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. www.pdecomp.net Includes a spectrum of applications in science, engineering, applied mathematicsPresents a combination of numerical and analytical methodsProvides transportable computer codes in Matlab and Maple
Author(s): Graham W. Griffiths, William E. Schiesser
Edition: 1
Publisher: Elsevier, Academic Press
Year: 2011
Language: English
Pages: 445
Tags: Математика;Вычислительная математика;
Traveling Wave Analysis of Partial Differential Equations......Page 3
Dedication......Page 5
Preface......Page 6
1 Introduction to Traveling Wave Analysis......Page 9
2 Linear Advection Equation......Page 15
3 Linear Diffusion Equation......Page 54
4 A Linear Convection Diffusion Reaction Equation......Page 63
5 Diffusion Equation with Nonlinear Source Terms......Page 72
6 Burgers–Huxley Equation......Page 116
7 Burgers–Fisher Equation......Page 128
8 Fisher–Kolmogorov Equation......Page 139
9 Fitzhugh–Nagumo Equation......Page 151
10 Kolmogorov–Petrovskii–Piskunov Equation......Page 177
11 Kuramoto–Sivashinsky Equation......Page 189
12 Kawahara Equation......Page 201
13 Regularized Long-Wave Equation......Page 243
14 Extended Bernoulli Equation......Page 265
15 Hyperbolic Liouville Equation......Page 278
16 Sine-Gordon Equation......Page 296
17 Mth-Order Klein–Gordon Equation......Page 311
18 Boussinesq Equation......Page 341
19 Modified Wave Equation......Page 378
A Analytical Solution Methods forTraveling Wave Problems......Page 391
Index......Page 440
