Construction of Fundamental Solution for an Odd-Order Equation

Abstract

In previous papers, we obtained some delta-shaped partial solutions of odd-order equations with multiple characteristics and studied some of their properties. In this paper, we first obtain the necessary estimates at infinity for these solutions, and then construct a fundamental solution (FS) of an odd-order equation with multiple characteristics in a rectangular domain as the sum of these particular solutions. We show that the FS is a solution to an inhomogeneous equation with multiple characteristics in a rectangular domain. In turn, knowledge of FS allows us to construct a potential theory for its further use in solving boundary-value problems.