The specific basis of trigonometric functions in the problem of approximate solution of integral equations with the kernel of the kind K(x-t)

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:

1. 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;
2. 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;
3. 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.