On a special integral equation with an exponential parameter in the kernel

Abstract

Applying Laplace transform on a generalized acoustical radiosity equation results in a special Fredholm integral equation of the second kind with $\lambda =1$ being the integral coefficient. The kernel of the equation contains a varying exponential complex parameter. The values of the parameter that make $\lambda =1$ be an eigenvalue of the kernel are defined in this paper as $L$-eigenvalues of the kernel, and the corresponding eigenfunctions are called $L$-eigenfunctions. The interest of this study is on the properties of the $L$-eigenvalues, $L$-eigenfunctions and the residues of related function at the $L$-eigenvalues. A set of theorems with a series of lemmas as bases are given and proven. They construct an integrated ensemble to reveal the decay structure of the generalized acoustical radiosity system with finite nonzero initial excitation.