On a boundary value problem for the heat equation and a singular integral equation associated with it

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. The particular case of the corresponding homogeneous integral equation was investigated earlier in [1, 2] and it was shown that in a weight class of essentially bounded functions it has, along with a trivial solution, a family of non-trivial solutions up to a constant factor. In this paper we study the more general case of a nonhomogeneous integral equation for which a representation of the general solution is found with using the resolvent constructed by us. Estimates of the resolvent and of the solution of the boundary value problem are established. (C) 2021 Institute of Mathematics and Mathematical Modeling, Ministry of Education and Science of Republic of Kazakhstan. Published by Elsevier Inc.

Automated summary

This paper examines the solvability of a singular integral equation arising in the theory of boundary value problems for the heat equation in an infinite angular domain, and investigates the general solution representation of a nonhomogeneous integral equation. Estimates for the resolvent and the solution of the boundary value problem are also established in the study.