CONSTRUCTION OF A REGULARIZATION OF THE SOLUTION TO THE FREGHOLM INTEGRAL EQUATION OF THE FIRST KIND

Abstract

This article examines the Fredholm integral equation of the first kind with a discontinuous kernel and constructs a regularization of the solution using the principle of compressive reflections. The uniqueness of the solution is proven. Fredholm integral equations of the first kind with continuous kernels under certain conditions were considered on different function spaces. Thus, M.M. Lavrentiev proposed a normalization method for the Fredholm integral equation of the first kind with a continuous kernel. To prove the existence of solutions to linear Fredholm and Volterra integral equations of the first kind with one variable and sufficient conditions for obtaining them, A. M. Denisov, V. O. Sergeev and other authors used the method of differentiation with respect to given functions. In their works, M.M. Lavrentyev, M.I.Imanaliev and A. Asanov studied the solution of linear integral equations of the first kind in the space of functions C(G), and generalized Volterra integral equations of the first type with a non – smooth kernel. Such problems are reduced to integral equations of the first and third kind, where the kernel represents the properties of the medium, the free term is the result of measurements at the boundaries, and the desired function is an indicator of the medium at internal points. At the same time, the issues of uniqueness of the solution, as well as the construction of regularizing families of operators and evaluation of their effectiveness, come to the fore.