A note on non-separable solutions of linear partial differential equations

Abstract

A method has been presented for constructing non-separable solutions of homogeneous linear partial differential equations of the type F( D, D′) W = 0, where D = ∂ ∂x , D′ = ∂ ∂y , F(D,D′)= ∑ n r+s=0 C rsD rD′ s, where c rs are constants and n stands for the order of the equation. The method has also been extended for equations of the form Φ( D, D′, D″) W = 0, where D = ∂ ∂x , D′ = ∂ ∂y , D″ = ∂ ∂z and Φ(D,D′,D″)W= ∑ n r+s+t=0 C rstD rD′ sD tD″ s. As illustration, the method has been applied to obtain nonseparable solutions of the two and three dimensional Helmholtz equations.