Steady-State Solutions in an Algebra of Generalized Functions

Formulas for the solutions of initial value problems for ordinary differential equations with singular $δ^{(n)}$-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 $δ(0)$, in some of the formulas for the solutions.

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