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
Arvustused
"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
Muu info
Surveys new developments in analytical and numerical methods, and relates the two through a series of partial differential equations examples
| Preface |
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xi | |
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1 Introduction to Traveling Wave Analysis |
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1 | (6) |
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1 | (3) |
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Residual Function Solutions |
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4 | (2) |
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6 | (1) |
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2 Linear Advection Equation |
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7 | (40) |
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7 | (13) |
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Solutions with Sharp Gradients or Discontinuities |
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20 | (17) |
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37 | (7) |
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44 | (3) |
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3 Linear Diffusion Equation |
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47 | (10) |
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55 | (2) |
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4 A Linear Convection Diffusion Reaction Equation |
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57 | (10) |
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65 | (2) |
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5 Diffusion Equation with Nonlinear Source Terms |
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67 | (44) |
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102 | (4) |
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106 | (3) |
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109 | (2) |
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6 Burgers-Huxley Equation |
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111 | (12) |
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118 | (3) |
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121 | (2) |
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7 Burgers-Fisher Equation |
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123 | (12) |
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128 | (5) |
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133 | (2) |
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8 Fisher-Kolmogorov Equation |
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135 | (12) |
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141 | (5) |
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146 | (1) |
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9 Fitzhugh-Nagumo Equation |
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147 | (26) |
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164 | (7) |
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171 | (2) |
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10 Kolmogorov-Petrovskii-Piskunov Equation |
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173 | (12) |
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179 | (4) |
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183 | (2) |
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11 Kuramoto-Sivashinsky Equation |
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185 | (12) |
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192 | (3) |
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195 | (2) |
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197 | (42) |
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217 | (17) |
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234 | (3) |
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237 | (2) |
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13 Regularized Long Wave Equation |
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239 | (22) |
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254 | (6) |
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260 | (1) |
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14 Extended Bernoulli Equation |
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261 | (14) |
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261 | (12) |
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273 | (2) |
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15 Hyperbolic Liouville Equation |
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275 | (18) |
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284 | (8) |
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292 | (1) |
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293 | (16) |
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301 | (6) |
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307 | (2) |
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17 Mth-Order Klein-Gordon Equation |
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309 | (30) |
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336 | (2) |
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338 | (1) |
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339 | (38) |
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370 | (4) |
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374 | (3) |
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19 Modified Wave Equation |
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377 | (14) |
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387 | (2) |
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389 | (2) |
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Appendix: Analytical Solution Methods for Traveling Wave Problems |
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391 | (50) |
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391 | (1) |
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391 | (20) |
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411 | (7) |
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A.4 Riccati Equation Method |
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418 | (11) |
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429 | (1) |
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430 | (5) |
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A.7 Additional Solutions by Addition of Arbitrary Constants |
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435 | (1) |
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435 | (1) |
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A.9 Maple Built-in Procedure TWSolutions |
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436 | (5) |
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437 | (4) |
| Index |
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441 | |
Dr. William 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 16 books, and his ODE/DAE/PDE computer routines have been accessed by some 5,000 colleges and universities, corporations, and government agencies.
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