On The Solutions of Fractional Boundary Value Problems for a Combined Caputo Fractional Differential Equations

Abstract

In this paper, a new class of fractional boundary value problem with the combined Caputo derivative is proposed and the physical interpretation of this new derivative has been explained. Under some assumptions, the positive solutions of the fractional differential equation with the help of Leray-Schauder and Krasnoselskii's fixed point theorems in a cone have been investigated. Moreover, the solution of the fractional Maxwell models involving the combined Caputo derivative by using the extended Laplace transform is obtained. Finally, some examples are given to support theoretical findings.