Taylor Method(s)

Given the initial value problem,

.

In absence of an analytic solution, we want to calculate the value of for a given value of . One can do this by considering the Taylor series expansion about the initial value:

which, given the differential equation, can be rewritten as

where represents the total derivative of evaluated at ( ). The Taylor Methods truncate this expansion to obtain a formula to calculate . In practice, it is not advisable to choose such that , because that requires a large step size. Rather, one chooses a smaller step size and calculates intermediate -values consecutively until the value at is obtained. This also yields the intermediate values needed to plot the solution curve when necessary.

Depending on where the Taylor series is truncated, we find the following methods.

Euler's Method

Taylor's method of order n

Local truncation error