Finite Rank and Complete Solutions of Euler–Poisson–Darboux Equations

This article deals with so-called finite-rank solutions, originally introduced by Laplace for linear second-order partial differential equations (PDEs) in the plane. They consist of linear combinations of undetermined functions and their derivatives up to a certain order, referred to as their rank. The article presents an algorithmic method for determining finite-rank solutions for linear PDEs of arbitrary order and with any number of independent variables—representing a significant generalization of Laplace’s method. This approach is developed in detail for Euler–Poisson–Darboux equations with one, two, or three spatial variables. Several solutions are explicitly provided and compared with so-called complete solutions. Furthermore, the extension of this method to general linear PDEs is discussed.