Numerical Computation of Solutions
Up to this point we have examined PDEs from an analytical viewpoint, often seeking a formula for the solution. Now we devote a brief chapter to solving PDEs numerically. It is a fact that in industry and applied science PDEs are almost always solved numerically on a computer; most real-world problems are too complicated to solve analytically. And, even if a problem can be solved analytically, usually the solution is in the form of a difficult integral or an infinite series, requiring a numerical calculation anyway. This chapter presents a brief introduction to one method, the finite difference method. There are many other methods, for example, the finite element method, to mention only one. Numerical methods for PDEs have been and continue to be one of the most active research areas in applied mathematics, computer science, and the applied sciences as investigators seek faster and more accurate algorithms. Another feature is that numerical methods give tremendous insight into the basic nature and theory of PDEs.