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