Recursive construction of a sequence of solutions to homogeneous linear partial differential equations with constant coefficients

Abstract

For second order homogeneous partial difference equations with constant coefficients in n variables, it is always possible to construct a generating function of exponential form containing n − 1 arbitrary parameters, from which a sequence of solutions ( P i 1,…, i n − 1 ) can be derived recursively. The determination of a solution u = Σ⋯ Σai 1,…, i n − 1 Pi 1,…, i n − 1 that approximately satisfies given boundary conditions, for a given problem, is then possible by solving a linear least squares problem. Once the a i 1,…, i n − 1 are known, the evaluation of u and its partial derivatives can be done very easily, using a second recursion, that can be derived from the recursion used for the determination of the P i 1,…, i n − 1 . Some classical examples will be treated, for which the relevant formulae are given completely.