Fractional-order boundary value problem with Sturm-Liouville boundary conditions

TL;DR

This paper introduces a new approach to solving second-order boundary value problems using a novel conformable fractional derivative, establishing existence results with a fixed point theorem and comparing favorably to traditional derivatives.

Contribution

It develops a new framework with conformable fractional derivatives for boundary value problems and proves positive solutions using Green's functions and fixed point theory.

Findings

Existence of positive solutions established

Comparison shows advantages over Riemann-Liouville derivatives

New conformable derivative effectively reformulates boundary problems

Abstract

Using the new conformable fractional derivative, which differs from the Riemann-Liouville and Caputo fractional derivatives, we reformulate the second-order conjugate boundary value problem in this new setting. Utilizing the corresponding positive fractional Green's function, we apply a functional compression-expansion fixed point theorem to prove the existence of a positive solution. We then compare our results favorably to those based on the Riemann-Liouville fractional derivative.