Contents >> Applied Mathematics >> Numerical Methods >> Ordinary Differential Equations >> Euler’s method

 Ordinary differential equations - Euler's method Euler's method Let's consider the differential equation (1) with an initial condition Substituting into the equation (1), we’ll receive value of a derivative in a point : At small the following expression takes place: Designating , let’s rewrite the last equality in the form of: (2) Accepting now for a new initial point, precisely also we’ll receive In the general case we’ll have: (3) 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 derivative actually 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): (4) 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.

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