Taylor’s series expansion and the Finite Difference Method

Perhaps the easiest interpretation for a Finite Difference formulation of numerical integration comes from the Taylor’s series expansion. Given a continuous function f(x), the discretized locations on the curve of f(x) that are separated by a distance ‘h’ can be expanded as a Taylor’s series. If the  are  and  , the function at i+1 can be represented in terms of the value at i. Mathematically

 

Stated in simpler terms, the first order Taylor’s series expansion is a literal translation of the definition of the first derivative of a continuous function f(x). Having stated thusly, the expansion can be pursued on both the forward and backward directions. That is, the expansion at a location can also be defined. Such an expansion thus becomes

Rearranging the above two equations in an ODE format, one arrives at the Forward and Backward Finite Difference formulations. By eliminating from the above equations (by simply subtracting one from the other), the Central Difference formulation can be derived.

But what if the higher order terms are put to practical use with the Taylor’s expansion series? Inclusion of higher order terms results in

 

and

 

                 

By ignoring the triple-differential term and adding the above equations, a formulation can be derived for the 2^nd derivative, which turns out to be

To state the obvious, a Finite Difference formulation for a 2^nd order differential has been derived starting from the Taylor’s expansion series.