Ordinary differential equations - Euler's method
Let's consider the differential equation
with an initial condition
Substitutinginto the equation (1), we’ll receive value of a derivative in a point :
At smallthe following expression takes place:
Designating, let’s rewrite the last equality in the form of:
Accepting now for a new initial point, precisely also we’ll receive
In the general case we’ll have:
It also is Euler's method. The value refers to as step of integration. Using this method, we receive the approached values у , at as the derivativeactually does not remain to a constant on an interval in length. Therefore we receive a mistake in definition of value of function у , that greater, than is more. Euler's method is the elementary method of numerical integration of differential equations and systems. Its defects are a small accuracy and a regular accumulation of mistakes.
More exact is Euler's modified method with recalculation. Its essence that at the first we find from the formula (3) so-called «a gross approach» (prediction):
and then the recalculation gives us too approached, but more exact value (correction):
Actually the recalculation allows to consider, though approximately, a change of a derivative on a step of integration as its values in the beginning and in the end of a step (Fig. 1) are considered, and then their average is chosen. Euler's method with recalculation (4) is in essence the 2-nd order Runge-Kutta method . This will become obvious of the further.
Fig. 1. The geometric representation of Euler’s method with recalculation.