The justification of the truncation applied to homogeneous integral equations with Cauchy's kernel

Abstract

This paper is devoted to studying the influence of the truncation applied to a homogeneous Cauchy-type singular integral equation on its eigenvalues and eigenfunctions. This equation represents the class to which mixed Sturm-Liouville problems of the Dirichlet-Neuman type are reducible. The study is illustrated through considering a concrete problem. It has been shown that an eigenvalue of the truncated equation tends to the exact corresponding one on increasing the order of the truncation set on that equation. As for a truncated eigenfunction, it leads to the designation of the exact one in the limit where the corresponding eigenvalue is sufficiently precise. An illustrative example is given.