The numerical solution of ordinary and partial differential equations / Granville Sewell.

Format:

Book

Author/Creator:

Sewell, Granville.

Series:

Pure and applied mathematics (John Wiley & Sons : Unnumbered)

Pure and applied mathematics; a Wiley-Interscience series of texts, monographs, and tracts

Language:

English

Subjects (All):

Differential equations--Numerical solutions--Data processing.

Differential equations.

Differential equations, Partial--Numerical solutions--Data processing.

Differential equations, Partial.

Differential equations, Partial--Numerical solutions.

Physical Description:

xii, 333 pages : illustrations ; 25 cm.

Edition:

Second edition.

Place of Publication:

Hoboken, N.J. : Wiley-Interscience/John Wiley, [2005]

Summary:

The Second Edition of this popular text provides an insightful introduction to the use of finite difference and finite element methods for the computational solution of ordinary and partial differential equations. Readers gain a thorough understanding of the theory underlying the methods presented in the text. The author emphasizes the practical steps involved in implementing the methods, culminating in readers learning how to write programs using FORTRAN90 and MATLAB to solve ordinary and partial differential equations.

The book begins with a review of direct methods for the solution of linear systems, with an emphasis on the special features of the linear systems that arise when differential equations are solved. The following four chapters introduce and analyze the more commonly used finite difference methods for solving a variety of problems, including ordinary and partial differential equations and initial value and boundary value problems. The techniques presented in these chapters, with the aid of carefully developed exercises and numerical examples, can be easily mastered by readers.

The final chapter of the text presents the basic theory underlying the finite element method. Following the guidance offered in this chapter, readers gain a solid understanding of the method and discover how to use it to solve many problems.

A special feature of the Second Edition is Appendix A, which describes a finite element program, PDE2D, developed by the author. Readers discover how PDE2D can be used to solve difficult partial differential equation problems, including nonlinear time-dependent and steady-state systems, and linear eigenvalue systems in 1D intervals, general 2D regions, and a wide range of simple 3D regions. The software itself is available to instructors who adopt the text to share with their students.

Contents:

0 Direct Solution of Linear Systems 1

0.1 General Linear Systems 1

0.2 Systems Requiring No Pivoting 5

0.3 The LU Decomposition 8

0.4 Banded Linear Systems 11

0.5 Sparse Direct Methods 17

1 Initial Value Ordinary Differential Equations 27

1.1 Euler's Method 28

1.2 Truncation Error, Stability, and Convergence 30

1.3 Multistep Methods 35

1.4 Adams Multistep Methods 39

1.5 Backward Difference Methods for Stiff Problems 46

1.6 Runge-Kutta Methods 51

2 The Initial Value Diffusion Problem 62

2.1 An Explicit Method 65

2.2 Implicit Methods 70

2.3 A One-Dimensional Example 74

2.4 Multidimensional Problems 77

2.5 A Diffusion-Reaction Example 83

3 The Initial Value Transport and Wave Problems 91

3.1 Explicit Methods for the Transport Problem 97

3.2 The Method of Characteristics 103

3.3 An Explicit Method for the Wave Equation 108

3.4 A Damped Wave Example 113

4 Boundary Value Problems 120

4.1 Finite Difference Methods 123

4.2 A Nonlinear Example 125

4.3 A Singular Example 127

4.4 Shooting Methods 129

4.5 Multidimensional Problems 133

4.6 Successive Overrelaxation 137

4.7 Successive Overrelaxation Examples 140

4.8 The Conjugate-Gradient Method 150

4.9 Systems of Differential Equations 156

4.10 The Eigenvalue Problem 160

4.11 The Inverse Power Method 164

5 The Finite Element Method 174

5.1 The Galerkin Method 174

5.2 Example Using Piecewise Linear Trial Functions 179

5.3 Example Using Cubic Hermite Trial Functions 182

5.4 A Singular Example and The Collocation Method 192

5.5 Linear Triangular Elements 199

5.6 An Example Using Triangular Elements 202

5.7 Time-Dependent Problems 206

5.8 A One-Dimensional Example 209

5.9 Time-Dependent Example Using Triangles 213

5.10 The Eigenvalue Problem 217

5.11 Eigenvalue Examples 219

Appendix A Solving PDEs with PDE2D 235

A.1 History 235

A.2 The PDE2D Interactive User Interface 236

A.3 One-Dimensional Steady-State Problems 239

A.4 Two-Dimensional Steady-State Problems 241

A.5 Three-Dimensional Steady-State Problems 248

A.6 Nonrectangular 3D Regions 251

A.7 Time-Dependent Problems 258

A.8 Eigenvalue Problems 261

A.9 The PDE2D Parallel Linear System Solvers 262

Appendix B The Fourier Stability Method 282

Appendix C MATLAB Programs 288

Appendix D Can "ANYTHING" Happen in an Open System? 316.

Notes:

Includes bibliographical references (pages 327-329) and index.

ISBN:

0471735809

OCLC:

57694797

Publisher Number:

9780471735809