12. Method of degenerated kernels for integral equations

12. Method of degenerated kernels for integral equations

Consider a second kind Fredholm integral equation:

(1)

The idea of the method of degenerated kernels consists of the following: we suggest that the kernel K(x,s) is degenerated, i.e. it can be represented as following:

(2)

Substituting (2) into (1) we obtain:



(2')

where the integrals are constant coefficients as they don’t depend on any parameter. That’s why we designate them as follows:

(3)

Then our equation (2’) takes the form

(4)

where c1,..,cn are unknowns are to be defined.

Substituting the formula (4) into the equations (3) we will obtain:

(5)

Taking designations



we get the following system of linear algebraic equations with n unknowns c1,...,cn:

(6)

Solving system (6) by some direct method (Gauss) or approximate method (simple iteration method) we find the coefficients c1,..,cn. Then our solution of integral Fredholm equation (1) is found by the formula

.

If the kernel K(x,s) is not degenerated then by means of Taylor/s formula we replace the kernel K(x,s) by the Taylors formula.