The Numerical Solution Of Ordinary And Partial Differential Equations 2nd Edition by Granville Sewell ISBN 0471742007 9780471742005
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ISBN 10: 0471742007
ISBN 13: 9780471742005
Author: Granville Sewell
Learn to write programs to solve ordinary and partial differential equations
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 themethods 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(r) 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 easilymastered 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.
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 Multidimensional 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 Multidimensional Problems
4.6 Successive Overrelaxation
4.7 Successive Overrelaxation 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
5.2 Example Using Piecewise Linear Trial Functions
5.3 Example Using Cubic Hermite Trial Functions
5.4 A Singular Example and The Collocation Method
5.5 Linear Triangular Elements
5.6 An Example Using Triangular Elements
5.7 Time-Dependent Problems.
5.8 A One-Dimensional Example
5.9 Time-Dependent Example Using Triangles
5.10 The Eigenvalue Problem
5.11 Eigenvalue Examples
5.12 Problems
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Tags: Granville Sewell, The Numerical Solution, Ordinary And Partial, Differential Equations