Euler's Method for ODE's

The first method we shall study for solving differential equations is called Euler's method, it serves to illustrate the concepts involved in the advanced methods. It has limited use because of the larger error that is accumulated with each successive step.However, it is important to study Euler's method because the remainder term and error analysis is easier to understand.

Theorem(Euler's Method)

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
.

Euler's method uses the formulas
,and


for


as an approximate solution to the differential equation using the discrete set of points
.

Error analysis for Euler's Method

When we obtained the formula
for Euler's method, the neglected term for each step has the form
.If this was the only error at each step, then at the end of the interval

, after
steps have been made, the accumulated error would be






.

The error is more complicated, but this estimate predominates.

Theorem (Precision of Euler's Method)

Assume that
is the solution to the I.V.P.
with
.If
and
is the sequence of approximations generated by Euler's method, then at each step, the local trunctaion 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


.

Remark.The global truncation error
is used to study the behavior of the error for various step sizes.It can be used to give us an idea of how much computing effort must be done to obtain an accurate approximation.

Numerical methods used in this module.Use Euler's method and the modified Euler's method. Construct numerical solutions of order
and
, respectively.The theory for the modified Euler method is not presented at this time, we are to trust that its development is similar, but the order for the error is better and is known to be
.

Algorithm (Euler's Method).To approximate the solution of the initial value problem
with
over
at a discrete set of points using the formulas


,and
for
.

Mathematica Subroutine (Euler's Method).

Algorithm (Modified Euler's Method).To approximate the solution of the initial value problem
with
over
at a discrete set of points using the formulas


,and
for
.

Mathematica Subroutine (Modified Euler's Method).