Sequences of related linear PDEs

Abstract

We introduce a transformation theory to generate chains of linear PDEs and related solution sequences. Our procedure depends on representing a given linear PDE in terms of a special equivalent system of two coupled linear PDEs where the auxiliary dependent variable satisfies the next PDE of a sequence. The solution of a PDE with variable coefficient depending on n + 1 constants α 1, α 2, …, α n + 1} is obtained from any solution of a PDE of the same type with variable coefficient depending on n constants {; α 1, α 2…, α n } by a simple Bäcklund transformation. Each sequence contains two inclusive chains since the PDE with n constants is a special case of the PDE with n + 2 constants. We generate solutions of wave equations with wave speeds C( x; α 1, α 2, …, α n ), Fokker-Planck equations with drifts F( x; α 1, α 2, … α n ), and diffusion equations with diffusivities K( x; α 1 α 2, …, α n ), where { α 1, α 2, …, α n } are arbitrary constants, n = 1, 2, … New explicit general solutions are obtained for a class of wave equations with wave speeds depending on three parameters.