APPROACHES TO THE SOLUTION OF THE CONVOLUTION EQUATION VOLTERRA I KIND WITH MANY VARIABLE IN SPECIAL CASES

Abstract

We consider a two-dimensional Volterra convolution equation of the first kind in the so-called special case. The goal of the work is to construct a continuous approximation to the exact solution by compiling a regularizing singularly perturbed type of equation, which is a Fredholm equation of the second kind and can be solved using a computer. We denote the solutions of the original and its regularizing equations by φ(x,y) and , respectively. Three theorems have been proven; we present their statement briefly: 1) if ) (D),   (D),, then according to the  norm:  ε→0;  2) if a) the solution φ(x,y) together with the partial derivatives of the first order are continuous, and φ(x,0)=φ(0,y)=0, then under the continuity condition a(x,y) we have  ε→0, where ; 3) if the functions a(x,y) and φ(x,y) both satisfy the above condition a), then  ε→0, where  is continuous solution of a certain Fredholm equation of the second kind. The proven theorems, although small, still make a certain contribution to the development of the theory of equations of Volterra type. The results of the work can be used in the approximate solution of these equations, which are often encountered in practice. It would be interesting to consider in a similar formulation linear equations of the first kind of Volterra type, as well as Fredholm type, which are not necessarily convolutional.