Closed-Form Solution of the Differential Equation (∂2∂x∂y+ax∂∂x+by∂∂y+cxy+∂∂t)P=0 by Normal-Ordering Exponential Operators

Abstract

A closed-form solution to Lambropoulos' partial differential equation (∂2∂x∂y+ax∂∂x+by∂∂y+cxy+∂∂t)P(x, y, t)=0,subject to the initial condition P(x, y, 0) = Φ(x, y), is presented. The applicability of the normal-ordering method to a class of partial differential equations is briefly discussed.