Matrix method of polynomial solutions to constant coefficient PDE's

TL;DR

This paper presents a matrix-based approach to explicitly construct polynomial solutions to systems of constant coefficient linear PDEs, enabling analysis of solution spaces and their properties.

Contribution

The paper introduces a novel matrix method for constructing polynomial solutions to constant coefficient PDEs, linking differential operator null-spaces to block matrix null-spaces.

Findings

Matrix method effectively determines solution space dimensions.

Application to Laplace, Helmholtz, Poisson equations demonstrates practical utility.

Method allows basis construction for polynomial solution spaces.

Abstract

In the paper, we introduce a matrix method to constructively determine spaces of polynomial solutions (in general, multiplied by exponentials) to a system of constant coefficient linear PDE's with polynomial (multiplied by exponentials) right-hand sides. The matrix method reduces the funding of a polynomial subspace of null-space of a differential operator to the funding of the null-space of a block matrix. Using this matrix approach, we investigate some linear algebra properties of the spaces of polynomial solutions. In particular, for a solution space containing polynomials up to some arbitrarily large degree, we can determine dimension and basis of the space. Some examples of polynomial (multiplied by exponential, in general) solutions of the Laplace, Helmholtz, Poisson equations are considered.