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

摘要

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 (Pi1,…, in − 1) can be derived recursively.The determination of a solution u = Σ⋯Σai1,…, in − 1Pi1,…, in − 1 that approximately satisfies given boundary conditions, for a given problem, is then possible by solving a linear least squares problem. Once the ai1,…, in − 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 Pi1,…, in − 1.Some classical examples will be treated, for which the relevant formulae are given completely.