Introduction to the Theory of $\mathcal{A}$-ODEs

TL;DR

This paper develops a comprehensive theory for differential equations over finite-dimensional real algebras, establishing existence, uniqueness, and solution methods, including extensions and algebraic techniques, for both nondegenerate and degenerate cases.

Contribution

It introduces the first systematic framework for $ ext{A}$-ODEs, including existence, uniqueness, solution methods, and the impact of zero-divisors, extending classical ODE theory to algebraic settings.

Findings

Proved existence and uniqueness theorems for $ ext{A}$-ODEs.

Derived Abel's formula and properties of the Wronskian in this context.

Presented three methods for solving nondegenerate $ ext{A}$-ODEs.

Abstract

We study the theory of ordinary differential equations over a commutative finite dimensional real associative unital algebra $\mathcal{A}$. We call such problems $\mathcal{A}$-ODEs. If a function is real differentiable and its differential is in the regular representation of $\mathcal{A}$ then we say the function is $\mathcal{A}$-differentiable. In this paper, we prove an existence and uniqueness theorem, derive Abel's formula for the Wronskian and establish the existence of a fundamental solution set for many $\mathcal{A}$-ODEs. We show the Wronskian of a fundamental solution set cannot be a divisor of zero. Three methods to solve nondegenerate constant coefficient $\mathcal{A}$-ODE are given. First, we show how zero-divisors complicate solution by factorization of operators. Second, isomorphisms to direct product are shown to produce interesting solutions. Third, our extension technique is shown to solve any nondegenerate $\mathcal{A}$-ODE; we find a fundamental solution set by selecting the component functions of the exponential on the characteristic extension algebra. The extension technique produces all of the elementary functions seen in the usual analysis by a bit of abstract algebra applied to the appropriate exponential function. On the other hand, we show how zero-divisors destroy both existence and uniqueness in degenerate $\mathcal{A}$-ODEs. We also study the Cauchy Euler problem for $\mathcal{A}$-Calculus and indicate how we may solve first order $\mathcal{A}$-ODEs.