Positive Green's functions for some fractional-order boundary value problems

TL;DR

This paper introduces positive Green's functions for fractional boundary value problems using the conformable fractional derivative, providing a new approach to analyze such problems with various boundary conditions.

Contribution

It develops Green's functions for fractional boundary value problems using the conformable derivative, establishing their positivity under certain conditions, which is a novel approach.

Findings

Green's functions are positive under certain assumptions

The conformable fractional derivative differs from classical derivatives

Boundary value problems are reformulated with new Green's functions

Abstract

We use the newly introduced conformable fractional derivative, which is different from the Caputo and Riemann-Liouville fractional derivatives, to reformulate several common boundary value problems, including those with conjugate, right-focal, and Lidstone conditions. With the fractional differential equation and fractional boundary conditions established, we find the corresponding Green's functions and prove their positivity under appropriate assumptions.