Spectral generalized function method for solving homogeneous partial differential equations with constant coefficients

Abstract

A method based on a generalized function in Fourier space gives analytical solutions to homogeneous partial differential equations with constant coefficients of any order in any number of dimensions. The method exploits well-known properties of the Dirac delta, reducing the differential mathematical problem into the factorization of an algebraic expression that finally has to be integrated.  In particular, the method was utilized to solve the most general homogeneous second order partial differential equation in Cartesian coordinates, finding a general solution for non-parabolic partial differential equations, which can be seen as a generalization of d'Alambert solution.  We found that the traditional classification, i.e., parabolic, hyperbolic and elliptic, is not necessary reducing the classification to only parabolic and non-parabolic cases. We put special attention for parabolic partial differential equations, analyzing the general 1D homogeneous solution of the Photoacoustic and Photothermal equations in the frequency and time domain.  Finally, we also used the method to solve Helmholtz equation in cylindrical coordinates, showing that it can be used in other coordinates systems.