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.
- 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
Author(s): Graham W. Griffiths, William E. Schiesser
Edition: 1
Publisher: Academic Press
Year: 2011
Language: English
Pages: 461
cover......Page 1
Traveling Wave Analysis of Partial Differential Equations......Page 3
Copyright......Page 4
Dedication......Page 5
Preface......Page 6
Traveling Wave Solutions......Page 9
Residual Function Solutions......Page 12
References......Page 14
Smooth Solutions......Page 15
Solutions with Sharp Gradients or Discontinuities......Page 28
Appendix......Page 45
How They Work......Page 46
Limiter Functions......Page 48
References......Page 52
3.
Linear Diffusion Equation......Page 54
Reference......Page 62
4.
A Linear Convection Diffusion Reaction Equation......Page 63
Reference......Page 71
Case 1: Linear PDE, Dirichlet BCs (ncase=1, nbc=1, a=0.1, b=c=0)......Page 72
Case 2: Linear PDE, Neumann BCs (ncase=1, nbc=2, a=0.1, b=c=0)......Page 84
Case 3: Linear PDE, Third-Type BCs (ncase=1, nbc=3, a=0.1, b=c=0)......Page 87
Case 4: Linear PDE, Nonlinear Third-Type BCs (ncase=1, nbc=4, a=0.1, b=c=0)......Page 89
Case 5: Linear PDE, Analytical Neumann BCs (ncase=1, nbc=5, a=0.1, b=c=0)......Page 91
Case 6: Nonlinear PDE, Dirichlet BCs (ncase=2, nbc=1, a=b=c=1)......Page 95
Case 7: Nonlinear PDE, Neumann BCs (ncase=2, nbc=2, a=b=c=1)......Page 97
Case 8: Nonlinear PDE, Third-type BCs (ncase=2, nbc=3, a=b=c=1)......Page 99
Case 9: Nonlinear PDE, Nonlinear Third-Type BCs (ncase=2, nbc=4, a=b=c=1)......Page 103
Case 10: Nonlinear PDE, Analytical Neumann BCs (ncase=2, nbc=5, a=b=c=1)......Page 106
Appendix 1......Page 107
Appendix 2......Page 111
References......Page 114
6.
Burgers–Huxley Equation......Page 116
Appendix......Page 123
References......Page 126
7.
Burgers–Fisher Equation......Page 128
Appendix......Page 133
Reference......Page 138
8.
Fisher–Kolmogorov Equation......Page 139
Appendix......Page 145
References......Page 150
9.
Fitzhugh–Nagumo Equation......Page 151
The Coupled Fitzhugh–Nagumo Equations......Page 168
Analytical Solution of the Single Fitzhugh–Nagumo Equation......Page 170
References......Page 176
10.
Kolmogorov–Petrovskii–Piskunov Equation......Page 177
Appendix......Page 183
References......Page 187
11.
Kuramoto–Sivashinsky Equation......Page 189
Appendix......Page 196
References......Page 199
12.
Kawahara Equation......Page 201
Appendix 1......Page 221
Appendix 2......Page 238
References......Page 241
13.
Regularized Long-Wave Equation......Page 243
Brief Background Information......Page 258
Solution Using tanh Method......Page 262
Analytical Traveling Wave Solution for Coupled RLW Equations......Page 263
References......Page 264
14.
Extended Bernoulli Equation......Page 265
Appendix......Page 275
References......Page 277
15.
Hyperbolic Liouville Equation......Page 278
Appendix......Page 287
References......Page 295
16.
Sine-Gordon Equation......Page 296
Appendix......Page 304
References......Page 310
Case 1: Linear PDE (α = –c2, β = γ = f(x,t)=0)......Page 311
Case 2: Nonlinear PDE (ncase=2, m=2, r=3, α = –1, β = 0, γ = 1, f(x,t)≠ 0)......Page 325
Case 3: Nonlinear PDE (ncase=3, m=3, α = –2.5, β = 1, γ = 1.5, c=0.5, f(x,t)=0)......Page 331
Appendix......Page 338
References......Page 340
18.
Boussinesq Equation......Page 341
Case 1: Direct Calculation of ∂4u/∂x4 via u4x11p......Page 342
Case 2: Calculation of ∂4u/∂x4 via two-stage Differentiation of ∂2u/∂x2......Page 356
Case 3: Calculation of ∂4u/∂x4 via Four-Stage Differentiation of ∂u/∂x......Page 361
Case 4: Preceding Problem with Second Derivative BCs Replaced by (First Derivative) Neumann BCs......Page 365
Appendix......Page 372
Analytical Solution Using the Direct Integration Method......Page 374
Analytical Solution Using the Riccati-Based Method......Page 375
References......Page 376
19.
Modified Wave Equation......Page 378
Appendix......Page 388
Reference......Page 390
Tanh Method......Page 391
Example - KdV Equation......Page 393
A Maple tanh Method Procedure......Page 397
Extension to Coupled Equations......Page 403
Extension to higher spatial dimensions......Page 407
Exp Method......Page 411
Example - KdV Equation......Page 412
A Maple exp Method Procedure......Page 414
Introduction......Page 418
ODE Example......Page 419
Application to the Solution of PDEs......Page 420
Example—KdV Equation......Page 422
A Maple Riccati Method Procedure......Page 424
Direct Integration......Page 429
Factoring......Page 430
Factoring Operators......Page 431
Factorization Method for ODEs with Polynomial Nonlinearity......Page 432
Other Methods......Page 435
Maple Built-In Procedure TWSolutions......Page 436
References......Page 437
B......Page 440
F......Page 441
I......Page 442
K......Page 443
N......Page 444
S......Page 445
W......Page 446
