Fractional differential equations with mixed boundary conditions

TL;DR

This paper investigates the existence and uniqueness of solutions for a fractional differential equation of order between 2 and 3 with mixed boundary conditions, establishing equivalence with a Volterra equation and providing illustrative examples.

Contribution

It introduces new results on the existence and uniqueness of solutions for fractional differential equations with mixed boundary conditions, including equivalence with Volterra equations.

Findings

Proved equivalence between the boundary value problem and a Volterra integral equation.

Established conditions for the existence and uniqueness of solutions.

Provided illustrative examples demonstrating the theoretical results.

Abstract

In this paper, we discuss the existence and uniqueness of solutions of a boundary value problem for a fractional differential equation of order $\alpha\in(2,3)$, involving a general form of fractional derivative. First, we prove an equivalence between the Cauchy problem and the Volterra equation. Then, two results on the existence of solutions are proven, and we end with some illustrative examples.