The estimation of the error resulting due to the truncation applied to homogeneous integral equations with Cauchy's Kernel

Abstract

Many mixed Sturm-Liouville problems can be formulated in a standard form of a discrete Riemann problem. When it is concerned with the Holder-continuous entity at the points where the boundary condition changes, the discrete problem can be transformed to a homogeneous integral equation with Cauchy's kernel. This equation can in general be approximately solved, and namely through truncation. This gives rise to several questions about the justification of the truncation applied to a homogeneous operator as well as the influence of this truncation on the eigenfunctions and eigenvalues. In this paper, it has been shown that, provided an eigenvalue of this integral equation is precisely given, the corresponding solution of the truncated integral equations tends to the unique solution of the exact equation on increasing the truncations order indefinitely, and the relative error is estimated. As for the influence of the truncation on the eigenvalues of that equation, it may constitute the subject of another study.