Particular solutions of products of Helmholtz-type equations using the Matern function

Abstract

We derive closed-form particular solutions for Helmholtz-type partial differential equations. These are derived explicitly using the Matern basis functions. The derivation of such particular solutions is further extended to the cases of products of Helmholtz-type operators in two and three dimensions. The main idea of the paper is to link the derivation of the particular solutions to the known fundamental solutions of certain differential operators. The newly derived particular solutions are used, in the context of the method of particular solutions, to solve boundary value problems governed by a certain class of products of Helmholtz-type equations. The leave-one-out cross validation (LOOCV) algorithm is employed to select an appropriate shape parameter for the Matern basis functions. Three numerical examples are given to demonstrate the effectiveness of the proposed method.