A Noncommutative Mikusinski Calculus

TL;DR

This paper develops a noncommutative Mikusinski calculus by constructing a ring of fractions over boundary problems for differential equations, enabling symbolic calculus with boundary conditions.

Contribution

It introduces a noncommutative localization approach to extend Mikusinski calculus to boundary value problems, incorporating boundary conditions into the symbolic framework.

Findings

Constructed a left ring of fractions for boundary problems

Developed a module of generalized functions including boundary conditions

Established a symbolic calculus extending classical Mikusinski calculus

Abstract

We set up a left ring of fractions over a certain ring of boundary problems for linear ordinary differential equations. The fraction ring acts naturally on a new module of generalized functions. The latter includes an isomorphic copy of the differential algebra underlying the given ring of boundary problems. Our methodology employs noncommutative localization in the theory of integro-differential algebras and operators. The resulting structure allows to build a symbolic calculus in the style of Heaviside and Mikusinski, but with the added benefit of incorporating boundary conditions where the traditional calculi allow only initial conditions.