What is Euler’s modified method?

Overview

This method was given by Leonhard Euler. Euler’s method is the first order numerical methods for solving ordinary differential equations with given initial value.

It is the basic explicit method for numerical integration of the ODE’s.

Euler method

The general first order differential equation 

With the initial condition

In this method the solution is in the form of tabulated values.

Integrating both sides of the equation (i) we get

Assuming that in  ,this  gives Euler’s formula

In general formula

Error estimate for the method

Example: Use Euler’s procedure to  find y(0.4) from the differential equation

Sol:

Given equation  dy/dx = xy

Here

 We  break  the  interval in four steps.

 So that

By Euler’s formula

For n=0 in equation (i) we get, the first approximation

n=1 in equation (i) we obtain

Put=2 in equation (i) we  get, the third approximation

Put n=3 in equation (i) we  get, the fourth approximation

Hence y(0.4)  =1.061106.

Modified Euler’s  Method:

Instead of  approximating f(x, y) by as in Euler’s method.  In  the  modified Euler’s  method we  have the iteration formula

Where is the nth approximation to y1 .The iteration started with the Euler’s  formula

Example: Use modified  Euler’s method  to compute y  for  x=0.05.  Given that

Result correct to three decimal places.

Sol:

Given  equation   dy/dx = x + y

Here f(x, y) = x + y with y(0) = 1

Take h = 0.05 – 0 = 0.05

By modified Euler’s  formula the  initial iteration is

The iteration  formula by  modified Euler’s method  is

For n=0  in equation  (i)  we get

Where   as  above

For n=1  in equation  (i)  we get

 For n=3  in equation  (i)  we get

Since  third and  fourth approximation are equal .

Hence y=1.0526  at x =  0.05  correct  to  three decimal places.

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