8.5 Section 2 Background .1 Cosmology .11 Math Section 8.7

Book 9 Computational Mathematics Chapter 8 Ordinary Differential Equation Initial Value Problems

Section 6 Modification of Euler's Method

A first improvement of Euler's method for solving Problems (5.1) and (5.2) involves a preliminary evaluation of

(6.1)

followed by setting

6.2)

When using this method, the so called Euler-Cauchy method ([18]), one must first find the rough approximation

then

and, finally, set

The remainder terms of Eulers methods are O(h3) ( [26]). The error at the point xn can be estimated by computation of y*n, using h/2. Then

where y(x) is the exact solution.

Example 1: Use these two Euler methods to integrate the equation

(4.5)

with the initial condition y(0) = l and h = 0.2. Compare their results and also with the exact solution.

Solution: The results of Euler's first improved method are given in the table

which is constructed as follows:

Enter into the first row i =0, x0 = 0, y0 = 1. Compute f0 = f(x0,y0), whence, by (6.1) with x? = 0.1,

Next, find f(x?,y?) = 0.9182 and

By (6.2), one has

Using this result, enter into the second row i = 1, xl = 0.2, yl = 1.1836 and find h ? f (xl,yl) = 0.0846. Then compute from (6.1) for x3/2 = 0.3

Since f(x3/2 ,y3/2) = 0.7942 and h?f(x3/2 ,y3/2) = 0.2?0.7942 = 0.1590, Equation (6.2) yields

etc.

Second improved method of Euler-Cauchy:

The results of the computations by this method are presented in the next table:

It is completed by entering into the first row i = 0, x0 = 0, y0 = 1, f0 = f(x0,y0) = 1. Equation (6.3) then yields . Enter into the corresponding columns of the first row. Then find and obtain from (6.4)

In the second row: i = 1, x1= 0.2, y1 = 1.1867 , whence

z = I

Now fill the table in column by column.

The improved Euler methods are more accurate when compared with the ordinary Euler method, as can be seen from the next table which presents the values of the exact solution of Equation (6.5) and of the approximate solutions contained in the last two tables.

Exercises


11.
Solve
Problem 11 of Section IV by both improved Euler methods for h1 = 20 and h2 = 10.