On the Production of Nonseparable Solutions of Linear Partial Differential Equations

Abstract

A particular means of producing nonseparable solutions to a linear partial differential equation from given separable solutions to the same partial differential equation is exposed and extended. The method, involving symmetry operators, is straightforward, requiring only basic partial differentiation. Examples are presented for important standard second-order linear partial differential equations, with particular attention being given to Helmholtz equations, for which previously known ad hoc solutions are derived within the current methodology. Although the examples presented are restricted to second-order linear partial differential equations, the method is developed in sufficient generality to cover all linear partial differential equations and to make comparison with previous work meaningful. Indeed, comparison with previous work enables the development of certain novel results.