Matrix method of polynomial solutions to constant coefficient PDE's

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.