An Extension to an Algebraic Method for Linear Time-Invariant System and Network Theory: The full AC-Calculus

TL;DR

This paper introduces an algebraic extension to the AC-calculus inspired by phasor analysis, enabling purely algebraic solutions for inhomogeneous linear differential equations in system and network theory.

Contribution

It develops a complex structure on elementary functions to algebraize inhomogeneous linear ODEs, extending the AC-calculus for more effective solutions.

Findings

Provides an algebraic method for solving inhomogeneous linear ODEs

Extends the AC-calculus framework to broader function spaces

Enables algebraic computation of particular solutions

Abstract

Being inspired by phasor analysis in linear circuit theory, and its algebraic counterpart - the AC-(operational)-calculus for sinusoids developed by W. Marten and W. Mathis - we define a complex structure on several spaces of real-valued elementary functions. This is used to algebraize inhomogeneous linear ordinary differential equations with inhomogenities stemming from these spaces. Thus we deduce an effective method to calculate particular solutions of these ODEs in a purely algebraic way.