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

## Books

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.

Preface xiii

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

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