 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 () 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.  