European Journal of Molecular & Clinical Medicine

Abstract

The behaviour of the eigenvalues of Dirichlet Boundary-Domain Integro-Differential
Equations (BDIDEs) with a reduced number of collocation points has never been
discussed theoretically. The uncertainty of the behaviour of the eigenvalues of Dirichlet
BDIDEs will prohibit the use of iteration methods in solving the BDIDEs system of
equations. The purpose of this paper is to demonstrate the spectral properties of matrix
operator obtained from the discretized Dirichlet BDIDEs with reduced number of
collocation points. We calculate the eigenvalues of the matrix operator, numerically. The
discussions of the spectral properties are based on the eigenvalues of the discretized
BDIDEs that are obtained numerically. In our numerical test, the attribution of the eigenvalues
for matrix operator obtained numerically for the discretized BDIDEs with reduced
number of collocation points exceeds 1. The findings demonstrate that it is utterly
impracticable to solve the system yielded from the discretized Dirichlet BDIDEs with a
reduced number of collocation points with an iterative method. The theoretical explanation
of why this behaviour occurs is also provided. With this result of the eigenvalues attained,
the matrix equations yielded from the discretized BDIDEs with a reduced number of
collocation points can purely be solved by direct methods.