Open AccessWe consider the linear Fredholm integral equation of the second kind, where the kernel and the free term are smooth functions. We find the unknown function in this class as well.
Exact and approximate methods for the solution of linear Fredholm integral equations of the second kind are well developed. However, classical methods do not take into account the structural properties of the kernel and the free term of equation.
In this paper we develop and justify a special variant of the moment method to solve this equation, which takes into account the differential properties of initial data. The proposed paper furthers studies of N.S Gabbasov, I.P. Kasakina, and S.A Solov’eva. We use approximation theory, version of the general theory of approximate methods of analysis that Gabdulkhayev B.G suggested, and methods of functional analysis to prove theorems. In addition, we use N.S. Gabbasov’s ideas and methods in papers that are devoted to the Fredholm equations of the first kind, as well as N.S. Gabbasov and S.A Solov’eva’s investigations on the Fredholm equations of the third kind in the space of distributions.
The first part of the paper provides a description of the basic function space and elements of the theory of approximation in it.
In the second part we propose and theoretically justify a generalized moment method. We have demonstrated that the improvement of differential properties of the initial data improves the approximation accuracy. Since, in practice, the approximate equations are solved, as a rule, only approximately, we prove the stability and causality of the proposed method. The resulting estimate of the paper is in good agreement with the estimate for the ordinary moment method for equations of the second kind in the space of continuous functions.
In the final section we have shown that a developed method is optimal in order of accuracy among all polynomial projection methods to solve Fredholm integral equations of the second kind in the space of smooth functions.
We recommend using a developed method in case when the initial data are continuously differentiable functions, and moreover, the accuracy of the approximate solution is necessary to estimate by the norm of the space of smooth functions.
Similarly, we can develop other polynomial and spline methods for the approximate solution.