Spline Collocation Methods for Partial Differential Equations: With Applications in R

A comprehensive approach to numerical partial differential equations Spline Collocation Methods for Partial Differential Equations combines the collocation analysis of partial differential equations (PDEs) with the method of lines (MOL) in order to simplify the solution process. Using a series of example applications, the author delineates the main features of the approach in detail, including an established mathematical framework. The book also clearly demonstrates that spline collocation can offer a comprehensive method for numerical integration of PDEs when it is used with the MOL in which spatial (boundary value) derivatives are approximated with splines, including the boundary conditions. R, an open-source scientific programming system, is used throughout for programming the PDEs and numerical algorithms, and each section of code is clearly explained. As a result, readers gain a complete picture of the model and its computer implementation without having to fill in the details of the numerical analysis, algorithms, or programming. The presentation is not heavily mathematical, and in place of theorems and proofs, detailed example applications are provided. Appropriate for scientists, engineers, and applied mathematicians, Spline Collocation Methods for Partial Differential Equations: Introduces numerical methods by first presenting basic examples followed by more complicated applications Employs R to illustrate accurate and efficient solutions of the PDE models Presents spline collocation as a comprehensive approach to the numerical integration of PDEs and an effective alternative to other, well established methods Discusses how to reproduce and extend the presented numerical solutions Identifies the use of selected algorithms, such as the solution of nonlinear equations and banded or sparse matrix processing Features a companion website that provides the related R routines Spline Collocation Methods for Partial Differential Equations is a valuable reference and/or self-study guide for academics, researchers, and practitioners in applied mathematics and engineering, as well as for advanced undergraduates and graduate-level students.

Table of Contents:
Preface xiii About the CompanionWebsite xv 1 Introduction 1 1.1 Uniform Grids 2 1.2 Variable Grids 18 1.3 Stagewise Differentiation 24 Appendix A1 – Online Documentation for splinefun 27 Reference 30 2 One-Dimensional PDEs 31 2.1 Constant Coefficient 31 2.1.1 Dirichlet BCs 32 2.1.1.1 Main Program 33 2.1.1.2 ODE Routine 40 2.1.2 Neumann BCs 43 2.1.2.1 Main Program 44 2.1.2.2 ODE Routine 46 2.1.3 Robin BCs 49 2.1.3.1 Main Program 50 2.1.3.2 ODE Routine 55 2.1.4 Nonlinear BCs 60 2.1.4.1 Main Program 61 2.1.4.2 ODE Routine 63 2.2 Variable Coefficient 64 2.2.1 Main Program 67 2.2.2 ODE Routine 71 2.3 Inhomogeneous, Simultaneous, Nonlinear 76 2.3.1 Main Program 78 2.3.2 ODE routine 85 2.3.3 Subordinate Routines 88 2.4 First Order in Space and Time 94 2.4.1 Main Program 96 2.4.2 ODE Routine 101 2.4.3 Subordinate Routines 105 2.5 Second Order in Time 107 2.5.1 Main Program 109 2.5.2 ODE Routine 114 2.5.3 Subordinate Routine 117 2.6 Fourth Order in Space 120 2.6.1 First Order in Time 120 2.6.1.1 Main Program 121 2.6.1.2 ODE Routine 125 2.6.2 Second Order in Time 138 2.6.2.1 Main Program 140 2.6.2.2 ODE Routine 143 References 155 3 Multidimensional PDEs 157 3.1 2D in Space 157 3.1.1 Main Program 158 3.1.2 ODE Routine 163 3.2 3D in Space 170 3.2.1 Main Program, Case 1 170 3.2.2 ODE Routine 174 3.2.3 Main Program, Case 2 183 3.2.4 ODE Routine 187 3.3 Summary and Conclusions 193 4 Navier–Stokes, Burgers’ Equations 197 4.1 PDE Model 197 4.2 Main Program 198 4.3 ODE Routine 203 4.4 Subordinate Routine 205 4.5 Model Output 206 4.6 Summary and Conclusions 208 Reference 209 5 Korteweg–de Vries Equation 211 5.1 PDE Model 211 5.2 Main Program 212 5.3 ODE Routine 225 Contents ix 5.4 Subordinate Routines 228 5.5 Model Output 234 5.6 Summary and Conclusions 238 References 239 6 Maxwell Equations 241 6.1 PDE Model 241 6.2 Main Program 243 6.3 ODE Routine 248 6.4 Model Output 252 6.5 Summary and Conclusions 252 Appendix A6.1. Derivation of the Analytical Solution 257 Reference 259 7 Poisson–Nernst–Planck Equations 261 7.1 PDE Model 261 7.2 Main Program 265 7.3 ODE Routine 271 7.4 Model Output 276 7.5 Summary and Conclusions 284 References 286 8 Fokker–Planck Equation 287 8.1 PDE Model 287 8.2 Main Program 288 8.3 ODE Routine 293 8.4 Model Output 295 8.5 Summary and Conclusions 301 References 303 9 Fisher–Kolmogorov Equation 305 9.1 PDE Model 305 9.2 Main Program 306 9.3 ODE Routine 311 9.4 Subordinate Routine 313 9.5 Model Output 314 9.6 Summary and Conclusions 316 Reference 316 10 Klein–Gordon Equation 317 10.1 PDE Model, Linear Case 317 10.2 Main Program 318 10.3 ODE Routine 323 10.4 Model Output 326 10.5 PDE Model, Nonlinear Case 328 10.6 Main Program 330 10.7 ODE Routine 335 10.8 Subordinate Routines 338 10.9 Model Output 339 10.10 Summary and Conclusions 342 Reference 342 11 Boussinesq Equation 343 11.1 PDE Model 343 11.2 Main Program 344 11.3 ODE Routine 350 11.4 Subordinate Routines 354 11.5 Model Output 355 11.6 Summary and Conclusions 358 References 358 12 Cahn–Hilliard Equation 359 12.1 PDE Model 359 12.2 Main Program 360 12.3 ODE Routine 366 12.4 Model Output 369 12.5 Summary and Conclusions 379 References 379 13 Camassa–Holm Equation 381 13.1 PDE Model 381 13.2 Main Program 382 13.3 ODE Routine 388 13.4 Model Output 391 13.5 Summary and Conclusions 394 13.6 Appendix A13.1: Second Example of a PDE with a Mixed Partial Derivative 395 13.7 Main Program 395 13.8 ODE Routine 398 13.9 Model Output 400 Reference 403 14 Burgers–Huxley Equation 405 14.1 PDE Model 405 14.2 Main Program 406 14.3 ODE Routine 411 14.4 Subordinate Routine 416 14.5 Model Output 417 14.6 Summary and Conclusions 422 References 422 15 Gierer–Meinhardt Equations 423 15.1 PDE Model 423 15.2 Main Program 424 15.3 ODE Routine 429 15.4 Model Output 432 15.5 Summary and Conclusions 437 Reference 440 16 Keller–Segel Equations 441 16.1 PDE Model 441 16.2 Main Program 443 16.3 ODE Routine 449 16.4 Subordinate Routines 453 16.5 Model Output 453 16.6 Summary and Conclusions 458 Appendix A16.1. Diffusion Models 458 References 459 17 Fitzhugh–Nagumo Equations 461 17.1 PDE Model 461 17.2 Main Program 462 17.3 ODE Routine 467 17.4 Model Output 470 17.5 Summary and Conclusions 475 Reference 475 18 Euler–Poisson–Darboux Equation 477 18.1 PDE Model 477 18.2 Main Program 478 18.3 ODE Routine 483 18.4 Model Output 488 18.5 Summary and Conclusions 493 References 493 19 Kuramoto–Sivashinsky Equation 495 19.1 PDE Model 495 19.2 Main Program 496 19.3 ODE Routine 503 19.4 Subordinate Routines 506 19.5 Model Output 508 19.6 Summary and Conclusions 513 References 514 20 Einstein–Maxwell Equations 515 20.1 PDE Model 515 20.2 Main Program 516 20.3 ODE Routine 521 20.4 Model Output 526 20.5 Summary and Conclusions 533 Reference 536 A Differential Operators in Three Orthogonal Coordinate Systems 537 References 539 Index 541