WILLIAM E. SCHIESSER, PhD, ScD (hon.), is Emeritus McCann Professor of Biomolecular and Chemical Engineering and Professor of Mathematics at Lehigh University. He is the author, coauthor or coeditor of 17 books, including Method of Lines PDE Analysis in Biomedical Science and Engineering; Differential Equation Analysis in Biomedical Science and Engineering: Ordinary Differential Equation Applications with R and Differential Equation Analysis in Biomedical Science and Engineering: Partial Differential Equation Applications with R, published by Wiley.

• Preface xiiiAbout the CompanionWebsite xv1 Introduction 11.1 Uniform Grids 21.2 Variable Grids 181.3 Stagewise Differentiation 24Appendix A1 – Online Documentation for splinefun 27Reference 302 One-Dimensional PDEs 312.1 Constant Coefficient 312.1.1 Dirichlet BCs 322.1.1.1 Main Program 332.1.1.2 ODE Routine 402.1.2 Neumann BCs 432.1.2.1 Main Program 442.1.2.2 ODE Routine 462.1.3 Robin BCs 492.1.3.1 Main Program 502.1.3.2 ODE Routine 552.1.4 Nonlinear BCs 602.1.4.1 Main Program 612.1.4.2 ODE Routine 632.2 Variable Coefficient 642.2.1 Main Program 672.2.2 ODE Routine 712.3 Inhomogeneous, Simultaneous, Nonlinear 762.3.1 Main Program 782.3.2 ODE routine 852.3.3 Subordinate Routines 882.4 First Order in Space and Time 942.4.1 Main Program 962.4.2 ODE Routine 1012.4.3 Subordinate Routines 1052.5 Second Order in Time 1072.5.1 Main Program 1092.5.2 ODE Routine 1142.5.3 Subordinate Routine 1172.6 Fourth Order in Space 1202.6.1 First Order in Time 1202.6.1.1 Main Program 1212.6.1.2 ODE Routine 1252.6.2 Second Order in Time 1382.6.2.1 Main Program 1402.6.2.2 ODE Routine 143References 1553 Multidimensional PDEs 1573.1 2D in Space 1573.1.1 Main Program 1583.1.2 ODE Routine 1633.2 3D in Space 1703.2.1 Main Program, Case 1 1703.2.2 ODE Routine 1743.2.3 Main Program, Case 2 1833.2.4 ODE Routine 1873.3 Summary and Conclusions 1934 Navier–Stokes, Burgers’ Equations 1974.1 PDE Model 1974.2 Main Program 1984.3 ODE Routine 2034.4 Subordinate Routine 2054.5 Model Output 2064.6 Summary and Conclusions 208Reference 2095 Korteweg–de Vries Equation 2115.1 PDE Model 2115.2 Main Program 2125.3 ODE Routine 225Contents ix5.4 Subordinate Routines 2285.5 Model Output 2345.6 Summary and Conclusions 238References 2396 Maxwell Equations 2416.1 PDE Model 2416.2 Main Program 2436.3 ODE Routine 2486.4 Model Output 2526.5 Summary and Conclusions 252Appendix A6.1. Derivation of the Analytical Solution 257Reference 2597 Poisson–Nernst–Planck Equations 2617.1 PDE Model 2617.2 Main Program 2657.3 ODE Routine 2717.4 Model Output 2767.5 Summary and Conclusions 284References 2868 Fokker–Planck Equation 2878.1 PDE Model 2878.2 Main Program 2888.3 ODE Routine 2938.4 Model Output 2958.5 Summary and Conclusions 301References 3039 Fisher–Kolmogorov Equation 3059.1 PDE Model 3059.2 Main Program 3069.3 ODE Routine 3119.4 Subordinate Routine 3139.5 Model Output 3149.6 Summary and Conclusions 316Reference 31610 Klein–Gordon Equation 31710.1 PDE Model, Linear Case 31710.2 Main Program 31810.3 ODE Routine 32310.4 Model Output 32610.5 PDE Model, Nonlinear Case 32810.6 Main Program 33010.7 ODE Routine 33510.8 Subordinate Routines 33810.9 Model Output 33910.10 Summary and Conclusions 342Reference 34211 Boussinesq Equation 34311.1 PDE Model 34311.2 Main Program 34411.3 ODE Routine 35011.4 Subordinate Routines 35411.5 Model Output 35511.6 Summary and Conclusions 358References 35812 Cahn–Hilliard Equation 35912.1 PDE Model 35912.2 Main Program 36012.3 ODE Routine 36612.4 Model Output 36912.5 Summary and Conclusions 379References 37913 Camassa–Holm Equation 38113.1 PDE Model 38113.2 Main Program 38213.3 ODE Routine 38813.4 Model Output 39113.5 Summary and Conclusions 39413.6 Appendix A13.1: Second Example of a PDE with a Mixed Partial Derivative 39513.7 Main Program 39513.8 ODE Routine 39813.9 Model Output 400Reference 40314 Burgers–Huxley Equation 40514.1 PDE Model 40514.2 Main Program 40614.3 ODE Routine 41114.4 Subordinate Routine 41614.5 Model Output 41714.6 Summary and Conclusions 422References 42215 Gierer–Meinhardt Equations 42315.1 PDE Model 42315.2 Main Program 42415.3 ODE Routine 42915.4 Model Output 43215.5 Summary and Conclusions 437Reference 44016 Keller–Segel Equations 44116.1 PDE Model 44116.2 Main Program 44316.3 ODE Routine 44916.4 Subordinate Routines 45316.5 Model Output 45316.6 Summary and Conclusions 458Appendix A16.1. Diffusion Models 458References 45917 Fitzhugh–Nagumo Equations 46117.1 PDE Model 46117.2 Main Program 46217.3 ODE Routine 46717.4 Model Output 47017.5 Summary and Conclusions 475Reference 47518 Euler–Poisson–Darboux Equation 47718.1 PDE Model 47718.2 Main Program 47818.3 ODE Routine 48318.4 Model Output 48818.5 Summary and Conclusions 493References 49319 Kuramoto–Sivashinsky Equation 49519.1 PDE Model 49519.2 Main Program 49619.3 ODE Routine 50319.4 Subordinate Routines 50619.5 Model Output 50819.6 Summary and Conclusions 513References 51420 Einstein–Maxwell Equations 51520.1 PDE Model 51520.2 Main Program 51620.3 ODE Routine 52120.4 Model Output 52620.5 Summary and Conclusions 533Reference 536A Differential Operators in Three Orthogonal Coordinate Systems 537References 539Index 541

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WILLIAM E. SCHIESSER, PhD, ScD (hon.), is Emeritus McCann Professor of Biomolecular and Chemical Engineering and Professor of Mathematics at Lehigh University. He is the author, coauthor or coeditor of 17 books, including Method of Lines PDE Analysis in Biomedical Science and Engineering; Differential Equation Analysis in Biomedical Science and Engineering: Ordinary Differential Equation Applications with R and Differential Equation Analysis in Biomedical Science and Engineering: Partial Differential Equation Applications with R, published by Wiley.

• Preface xiiiAbout the CompanionWebsite xv1 Introduction 11.1 Uniform Grids 21.2 Variable Grids 181.3 Stagewise Differentiation 24Appendix A1 – Online Documentation for splinefun 27Reference 302 One-Dimensional PDEs 312.1 Constant Coefficient 312.1.1 Dirichlet BCs 322.1.1.1 Main Program 332.1.1.2 ODE Routine 402.1.2 Neumann BCs 432.1.2.1 Main Program 442.1.2.2 ODE Routine 462.1.3 Robin BCs 492.1.3.1 Main Program 502.1.3.2 ODE Routine 552.1.4 Nonlinear BCs 602.1.4.1 Main Program 612.1.4.2 ODE Routine 632.2 Variable Coefficient 642.2.1 Main Program 672.2.2 ODE Routine 712.3 Inhomogeneous, Simultaneous, Nonlinear 762.3.1 Main Program 782.3.2 ODE routine 852.3.3 Subordinate Routines 882.4 First Order in Space and Time 942.4.1 Main Program 962.4.2 ODE Routine 1012.4.3 Subordinate Routines 1052.5 Second Order in Time 1072.5.1 Main Program 1092.5.2 ODE Routine 1142.5.3 Subordinate Routine 1172.6 Fourth Order in Space 1202.6.1 First Order in Time 1202.6.1.1 Main Program 1212.6.1.2 ODE Routine 1252.6.2 Second Order in Time 1382.6.2.1 Main Program 1402.6.2.2 ODE Routine 143References 1553 Multidimensional PDEs 1573.1 2D in Space 1573.1.1 Main Program 1583.1.2 ODE Routine 1633.2 3D in Space 1703.2.1 Main Program, Case 1 1703.2.2 ODE Routine 1743.2.3 Main Program, Case 2 1833.2.4 ODE Routine 1873.3 Summary and Conclusions 1934 Navier–Stokes, Burgers’ Equations 1974.1 PDE Model 1974.2 Main Program 1984.3 ODE Routine 2034.4 Subordinate Routine 2054.5 Model Output 2064.6 Summary and Conclusions 208Reference 2095 Korteweg–de Vries Equation 2115.1 PDE Model 2115.2 Main Program 2125.3 ODE Routine 225Contents ix5.4 Subordinate Routines 2285.5 Model Output 2345.6 Summary and Conclusions 238References 2396 Maxwell Equations 2416.1 PDE Model 2416.2 Main Program 2436.3 ODE Routine 2486.4 Model Output 2526.5 Summary and Conclusions 252Appendix A6.1. Derivation of the Analytical Solution 257Reference 2597 Poisson–Nernst–Planck Equations 2617.1 PDE Model 2617.2 Main Program 2657.3 ODE Routine 2717.4 Model Output 2767.5 Summary and Conclusions 284References 2868 Fokker–Planck Equation 2878.1 PDE Model 2878.2 Main Program 2888.3 ODE Routine 2938.4 Model Output 2958.5 Summary and Conclusions 301References 3039 Fisher–Kolmogorov Equation 3059.1 PDE Model 3059.2 Main Program 3069.3 ODE Routine 3119.4 Subordinate Routine 3139.5 Model Output 3149.6 Summary and Conclusions 316Reference 31610 Klein–Gordon Equation 31710.1 PDE Model, Linear Case 31710.2 Main Program 31810.3 ODE Routine 32310.4 Model Output 32610.5 PDE Model, Nonlinear Case 32810.6 Main Program 33010.7 ODE Routine 33510.8 Subordinate Routines 33810.9 Model Output 33910.10 Summary and Conclusions 342Reference 34211 Boussinesq Equation 34311.1 PDE Model 34311.2 Main Program 34411.3 ODE Routine 35011.4 Subordinate Routines 35411.5 Model Output 35511.6 Summary and Conclusions 358References 35812 Cahn–Hilliard Equation 35912.1 PDE Model 35912.2 Main Program 36012.3 ODE Routine 36612.4 Model Output 36912.5 Summary and Conclusions 379References 37913 Camassa–Holm Equation 38113.1 PDE Model 38113.2 Main Program 38213.3 ODE Routine 38813.4 Model Output 39113.5 Summary and Conclusions 39413.6 Appendix A13.1: Second Example of a PDE with a Mixed Partial Derivative 39513.7 Main Program 39513.8 ODE Routine 39813.9 Model Output 400Reference 40314 Burgers–Huxley Equation 40514.1 PDE Model 40514.2 Main Program 40614.3 ODE Routine 41114.4 Subordinate Routine 41614.5 Model Output 41714.6 Summary and Conclusions 422References 42215 Gierer–Meinhardt Equations 42315.1 PDE Model 42315.2 Main Program 42415.3 ODE Routine 42915.4 Model Output 43215.5 Summary and Conclusions 437Reference 44016 Keller–Segel Equations 44116.1 PDE Model 44116.2 Main Program 44316.3 ODE Routine 44916.4 Subordinate Routines 45316.5 Model Output 45316.6 Summary and Conclusions 458Appendix A16.1. Diffusion Models 458References 45917 Fitzhugh–Nagumo Equations 46117.1 PDE Model 46117.2 Main Program 46217.3 ODE Routine 46717.4 Model Output 47017.5 Summary and Conclusions 475Reference 47518 Euler–Poisson–Darboux Equation 47718.1 PDE Model 47718.2 Main Program 47818.3 ODE Routine 48318.4 Model Output 48818.5 Summary and Conclusions 493References 49319 Kuramoto–Sivashinsky Equation 49519.1 PDE Model 49519.2 Main Program 49619.3 ODE Routine 50319.4 Subordinate Routines 50619.5 Model Output 50819.6 Summary and Conclusions 513References 51420 Einstein–Maxwell Equations 51520.1 PDE Model 51520.2 Main Program 51620.3 ODE Routine 52120.4 Model Output 52620.5 Summary and Conclusions 533Reference 536A Differential Operators in Three Orthogonal Coordinate Systems 537References 539Index 541

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WILLIAM E. SCHIESSER, PhD, ScD (hon.), is Emeritus McCann Professor of Biomolecular and Chemical Engineering and Professor of Mathematics at Lehigh University. He is the author, coauthor or coeditor of 17 books, including Method of Lines PDE Analysis in Biomedical Science and Engineering; Differential Equation Analysis in Biomedical Science and Engineering: Ordinary Differential Equation Applications with R and Differential Equation Analysis in Biomedical Science and Engineering: Partial Differential Equation Applications with R, published by Wiley.

• Preface xiiiAbout the CompanionWebsite xv1 Introduction 11.1 Uniform Grids 21.2 Variable Grids 181.3 Stagewise Differentiation 24Appendix A1 – Online Documentation for splinefun 27Reference 302 One-Dimensional PDEs 312.1 Constant Coefficient 312.1.1 Dirichlet BCs 322.1.1.1 Main Program 332.1.1.2 ODE Routine 402.1.2 Neumann BCs 432.1.2.1 Main Program 442.1.2.2 ODE Routine 462.1.3 Robin BCs 492.1.3.1 Main Program 502.1.3.2 ODE Routine 552.1.4 Nonlinear BCs 602.1.4.1 Main Program 612.1.4.2 ODE Routine 632.2 Variable Coefficient 642.2.1 Main Program 672.2.2 ODE Routine 712.3 Inhomogeneous, Simultaneous, Nonlinear 762.3.1 Main Program 782.3.2 ODE routine 852.3.3 Subordinate Routines 882.4 First Order in Space and Time 942.4.1 Main Program 962.4.2 ODE Routine 1012.4.3 Subordinate Routines 1052.5 Second Order in Time 1072.5.1 Main Program 1092.5.2 ODE Routine 1142.5.3 Subordinate Routine 1172.6 Fourth Order in Space 1202.6.1 First Order in Time 1202.6.1.1 Main Program 1212.6.1.2 ODE Routine 1252.6.2 Second Order in Time 1382.6.2.1 Main Program 1402.6.2.2 ODE Routine 143References 1553 Multidimensional PDEs 1573.1 2D in Space 1573.1.1 Main Program 1583.1.2 ODE Routine 1633.2 3D in Space 1703.2.1 Main Program, Case 1 1703.2.2 ODE Routine 1743.2.3 Main Program, Case 2 1833.2.4 ODE Routine 1873.3 Summary and Conclusions 1934 Navier–Stokes, Burgers’ Equations 1974.1 PDE Model 1974.2 Main Program 1984.3 ODE Routine 2034.4 Subordinate Routine 2054.5 Model Output 2064.6 Summary and Conclusions 208Reference 2095 Korteweg–de Vries Equation 2115.1 PDE Model 2115.2 Main Program 2125.3 ODE Routine 225Contents ix5.4 Subordinate Routines 2285.5 Model Output 2345.6 Summary and Conclusions 238References 2396 Maxwell Equations 2416.1 PDE Model 2416.2 Main Program 2436.3 ODE Routine 2486.4 Model Output 2526.5 Summary and Conclusions 252Appendix A6.1. Derivation of the Analytical Solution 257Reference 2597 Poisson–Nernst–Planck Equations 2617.1 PDE Model 2617.2 Main Program 2657.3 ODE Routine 2717.4 Model Output 2767.5 Summary and Conclusions 284References 2868 Fokker–Planck Equation 2878.1 PDE Model 2878.2 Main Program 2888.3 ODE Routine 2938.4 Model Output 2958.5 Summary and Conclusions 301References 3039 Fisher–Kolmogorov Equation 3059.1 PDE Model 3059.2 Main Program 3069.3 ODE Routine 3119.4 Subordinate Routine 3139.5 Model Output 3149.6 Summary and Conclusions 316Reference 31610 Klein–Gordon Equation 31710.1 PDE Model, Linear Case 31710.2 Main Program 31810.3 ODE Routine 32310.4 Model Output 32610.5 PDE Model, Nonlinear Case 32810.6 Main Program 33010.7 ODE Routine 33510.8 Subordinate Routines 33810.9 Model Output 33910.10 Summary and Conclusions 342Reference 34211 Boussinesq Equation 34311.1 PDE Model 34311.2 Main Program 34411.3 ODE Routine 35011.4 Subordinate Routines 35411.5 Model Output 35511.6 Summary and Conclusions 358References 35812 Cahn–Hilliard Equation 35912.1 PDE Model 35912.2 Main Program 36012.3 ODE Routine 36612.4 Model Output 36912.5 Summary and Conclusions 379References 37913 Camassa–Holm Equation 38113.1 PDE Model 38113.2 Main Program 38213.3 ODE Routine 38813.4 Model Output 39113.5 Summary and Conclusions 39413.6 Appendix A13.1: Second Example of a PDE with a Mixed Partial Derivative 39513.7 Main Program 39513.8 ODE Routine 39813.9 Model Output 400Reference 40314 Burgers–Huxley Equation 40514.1 PDE Model 40514.2 Main Program 40614.3 ODE Routine 41114.4 Subordinate Routine 41614.5 Model Output 41714.6 Summary and Conclusions 422References 42215 Gierer–Meinhardt Equations 42315.1 PDE Model 42315.2 Main Program 42415.3 ODE Routine 42915.4 Model Output 43215.5 Summary and Conclusions 437Reference 44016 Keller–Segel Equations 44116.1 PDE Model 44116.2 Main Program 44316.3 ODE Routine 44916.4 Subordinate Routines 45316.5 Model Output 45316.6 Summary and Conclusions 458Appendix A16.1. Diffusion Models 458References 45917 Fitzhugh–Nagumo Equations 46117.1 PDE Model 46117.2 Main Program 46217.3 ODE Routine 46717.4 Model Output 47017.5 Summary and Conclusions 475Reference 47518 Euler–Poisson–Darboux Equation 47718.1 PDE Model 47718.2 Main Program 47818.3 ODE Routine 48318.4 Model Output 48818.5 Summary and Conclusions 493References 49319 Kuramoto–Sivashinsky Equation 49519.1 PDE Model 49519.2 Main Program 49619.3 ODE Routine 50319.4 Subordinate Routines 50619.5 Model Output 50819.6 Summary and Conclusions 513References 51420 Einstein–Maxwell Equations 51520.1 PDE Model 51520.2 Main Program 51620.3 ODE Routine 52120.4 Model Output 52620.5 Summary and Conclusions 533Reference 536A Differential Operators in Three Orthogonal Coordinate Systems 537References 539Index 541

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