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

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 *y*_{i}.
Proceeding from the rough approximation

(7.1)

form the iterative process

(7.2)

Repeat it until the decimal digits of subsequent approximations *y*_{i+1}^{(k)}*,y*_{i+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* = 2*e*^{x} - *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.