A Boundary Value Problem with Caputo–Hadamard Fractional Derivative: Analysis and Numerical Solution

Abstract

We investigate a boundary-value problem governed by a fractional differential equation, which is non-linear. The fractional derivative is the combined Caputo-Hadamard fractional derivative. We establish the required conditions for the existence and uniqueness of solutions utilising the two standard fixed-point theorems, Banach fixed-point and Sadovskii fixed-point. Furthermore, we address and analyse the problem's stability through demanding conditions using the Ulam-Hyers and Ulam-Hyers-Rassias stability methods. To illustrate the theoretical results, we present an example that validates the existence, uniqueness, and stability criteria. The analytical solution obtained from the problem is discretised  with fractional rectangular $L_{\ln, 1}$ interpolationon on a non-uniform mesh.  The resulting system of non-linear equations is solved using the Newton-Raphson method, incorporating the Jacobian matrix to couple $\eta(t)$ and $\theta(t, \eta(t))$ non-linearly.  The stability and reliability of the suggested numerical approach are examined through illustrative examples.