A $\star$-Product Approach for Analytical and Numerical Solutions of Nonautonomous Linear Fractional Differential Equations

TL;DR

This paper introduces a new method using the $igstar$-product to solve nonautonomous linear fractional differential equations analytically and numerically, with potential for closed-form solutions.

Contribution

It develops a $igstar$-product framework for reformulating and discretizing solutions to fractional differential equations, offering a novel analytical and numerical approach.

Findings

The $igstar$-product reformulation facilitates solution derivation.

Discretization of the $igstar$-formalism enables numerical solutions.

In some cases, closed-form solutions are obtainable using this method.

Abstract

This article presents a novel solution method for nonautonomous linear ordinary fractional differential equations. The approach is based on reformulating the analytical solution using the $\star$-product, a generalization of the Volterra convolution, followed by an appropriate discretization of the resulting expression. Additionally, we demonstrate that, in certain cases, the $\star$-formalism enables the derivation of closed-form solutions, further highlighting the utility of this framework.