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

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*(*h*^{3}) ( [26]).
The error at the point *x*_{n} 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, *x*_{0 }= 0, *y*_{0 }= 1. Compute *f*_{0}
= *f*(*x*_{0},*y*_{0}), 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, *x*_{l }= 0.2, *y*_{l}
= 1.1836 and find *h* ? *f* (*x*_{l},*y*_{l}) = 0.0846. Then compute from (6.1) for *x*_{3/2}
= 0.3

Since *f*(*x*_{3/2 }*,y*_{3/2}) = 0.7942 and *h*?*f*(*x*_{3/2
}*,y*_{3/2}) = 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, *x*_{0} = 0, *y*_{0}
= 1, *f*_{0} = *f*(*x*_{0},*y*_{0}) = 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, *x*_{1}= 0.2,* y*_{1} = 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