Chapter 12. Solving Partial Differential Equations
Partial differential equations are incredibly important in scientific and engineering problem solving, and it’s no wonder a vast array of methods have been devised to solve them. As in the case of ordinary differential equations, you can divide problems involving partial differential equations into two broad classes: boundary value problems and initial value problems (including initial-boundary value problems).
The best examples for boundary value problems are elliptic equations of the form:
I’m using the standard subscript notation here to indicate partial derivatives, e.g., u x represents the first partial derivative of u with respect to x.
The first equation shown above is the well-known Laplace equation, while the second is the Poisson equation. Boundary conditions are required for these equations and can consist of specification of u, derivatives of u, or a mix of these two on the problem boundary, corresponding to Dirichlet, Neumann, and mixed conditions, respectively. The numerical solution to boundary value problems of this form generally consists of discretizing the problem domain using one of a variety of techniques, such as the finite difference method, finite element method, or boundary element method, and formulating a system of algebraic equations that can be solved for the unknown values of u at discrete ...