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...
