Choice of Parameters for Solving Linear Nonclassical Equations of the First Kind

Abstract

In this paper, we have chosen a regularization parameter for solving the linear Volterra integral equation of the first kind. Various questions for integral equations have been investigated in many papers. Even when the Volterra equation of the first type is an integral equation with an exact output, non-classical equations integrable by the limit are linear, and nonlinear integral equations are linear, and this is due to the need to develop new methods for the uniqueness of their solutions. But in this paper, we have obtained fundamental results for Fredholm integral equations of the first kind, where regularizing operators are constructed according to M. M. Lavrentiev for solving linear integral Fredholm equations. On the basis of the concept of the introduced derivative of a function with respect to an increasing function, linear Volterra integral equations of the first kind were studied. The aim of the study is to construct a regularizing operator and choose a regularization parameter. In the study, we have applied the concept of a derivative with respect to an increasing function, the regularization method according to M. M. Lavrentiev, methods of functional analysis, methods of transformation of equations, methods of integral and differential equations. The parameter for regularization is selected. Regularizing operator according to M. M. Lavrentiev is constructed and a uniqueness theorem is proved. The proposed methods can be used for the study of integral, integral-differential equations such as the Volterra integral equation of the first kind, as well as for the qualitative study of some applied processes in physics, ecology, medicine, geophysics, and the theory of control of complex systems. In connection with the application of integral equations, new areas are developing, for example, in the economic sciences, in some sections of biology, etc. They can be used in the further development of the theory of Volterra integral equations of the first kind. And also, when solving specific applied problems that lead to equations of the first kind.