What is a Finite Difference Equation?
A finite difference equation is a tool for numerically solving an ordinary or partial differential equation that would be difficult to solve analytically. Finite difference equations can be generally represented by Equation 1.
In Equation 1, represents the first order derivative of the function , represents the next point along the curve of the function , and is the truncation error between the points and . The truncation error is included because the result of the numeric solution to the differential equation will be slightly different from the result of the analytical solution. Equation 1 specifically represents a forward difference equation. There is also a backward difference equation that uses the point which is the previous point along the curve of the function. And there is also a central difference equation that uses both the next point, , and the previous point, , along the curve of the function. None of these equations is necessarily better than the other and so which to apply is dependent on the situation or data modeled.
Why is this Useful?
Sometimes it is near impossible to solve a differential equation analytically and in general it is easier to program a computer to solve the differential equation numerically. The reason this works is because the finite difference equation transforms the differential equation into a simpler set of arithmetic problems. The problem is basically stepped down to a less computer resource intensive form and the computer can analyze the data quicker with acceptable accuracy.