Point 2. Approximate integration of differential equations Successive approximations method

Let it’s necessary to solve Cauchy problem

, ( 7 )

. ( 8 )

Theorem 2. Cauchy problem (7), (8) is equivalent to the next integral equation

( 9 )

■a) If a function is a solution of Cauchy problem, then

.

Integrating the identity we have

and so the function is also a solution of the integral equation.

b) Let now a function is a solution of the integral equation, that is

.

Then , and after differentiation

.

Therefore this function is a solution of Cauchy problem.■

Let’s put

,

,

,

……………………………..

, ( 10 )

It can be proved that if a function and its partial derivative are continuous in some domain D of the -plane, then there is a function such that for any x from a certain interval

.

Passing to the limit as in the equality (10) we see that

and therefore the function is a solution of the integral equation (9) and of Cau-chy problem.

On this base the formula (10) represents an approximate value of a solution of Cauchy problem.

Note. A function represented by the formula (10) is called the nth approximation to an unknown solution of Cauchy problem.

Ex. 3. Find first three approximations to a solution of Cauchy problem

.

Here , and Cauchy problem is equivalent to the integral equation

.

Therefore

.

We can find the value of a solution of Cauchy problem with arbitrary accuracy.

Euler method

Let’s we find solution of Cauchy problem on a segment . We divide the segment into n equal parts of the length

by division points

.

Further we substitute the derivative of the unknown function

by the divided difference

and the differential equation (7) by the difference equation

,

from which

. ( 11 )

With the help of (11) we have

( 12.1 )

( 12.2 )

( 12.3 )

………………………………..

. ( 12.n )

If we join points , we’ll represent an approximate graph of the solution of Cauchy problem.

Ex. 4. Find an approximate solution of Cauchy problem on a segment .

Here Dividing the segment into 10 equal parts of the length we get the calculating formula

.

We represent all calculations with the help of the next table

0 0.0 1.00 1.00 1.00 1 0.1 1.00 1.01 1.01 2 0.2 1.01 1.02 1.03 3 0.3 1.03 1.03 1.06 4 0.4 1.06 1.04 1.10 5 0.5 1.10 1.05 1.16 6 0.6 1.16 1.06 1.23 7 0.7 1.23 1.07 1.31 8 0.8 1.31 1.08 1.42 9 0.9 1.42 1.09 1.55

Corresponding points of the graph of the approximate solution of Cauchy problem