Steady-State Solutions in an Algebra of Generalized Functions

TL;DR

This paper derives solutions for differential equations with singular driving terms within a Colombeau algebra, revealing solutions with potential physical relevance that extend beyond Schwartz distribution theory.

Contribution

It introduces a method to solve initial value problems with singular inputs using generalized functions, including infinite constants, expanding the scope of solutions beyond traditional distribution theory.

Findings

Solutions involve infinitely large constants like δ(0)

Some solutions may have physical interpretations

Extends solution methods beyond Schwartz distributions

Abstract

Formulas for the solutions of initial value problems for ordinary differential equations with singular $\delta^{(n)}$-like driving terms are derived in the framework of an algebra of generalized functions (of Colombeau type) over a field of generalized scalars. Some of the solutions might have physical meaning - such as of the electrical current after lightning or under superconductivity - but do not have counterparts in the theory of Schwartz distributions. What is somewhat unusual (compared with other similar works) is the involvement of infinitely large constants, such as $\delta(0)$, in some of the formulas for the solutions.