8.6 Section 2 Background .1 Cosmology .11 Math Section 8.8

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

Section 7 Iterative Euler Method

The Euler-Cauchy method for the solution of Problems (5.1) and (5.2) can be made still more accurate by introduction of an iterative process ([45]) for each yi. Proceeding from the rough approximation

(7.1)

form the iterative process

(7.2)

Repeat it until the decimal digits of subsequent approximations yi+1(k),yi+1(k+1) coincide, when

As a rule, provided h is sufficiently small, convergence will be fast. If after three or four steps the required number of decimal digits do not coincide, decrease h.

Example 1: Using the iterative method, find with an accuracy of 10-4 the value of y(0.1) of the solution of the differential equation

for the initial condition y(0) =1.

Solution: Let h = 0.05. By (7.1)

By (8.2),

As the required accuracy has been achieved, round off to four digits:

Reapply (7,1) for i = 1:

Then (7.2) yields

Thus,

The analytic solution is y = 2ex - x - 1, whence

Exercises

Apply Euler's iterative method to find on the interval [0,1] solutions of the following equations for the given initial conditions, using h = 0.1.