Method of approximations.

Problem of Koshie:

- O.D.E.

- initial condition

.

Let to find solution for ;

1-st approximation: ,

;

2-nd approximation:

;

…

.

Numerical methods.

Methods of Euler.

 

 

 

 

Put step: h; x_i=x₀ + ih, i= 0, 1, 2, …

The value of is tg of the corner of tangent at the point , and we consider the part of tangent instead of part of line :

 

 

Modifications of method of Euler.

1-st mod.

Consider point .

We find the value of tangent of corner in the middle of distance, then

.

2-nd mod.

Equation of higher order may be reduced to system.

Define:

We obtain system:

Express for last equation:

Apply, for example, method of Euler:

Method of Runghe-Kutta.

Method of Runghe-Kutta may be extrapolated for system: