Partial Differential Equations Involving Fractional-Order Integro-Differentiation Operators

Abstract

This study investigates partial differential equations (PDEs) involving fractional-order integro-differentiation operators, motivated by the limitations of classical PDEs in capturing memory and non-local behaviors. While traditional models assume local interactions and instantaneous responses, many real-world systems—such as those in anomalous diffusion and viscoelasticity—exhibit hereditary effects that require a more generalized framework. Addressing this knowledge gap, we formulate a time-space fractional PDE model incorporating the Caputo derivative for temporal dynamics, a fractional Laplacian for spatial non-locality, and an integral memory kernel to encode historical dependence. Through weak formulation and energy methods, the existence and stability of solutions are analytically established under suitable kernel assumptions. Numerical simulations, employing finite difference and spectral methods, reveal that solutions with fractional orders α < 1 exhibit subdiffusive behavior and long-memory effects, deviating significantly from classical Gaussian diffusion. The results underscore the influence of fractional parameters on propagation speed, memory decay, and equilibrium profiles. These findings extend the modeling capabilities of PDEs to a broader class of physical systems and open new directions for applying fractional calculus in geophysics, materials science, and biomedicine. Future research is suggested in the development of adaptive high-dimensional solvers, exploration of variable-order systems, and coupling with other physical processes.