Fractional Calculus on Manifolds: A Generalized Geometric Framework for Anomalous Diffusion

This study introduces an advanced geometric framework that extends fractional calculus to Riemannian manifolds, facilitating the modeling of anomalous diffusion in curved spaces. By formulating fractional differential operators—specifically, the fractional Laplace–Beltrami operator and non-local fractional gradients—adapted to the intrinsic geometry of manifolds, we bridge the gap between classical differential geometry and non-local calculus. Employing tools from spectral theory and functional analysis, we develop generalized models that naturally incorporate curvature and topological features of the underlying space. This approach is motivated by empirical observations of subdiffusion and superdiffusion phenomena in various contexts, including biological systems, financial markets, and quantum mechanics, where traditional integer-order models fall short. We also provide numerical schemes and simulation results to validate the theoretical framework, demonstrating its efficacy in capturing realistic anomalous diffusion patterns across diverse geometric domains.