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Although the Partial Differential Equations (PDE) models that are now studied are usually beyond traditional mathematical analysis, the numerical methods that are being developed and used require testing and validation. This is often done with PDEs that have known, exact, analytical solutions. The development of analytical solutions is also an active area of research, with many advances being reported recently, particularly traveling wave solutions for nonlinear evolutionary PDEs. Thus, the current development of analytical solutions directly supports the development of numerical methods by providing a spectrum of test problems that can be used to evaluate numerical methods.
This book surveys some of these new developments in analytical and numerical methods, and relates the two through a series of PDE examples. The PDEs that have been selected are largely "named'' since they carry the names of their original contributors. These names usually signify that the PDEs are widely recognized and used in many application areas. The authors’ intention is to provide a set of numerical and analytical methods based on the concept of a traveling wave, with a central feature of conversion of the PDEs to ODEs.
The Matlab and Maple software will be available for download from this website shortly.
- Includes a spectrum of applications in science, engineering, applied mathematics
- Presents a combination of numerical and analytical methods
- Provides transportable computer codes in Matlab and Maple
Scientists, Engineers, Applied Mathematicians, and Economists who use PDE models
1. Introduction to Traveling Wave Analysis
2. Linear Advection Equation
3. Linear Diffusion Equation
4. A Linear Convection Diffusion Reaction Equation
5. Diffusion Equation with Onlinear Source Terms
6. Burgers–Huxley Equation
7. Burgers–Fisher Equation
8. Fisher–Kolmogorov Equation
9. Fitzhugh–Nagumo Equation
10. Kolmogorov–Petrovskii–Piskunov Equation
11. Kuramoto–Sivashinsky Equation
12. Kawahara Equation
13. Regularized Long-Wave Equation
14. Extended Bernoulli Equation
15. Hyperbolic Liouville Equation
16. Sine-Gordon Equation
17. Mth-Order Klein–Gordon Equation
18. Boussinesq Equation
19. Modified Wave Equation
A. Analytical Solution Methods for Traveling Wave Problems
- No. of pages:
- © Academic Press 2011
- 9th December 2010
- Academic Press
- eBook ISBN:
- Hardcover ISBN:
The R routines are available from http://www.lehigh.edu/~wes1/pd_download Queries about the routines can be directed to firstname.lastname@example.org W.E. Schiesser is Emeritus McCann Professor of Chemical and Biomolecular Engineering and Professor of Mathematics at Lehigh University. He holds a PhD from Princeton University and a ScD (hon) from the University of Mons, Belgium. His research is directed toward numerical methods and associated software for ordinary, differential-algebraic and partial differential equations (ODE/DAE/PDEs), and the development of mathematical models based on ODE/DAE/PDEs. He is the author or coauthor of more than 14 books, and his ODE/DAE/PDE computer routines have been accessed by some 5,000 colleges and universities, corporations and government agencies.
"This book surveys some of the new developments in analytical and numerical computer solution methods for partial differential equations with applications to physical, chemical, and biological problems. The development of analytical solutions directly supports the development of numerical methods by providing a spectrum of test problems that can be used to evaluate numerical methods."--Zentralblatt MATH 1228-1
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