Journal of Applied & Computational Mathematics

In this paper we may use piece wise constant functions for the special type of system of second kind integro differential equation of the first order. The main problem is reduced to linear system of algebraic equations. Some numerical examples are dedicated for showing efficiency and validity of the method.

The perfectly matched layer (PML) is a widely used tool to truncate the infinite domain in numerical computations. It can also be used in the modal expansion of an open Pekeris waveguide. In a bounded waveguide with PML, the modal expansion consists of three kinds of modes. They are propagation modes, leaky modes and PML modes. The PML modes are introduced by the utilization of PML, and depend on the parameters of the PML. The validity of the modal expansion of the PML-truncated waveguide is discussed in this paper. It is proved that the expansion coefficient tends to zero when the index of modes tends to infinity, thus the truncation of the infinite sum is reasonable. Moreover, it is also proved that the numerical computation of the coefficients is stable.