An Operational Method in Partial Differential Equations

Abstract

Introduction. In the present paper an calculus for functions of n variables is developed, analogous to that of Mikusinski for functions of one variable [6], and applications of this calculus to linear partial differential equations with constant coefficients are given. As is known (see [13]), the Laplace transform method (and other transform methods) may be used to reduce partial differential equation to an ordinary differential equation. The advantage of Mikusinski's calculus over the Laplace transform method (besides being well justified) lies in the fact that it can be applied in cases in which Laplace transform does not exist. To apply his calculus in partial differential equations Mikusinski introduces the notion of an operational function1 and reduces the solution of partial differential equation in two variables to solution of an ordinary differential equation. Using generalized functions [3], [4] one usually obtains theorems about existence and uniqueness of solutions of partial differential equations without being able to find them explicitly. The calculus developed here has the same advantage over the Laplace transform method as Mikusinski's calculus. Although closely related to Mikusinski's calculus, it seems to us to comprise, in comparison with the latter, further step towards the algebraization of solutions of partial differential equations. This is achieved by omitting the notion of the operational used by Mikusinski and by developing and justifying purely algebraic method of solutions of partial differential equations. The paper consists of four sections. In ? 1, the basic definitions and theorems of the calculus are given. Except for the definitions of some new operators and theorems concerning these operators, the material of? 1 is partially known and is given here only for the sake of completeness. However we stress that the notion of an as introduced in this work depends on some region in n-dimensional Euclidean space Rn. In ? 2 sufficient conditions are found for an to be function (Theorem 1), and the notion of a function contained in an operator is introduced. The existence of such function and its uniqueness for some operators are also shown (Theorem 4). Similar to the notion of an operator, the notion of function contained in an also depends on the region in Rn.