Numerical Methods for Differential Equations: A Computational Approach(3 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.

Table of Contents:
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.