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Full Description
Mathematical modelling of physical and chemical systems is used extensively throughout science, engineering, and applied mathematics. To use mathematical models, one needs solutions to the model equations; this generally requires numerical methods. This book presents numerical methods and associated computer code in Matlab for the solution of a spectrum of models expressed as partial differential equations (PDEs). The authors focus on the method of lines (MOL), a well-established procedure for all major classes of PDEs, where the boundary value partial derivatives are approximated algebraically by finite differences. This reduces the PDEs to ordinary differential equations (ODEs) and makes the computer code easy to understand, implement, and modify. Also, the ODEs (via MOL) can be combined with any other ODEs that are part of the model (so that MOL naturally accommodates ODE/PDE models). This book uniquely includes a detailed line-by-line discussion of computer code related to the associated PDE model.
Contents
1. An introduction to the Method of Lines (MOL); 2. A one-dimensional, linear partial differential equation; 3. Green's function analysis; 4. Two nonlinear, variable coeffcient, inhomogeneous PDEs; 5. Euler, Navier-Stokes and Burgers equations; 6. The Cubic Schrödinger Equation (CSE); 7. The Korteweg-deVries (KdV) equation; 8. The linear wave equation; 9. Maxwell's equations; 10. Elliptic PDEs: Laplace's equation; 11. Three-dimensional PDE; 12. PDE with a mixed partial derivative; 13. Simultaneous, nonlinear, 2D PDEs in cylindrical coordinates; 14. Diffusion equation in spherical coordinates; Appendix 1: partial differential equations from conservation principles: the anisotropic diffusion equation; Appendix 2: order conditions for finite difference approximations; Appendix 3: analytical solution of nonlinear, traveling wave partial differential equations; Appendix 4: implementation of time varying boundary conditions; Appendix 5: the DSS library; Appendix 6: animating simulation results.
