Taylor Series Method for ODE's

The Taylorseries method is of general applicability and it is the standard to which we can compare the accuracy of the various other numerical methods for solving an I. V. P.It can be devised to have any specified degree of accuracy.

Theorem(Taylor SeriesMethod of Order n)

Assume thatf(t,y)is continuous and satisfies a Lipschits condition in the variabley,and consider theI. V. P. (initial value problem)


with
,over the interval
.

The Taylor series method uses the formulas
,and


for


where
is
evaluated at
,as an approximate solution to the differential equation using the discrete set of points
.

Theorem(Precisionof Taylor SeriesMethod of Order n)

Assume that
is the solution to the I.V.P.
with
.If
and
is the sequence of approximations generated by the Taylor series method of ordern, then at each step, the local truncation error is of the order
,and the overall global truncation error
is of the order


,for
.

The error at the right end of the interval is called the final global error


.

Algorithm (Taylor Series Method).

To compute a numerical approximation for the solution of the initial value problem
with
over
at a discrete set of points using the formulas


,and
,for


where
is
evaluated at
.

Mathematica Subroutine (Taylor SeriesMethod of Order n=4).