The Numerical Solution of Ordinary and Partial Differential Equations - 1st Edition - ISBN: 9780126374759, 9781483259147

The Numerical Solution of Ordinary and Partial Differential Equations

1st Edition

Authors: Granville Sewell
eBook ISBN: 9781483259147
Imprint: Academic Press
Published Date: 28th September 1988
Page Count: 284
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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.

Table of Contents


0. Direct Solution of Linear Systems

0.0 Introduction

0.1 General Linear Systems

0.2 Systems Requiring No Pivoting

0.3 The LU Decomposition

0.4 Banded Linear Systems

0.5 Sparse Direct Methods

0.6 Problems

1. Initial Value Ordinary Differential Equations

1.0 Introduction

1.1 Euler's Method

1.2 Truncation Error, Stability and Convergence

1.3 Multistep Methods

1.4 Adams Multistep Methods

1.5 Backward Difference Methods for Stiff Problems

1.6 Runge-Kutta Methods

1.7 Problems

2. The Initial Value Diffusion Problem

2.0 Introduction

2.1 An Explicit Method

2.2 Implicit Methods

2.3 A One-Dimensional Example

2.4 Multi-Dimensional Problems

2.5 A Diffusion-Reaction Example

2.6 Problems

3. The Initial Value Transport and Wave Problems

3.0 Introduction

3.1 Explicit Methods for the Transport Problem

3.2 The Method of Characteristics

3.3 An Explicit Method for the Wave Equation

3.4 A Damped Wave Example

3.5 Problems

4. Boundary Value Problems

4.0 Introduction

4.1 Finite Difference Methods

4.2 A Nonlinear Example

4.3 A Singular Example

4.4 Shooting Methods

4.5 Multi-Dimensional Problems

4.6 Successive Over-Relaxation

4.7 Successive Over-Relaxation Examples

4.8 The Conjugate Gradient Method

4.9 Systems of Differential Equations

4.10 The Eigenvalue Problem

4.11 The Inverse Power Method

4.12 Problems

5. The Finite Element Method

5.0 Introduction

5.1 The Galerkin Method for Boundary Value Problems

5.2 An Example Using Piecewise Linear Trial Functions

5.3 An Example Using Cubic Hermite Trial Functions

5.4 A Singular Example

5.5 Linear Triangular Elements

5.6 Examples Using Triangular Elements

5.7 Time-Dependent Problems

5.8 A One-Dimensional Example

5.9 A Time-Dependent Example Using Triangular Elements

5.10 The Eigenvalue Problem

5.11 Eigenvalue Examples

5.12 Problems

Appendix 1

Appendix 2




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© Academic Press 1988
Academic Press
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About the Author

Granville Sewell

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