Linear Partial Differential Equations in Module of Formal Generalized Functions over Commutative Ring

Abstract

We study the Cauchy problem for the linear differential equation ∂u/∂t = Fu in the module of formal generalized functions over a commutative ring, where F is a differential operator of finite or infinite order. We prove the well-posedness of the problem, find its fundamental solution, and obtain a representation of a unique solution to the Cauchy problem in the form of the convolution of the fundamental solution and initial condition. These results are specified for the heat equation, the simplest transport equation and the one-dimensional wave equation.