Solving partial differential equations numerically

Partial differential equations are differential equations that involve partial derivatives of functions in two or more variables, as opposed to ordinary derivatives in only a single variable. Partial differential equations are a vast topic, and could easily fill a series of books. A typical example of a partial differential equation is the (one-dimensional) heat equation:

Here, is a positive constant and is a function. The solution to this partial differential equation is a function , which represents the temperature of a rod, occupying the range , at a given time . To keep things simple, we will take , which amounts to saying that no heating/cooling is applied to the system, , and . In practice, we can rescale the problem to fix the constant , so this is not a restrictive problem. In this example, we will use boundary conditions:

These are equivalent to saying that the ends...