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Solution of system of the Fredholm integral equation of the first kind

The article is devoted to the study of a system of two inhomogeneous Fredholm integral equations of the first kind with two required functions depending on one variable. Integral equations describe the restoration of a blurred image, production costs, etc. Fredholm integral equations with one desired function have been considered in many works, but relatively few works have been devoted to systems of such equations. The questions of stability for the solution of systems and the construction of a regularizing system of equations were investigated, but the solution was not constructed in an explicit form. In this paper, the kernels depend on two variables. The case is considered: in the kernels and inhomogeneities, the variables are separated in the equations; these functions are decomposed on the basis of two functions on the interval of integration. Examples of basic functions are given. A condition is determined under which the system has a unique solution in the chosen basis, formulated as a theorem. The solution is found in the form of an expansion in this basis. To illustrate the results obtained, an example is considered