Green’s Function for the Cauchy Problem to the Dissipative Linear Evolution Equation of Arbitrary Order

This work addresses the Cauchy problem for a linear equation with a first-order time derivative t and an arbitrary-order spatial derivative x. This equation is a generalization of the linear heat equation of the second order in the case of arbitrary order with respect to spatial variable. The considered linear equation arises from the linearization of the Burgers hierarchy of equations. The Cauchy problem to a linear equation can be solved using the Green function method. The Green function is explicitly constructed for the case of dissipative and dispersive equations and is expressed in terms of generalized hypergeometric functions. The general formulas obtained for representing Green’s function are new. A discussion of specific cases of the equation is also provided.