Fractional Sequential Coupled Systems of Hilfer and Caputo Integro-Differential Equations with Non-Separated Boundary Conditions

Abstract

In studying boundary value problems and coupled systems of fractional order in (1,2], involving Hilfer fractional derivative operators, a zero initial condition is necessary. The consequence of this fact is that boundary value problems and coupled systems of fractional order with non-zero initial conditions cannot be studied. For example, such boundary value problems and coupled systems of fractional order are those including separated, non-separated, or periodic boundary conditions. In this paper, we propose a method for studying a coupled system of fractional order in (1,2], involving fractional derivative operators of Hilfer and Caputo with non-separated boundary conditions. More precisely, a sequential coupled system of fractional differential equations including Hilfer and Caputo fractional derivative operators and non-separated boundary conditions is studied in the present paper. As explained in the concluding section, the opposite combination of Caputo and Hilfer fractional derivative operators requires zero initial conditions. By using Banach’s fixed point theorem, the uniqueness of the solution is established, while by applying the Leray–Schauder alternative, the existence of solution is obtained. Numerical examples are constructed illustrating the main results.

Keywords

• Hilfer fractional derivative
• fractional integrals
• existence
• uniqueness
• fixed point theorems