Abstract

In this paper we will deal with the approximate solution of Fredholm's and Volterra's equations with the kernel of the kind *K*(*x-t*). We shall use the known algorithm for the search of the approximate solution in the form of a linear combination of preassigned basic functions

*φ*(*x*)* ≈ ∑nk=0 ckφk*(*x*),

with the help of Galerkin's method.

The principal matter of the paper is the choice of the specific basis {*φk*(*x*)} which:

- possesses high approximate properties, i.e., makes possible to find the approximate solution with a good accuracy, but with a small number of basic functions;
- makes possible (by using the inner properties of the functions
*φk*(*x*)) to easily transform the double integral by Galerkin's algorithm to a simple (of multiplicity 1) integral; - reduces the problem to a system of equations with a reducible matrix, i.e. reduces it to parallelizing an algorithm to two independent subsystems of equations if the kernel is
*K*(|*x-t*|).

In the Appendix we illustrate the use of the specific basis of functions by solving the integral Peierls equation.

File

smelov.pdf2.28 MB

Pages

93-100