Numerical Methods for Differential Equations: A Computational Approach (Engineering Mathematics)

With emphasis on modern techniques, Numerical Methods for Differential Equations: A Computational Approach covers the development and application of methods for the numerical solution of ordinary differential equations. Some of the methods are extended to cover partial differential equations. All techniques covered in the text are on a program disk included with the book, and are written in Fortran 90. These programs are ideal for students, researchers, and practitioners because they allow for straightforward application of the numerical methods described in the text. The code is easily modified to solve new systems of equations. Numerical Methods for Differential Equations: A Computational Approach also contains a reliable and inexpensive global error code for those interested in global error estimation. This is a valuable text for students, who will find the derivations of the numerical methods extremely helpful and the programs themselves easy to use. It is also an excellent reference and source of software for researchers and practitioners who need computer solutions to differential equations.

Differential Equations
Classification of Differential Equations
Linear Equations
Non-Linear Equations
Existence and Uniqueness of Solutions
Numerical Methods
Computer Programming
First Ideas and Single-Step Methods
Analytical and Numerical Solutions
A First Example
The Taylor Series Method
Runge-Kutta Methods
Second and Higher Order Equations
Error Considerations
Definitions
Local Truncation Error for the Taylor Series Method
Local Truncation Error for the Runge-Kutta Method
Local Truncation and Global Errors
Local Error and LTE
Runge-Kutta Methods
Error Criteria
A Third Order Formula
Fourth Order Formulae
Fifth and Higher Order Formulae
Rationale for Higher Order Formulae
Computational Examples
Step-Size Control
Steplength Prediction
Error Estimation
Local Extrapolation
Error Estimation with RK Methods
More Runge-Kutta Pairs
Application of RK Embedding
Dense Output
Construction of Continuous Extensions
Choice of Free Parameters
Higher-Order Formulae
Computational Aspects of Dense Output
Inverse Interpolation
Stability and Stiffness
Absolute Stability
Non-Linear Stability
Stiffness
Improving the Stability of RK Methods
Multistep Methods
The Linear Multistep Process
Selection of Parameters
A Third Order Implicit Formula
A Third Order Explicit Formula
Predictor-Corrector Schemes
Error Estimation
A Predictor-Corrector Program
Multistep Formulae from Quadrature
Quadrature Applied to Differential Equations
The Adams-Bashforth Formulae
The Adams-Moulton Formulae
Other Multistep Formulae
Varying the Step Size
Numerical Results
Stability of Multistep Methods
Some Numerical Experiments
Zero-Stability
Weak Stability Theory
Stability Properties of Some Formulae
Stability of Predictor-Corrector Pairs
Methods for Stiff Systems
Differentiation Formulae
Implementation of BDF Schemes
A BDF Program
Implicit Runge-Kutta Methods
A Semi-Implicit RK Program
Variable Coefficient Multistep Methods
Variable Coefficient Integrators
Practical Implementation
Step-Size Estimation
A Modified Approach
An Application of STEP90
Global Error Estimation
Classical Extrapolation
Solving for the Correction
An Example of Classical Extrapolation
The Correction Technique
Global Embedding
A Global Embedding Program
Second Order Equations
Transformation of the RK Process
A Direct Approach to the RKNG Processes
The Special Second Order Problem
Dense Output for RKN Methods
Multistep Methods
Partial Differential Equations
Finite Differences
Semi-Discretization of the Heat Equation
Highly Stable Explicit Schemes
Equations with Two Space Dimensions
Non-Linear Equations
Hyperbolic Equations
Appendix A: Programs for Single Step Methods
A Variable Step Taylor Method
An Embedded Runge-Kutta Program
A Sample RK Data File
An Alternative Runge-Kutta Scheme
Runge-Kutta with Dense Output
A Sample Continuous RK Data File
Appendix B: Multistep Programs
A Constant Steplength Program
A Variable Step Adams PC Scheme
A Variable Coefficient Multistep Package
Appendix C: Programs for Stiff Systems
A BDF Program
A Diagonally Implicit RK Program
Appendix D: Global Embedding Programs
The Gem Global Embedding Code
The GEM90 Package with Global Embedding
A Driver Program for GEM90
Appendix E: A Runge-Kutta Nyström Program
Bibliography
Index
Each chapter also includes an introduction and a section of exercise problems.