Numerical Solution of Ordinary and Partial Differential Equations (eBook)

Granville Sewell (Autor)

eBook Download: PDF

2014 | 1. Auflage
284 Seiten
Elsevier Science (Verlag)
978-1-4832-5914-7 (ISBN)

Infos

The Numerical Solution of Ordinary and Partial Differential Equations is an introduction to the numerical solution of ordinary and partial differential equations. Finite difference methods for solving partial differential equations are mostly classical low order formulas, easy to program but not ideal for problems with poorly behaved solutions or (especially) for problems in irregular multidimensional regions. FORTRAN77 programs are used to implement many of the methods studied. Comprised of six chapters, this book begins with a review of direct methods for the solution of linear systems, with emphasis on the special features of the linear systems that arise when differential equations are solved. The next four chapters deal with the more commonly used finite difference methods for solving a variety of problems, including both ordinary differential equations and partial differential equations, and both initial value and boundary value problems. The final chapter is an overview of the basic ideas behind the finite element method and covers the Galerkin method for boundary value problems. Examples using piecewise linear trial functions, cubic hermite trial functions, and triangular elements are presented. This monograph is appropriate for senior-level undergraduate or first-year graduate students of mathematics.

Inhalt

Front Cover 1
The Numerical Solution of Ordinary and Partial Differential Equations 4
Copyright Page 5
Table of Contents 8
Preface 12
Chapter 0. Direct Solution of Linear Systems 14
0.0 Introduction 14
0.1 General Linear Systems 14
0.2 Systems Requiring No Pivoting 18
0.3 The LU Decomposition 22
0.4 Banded Linear Systems 25
0.5 Sparse Direct Methods 31
0.6 Problems 37
Chapter 1. Initial Value Ordinary Differential Equations 42
1.0 Introduction 42
1.1 Euler's Method 44
1.2 Truncation Error, Stability and Convergence 45
1.3 Multistep Methods 51
1.4 Adams Multistep Methods 56
1.5 Backward Difference Methods for Stiff Problems 63
1.6 Runge-Kutta Methods 69
1.7 Problems 77
Chapter 2. The Initial Value Diffusion Problem 80
2.0 Introduction 80
2.1 An Explicit Method 84
2.2 Implicit Methods 89
2.3 A One-Dimensional Example 93
2.4 Multi-Dimensional Problems 96
2.5 A Diffusion-Reaction Example 102
2.6 Problems 106
Chapter 3. The Initial Value Transport and Wave Problems 110
3.0 Introduction 110
3.1 Explicit Methods for the Transport Problem 116
3.2 The Method of Characteristics 124
3.3 An Explicit Method for the Wave Equation 127
3.4 A Damped Wave Example 133
3.5 Problems 137
Chapter 4. Boundary Value Problems 142
4.0 Introduction 142
4.1 Finite Difference Methods 146
4.2 A Nonlinear Example 148
4.3 A Singular Example 151
4.4 Shooting Methods 152
4.5 Multi-Dimensional Problems 157
4.6 Successive Over-Relaxation 161
4.7 Successive Over-Relaxation Examples 165
4.8 The Conjugate Gradient Method 177
4.9 Systems of Differential Equations 183
4.10 The Eigenvalue Problem 187
4.11 The Inverse Power Method 191
4.12 Problems 196
Chapter 5. The Finite Element Method 202
5.0 Introduction 202
5.1 The Galerkin Method for Boundary Value Problems 203
5.2 An Example Using Piecewise Linear Trial Functions 207
5.3 An Example Using Cubic Hermite Trial Functions 212
5.4 A Singular Example 222
5.5 Linear Triangular Elements 229
5.6 Examples Using Triangular Elements 233
5.7 Time-Dependent Problems 240
5.8 A One-Dimensional Example 244
5.9 A Time-Dependent Example Using Triangular Elements 249
5.10 The Eigenvalue Problem 253
5.11 Eigenvalue Examples 255
5.12 Problems 260
Appendix 1: The Fourier Stability Method 266
Appendix 2: Parallel Algorithms 274
References 280
Index 282

Produkt-Details

• Erscheint lt. Verlag: 10.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• ISBN-10: 1-4832-5914-5 / 1483259145
• ISBN-13: 978-1-4832-5914-7 / 9781483259147