A Coupled Singular Fractional Integro-Differential System with Bessel Operator: Existence, Uniqueness, and Stability

This paper investigates a coupled system of two-dimensional singular fractional partial differential equations within a bounded domain. The system is defined by Neumann boundary conditions, non-local weighted integral constraints, memory terms, Caputo fractional derivatives, and a Bessel operator. Physically, the model captures memory effects, linear dissipation, and thermoelastic coupling to describe anomalous diffusion-wave dynamics and fractional viscoelastic thermoelasticity in cylindrically symmetric media. By developing a functional framework based on weighted Sobolev spaces and utilizing energy methods with integro-differential operators, we obtain essential a priori estimates. These estimates ensure that the solutions are unique and continuously dependent on the data. Furthermore, we establish the existence of solutions by demonstrating that the associated operator has a dense range and a trivial orthogonal complement. Ultimately, this study makes a substantial contribution to the well-posedness theory of singular fractional thermoelastic models subject to nonlocal conditions.