Regularization of the solution of non-classical linear Volterra equations of the first kind with initial condition

Models of many problems of an applied nature are reduced to equations by an integral equation, among which non-classical equations are of special interest and little studied. Integral equations play an important role in the section of the integrodifferential equation. With the help of them, modern sciences and technologies are developing, i.e. they are widely used in the branches of mathematics, are used in physics, in mechanical engineering, in radio engineering, in computer technology, geophysics, control theory, etc. New areas related to the application of integral equations are developing, for example, economic sciences, some sections of biology and management, etc. With the help of modern computer technology, it becomes possible to implement a variety of numerical theories and simulate complex processes. In the same way, many problems are brought to integral equations. In this case, a qualitative study of problem solving comes to the fore. However, equations with two variable limits of integration, which are called non-classical, are poorly understood. This is due to difficulties in constructing a resolvent and in compiling a relation for it, because an analytical representation in general has not yet been obtained, with the exception of some model cases. Therefore, such research decisions are relevant. In this paper, the solution and regularization of the nonlinear nonclassical Volterra integral equation of the first kind is considered. The linear nonclassical Volterra integral equation of the first kind is solved using a derivative and is determined by regularization. The theorem is formulated by the proven fact. An appropriate example will be used, which will fully reveal the solution and evaluation.