Method
10.2 Some Completely Discrete Schemes
10.3 Problems
11 Hyperbolic Equations
11.1 Characteristic Directions and Surfaces
11.2 The Wave Equation
11.3 First Order Scalar Equation
11.4 Symmetric Hyperbolic Systems
11.5 Problems
12 Finite Difference Methods for Hyperbolic Equations
12.1 First Order Scalar Equations
12.2 Symmetric Hyperbolic Systems
12.3 The Wendroff Box Scheme
12.4 Problems
13 The Finite Element Method for Hyperbolic Equations
13.1 The Wave Equation
13.2 First Order Hyperbolic Equations
13.3 Problems
14 Some Other Classes of Numerical Methods
14.1 Collocation Methods
14.2 Spectral Methods
14.3 Finite Volume Methods
14.4 Boundary Element Methods
14.5 Problems
A Some Tools from Mathematical Analysis
A.1 Abstract Linear Spaces
A.2 Function Spaces
A.3 The Fourier Transform
A.4 Problems
B Orientation on Numerical Linear Algebra
B.1 Direct Methods
B.2 Iterative Methods. Relaxation, Overrelaxation, and Acceleration
B.3 Preconditioned Conjugate Gradient Methods
B.4 Preconditioned Conjugate Gradient Methods
B.5 Multigrid and Domain Decomposition Methods
Bibliography
Index
1.1 Background
1.2 Notation and Mathematical Preliminaries
1.3 Physical Derivation of the Heat Equation
1.4 Problems
2 A Two-Point Boundary Value Problem
2.1 The Maximum Principle
2.2 Green's Function
2.3 Variational Formulation
2.4 Problems
3 Elliptic Equations
3.1 Preliminaries
3.2 A Maximum Principle
3.3 Dirichlet's Problem for a Disc. Poisson's Integral
3.4 Fundamental Solutions. Green's Function
3.5 Variational Formulation of the Dirichlet Problem
3.6 A Neumann Problem
3.7 Regularity
3.8 Problems
4 Finite Difference Methods for Elliptic Equations
4.1 A Two-Point Boundary Value Problem
4.2 Poisson's Equation
4.3 Problems
5 Finite Element Methods for Elliptic Equations
5.1 A Two-Point boundary Value Problem
5.2 A Model Problem in the Plane
5.3 Some Facts from Approximation Theory
5.4 Error Estimates
5.5 An A Posteriori Error Estimate
5.6 Numerical Integration
5.7 A Mixed Finite Element Method
5.8 Problems
6 The Elliptic Eigenvalue Problem
6.1 Eigenfunction Expansions
6.2 Numerical Solution of the Eigenvalue Problem
6.3 Problems
7 Initial-Value Problems for ODEs
7.1 The Initial Value Problem for a Linear System
7.2 Numerical Solution of ODEs
7.3 Problems
8 Parabolic Equations
8.1 The Pure Initial Value Problem
8.2 Solution by Eigenfunction Expansion
8.3 Variational Formulation. Energy Estimates
8.4 A Maximum Principle
8.5 Problems
9 Finite Difference Methods for Parabolic Problems
9.1 The Pure Initial Value Problem
9.2 The Mixed Initial-Boundary Value Problem
9.3 Problems
10 The Finite Element Method for a Parabolic Problem
10.1 The Semidiscrete Galerkin Finite Element Method
10.2 Some Completely Discrete Schemes
10.3 Problems
11 Hyperbolic Equations
11.1 Characteristic Directions and Surfaces
11.2 The Wave Equation
11.3 First Order Scalar Equation
11.4 Symmetric Hyperbolic Systems
11.5 Problems
12 Finite Difference Methods for Hyperbolic Equations
12.1 First Order Scalar Equations
12.2 Symmetric Hyperbolic Systems
12.3 The Wendroff Box Scheme
12.4 Problems
13 The Finite Element Method for Hyperbolic Equations
13.1 The Wave Equation
13.2 First Order Hyperbolic Equations
13.3 Problems
14 Some Other Classes of Numerical Methods
14.1 Collocation Methods
14.2 Spectral Methods
14.3 Finite Volume Methods
14.4 Boundary Element Methods
14.5 Problems
A Some Tools from Mathematical Analysis
A.1 Abstract Linear Spaces
A.2 Function Spaces
A.3 The Fourier Transform
A.4 Problems
B Orientation on Numerical Linear Algebra
B.1 Direct Methods
B.2 Iterative Methods. Relaxation, Overrelaxation, and Acceleration
B.3 Preconditioned Conjugate Gradient Methods
B.4 Preconditioned Conjugate Gradient Methods
B.5 Multigrid and Domain Decomposition Methods
Bibliography
Index
