The resolvent and the particular solution of one singular integral equation

Abstract

In this paper, we study the solvability of a singular integral equation arising in the theory of boundary value problems for the heat equation in an infinite angular domain. With the help of thermal potentials, the boundary value problem of heat conduction is reduced to a singular integral Volterra equation of the second kind. The corresponding homogeneous integral equation was investigated by us earlier in the previous paper of M.T. Jenaliyev and M.I. Ramazanov, and it was shown that in some weight class of essentially bounded functions it has, along with a trivial solution, a family of nontrivial solutions up to a constant factor. In this paper we study the nonhomogeneous integral equation, for which a representation of the general solution is found with the help of the resolvent constructed by us. Estimates of the resolvent are established.