Analysis of sequential \(\Phi \)-Hilfer fractional partial differential equations

Abstract

This paper presents a detailed analytical framework for a class of fractional partial differential equations involving the \(\Phi \)-Hilfer fractional derivative and Barbashin-type structures. By using a \(\Phi \)-Bielecki-type norm on a suitable Banach space, we establish a robust functional framework for modeling systems with memory and hereditary effects. Our formulation encompasses several well-known fractional derivatives, such as the Riemann–Liouville, Caputo, and Hadamard types, as special cases. We rigorously prove the existence and uniqueness of mild solutions using fixed-point theorems. In addition, we examine the generalized Ulam–Hyers–Rassias stability of solutions under perturbations. Illustrative examples and numerical results support the theoretical findings and demonstrate the effectiveness of our approach in analyzing complex dynamical systems influenced by nonlocal and history-dependent processes.