Qualitative analysis of the fractional Basset problem with boundary conditions via the Caputo–Fabrizio derivative

Abstract

The concept of fixed points serves as an effective and essential tool in analyzing nonlinear phenomena. This study investigates the existence and uniqueness of solutions for a class of Basset-type fractional differential equations with boundary conditions involving the Caputo–Fabrizio fractional derivative. These equations emerge from the generalized Basset force describing the motion of a sphere settling in a viscous fluid. Darbo’s fixed point theorem, combined with the measure of noncompactness, is applied to establish the existence of solutions. Uniqueness is ensured via Banach’s fixed point theorem. Additionally, stability analysis is performed using Ulam–Hyers and Ulam–Hyers–Rassias concepts. An illustrative example, supported by tables and figures, demonstrates the applicability of the theoretical results.