Representation of Solutions of Linear Homogeneous Caputo Fractional Differential Equations with Continuous Variable Coefficients

TL;DR

This paper develops series representations for solutions of linear homogeneous Caputo fractional differential equations with continuous variable coefficients, revealing how solution forms depend on derivative orders and coefficient distributions.

Contribution

It introduces a novel series representation for the fundamental solutions of such equations, accounting for variable coefficients and fractional derivative orders.

Findings

Series representation of fundamental solutions derived

Solution forms depend on derivative order distribution

Representation varies with coefficient and order configurations

Abstract

We consider the canonical fundamental systems of solutions of linear homogeneous Caputo fractional differential equations with continuous variable coefficients. Here we gained a series-representation of the canonical fundamental system by coefficients of the considered equations and the representation of solution to initial value problems using the canonical fundamental system. According to our results, the canonical fundamental system of solutions to linear homogeneous differential equation with Caputo fractional derivatives and continuous variable coefficients has different representations according to the distributions of the lowest order of the fractional derivatives in the equation and the distance from the highest order to its adjacent order of the fractional derivatives in the equation.