Numerical solution of differential equations

Ordinary differential equations.

Simplest form of O.D.E.

Analytic solution: .

Digital solution presents the table of function:

x0	x1	x2	…	xn	…
y0	y1	y2	…	yn	…

Method of approximations.

Problem of Koshie:

- O.D.E.

- initial condition

.

Let to find solution for ;

1-st approximation: ,

;

2-nd approximation:

;

…

.

Numerical methods.

M ethods 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.

C onsider point .

O

btain value

…

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:

Brink problem for differential equations.

Simplest case:

.

Boundary conditions

initial conditions: ending conditions:

- problem of 1-st sort;

- 2-nd sort;

or or - mixed sort.

Consider linear variant:

Linear two-points brink problem for 2-nd order differential equation.

Reducing of linear 2-points brink problem for 2-nd order differential equation to problem of Koshie.

Let find solution in form:

where

Consider 1-st boundary condition:

Let guarantee equivalence without depending on coefficient c:

if , then else

if , then .

Consider c provided 2-nd boundary condition:

Methods, based on finite differences.

finite value:

For boundary points:

, but error is great;

take order of Tailor:

Analogously: .

2-n derivative:

.

So, we obtain system for :

,

boundary points -

This linear system may be resolved by any method, but it may be applied reducing to Koshie problem.

Let

.

Put digital model:

Exclude :

for we have recurrence:

Analogously for :