Solvability and construction of solution of the first kind Fredholm integral equation

Abstract

The solvability and construction of the general solution of the the first kind Fredholm integral
equation are among the few studied problems in mathematics. There are various approaches to
solving this problem. Note the following methods for solving ill-posed problem: regularization
method, the method of successive approximations, the method of undetermined coefficients. The
purpose of this work to create a new method for solvability and construction of solution of integral
equation of the first kind. It follows from the foregoing, the study of the solvability and construction
of the solution of the Fredholm integral equation of the first kind is topical. In this paper the
solvability and construction of the solution matrix Fredholm integral equation of the first kind
is considered. Construction of an approximate solution of Fredholm integral equation of the first
kind. The results are valid for the matrix Fredholm integral equation of the first kind, like with
asymmetric core and symmetric. A new method for studying of solvability and construction of
a solution for Fredholm integral equation of the first kind is proposed. Necessary and sufficient
conditions for existence of solutions for a given right-hand side are obtained in two cases: when
the origin function belongs to the space L2; origin function belongs to a given set of L2: Solvability
conditions and the method of construction an approximate solution of the integral Fredholm
equation of the first kind are obtained.