A Compendium of Partial Differential Equation Models

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.

Spis treści:
1. An introduction to the Method of Lines (MOL)
2. A one-dimensional, linear partial differential equation
3. Greens 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. Maxwells equations
10. Elliptic PDEs, Laplaces 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.