INITIAL VALUE PROBLEM FOR FRACTIONAL-ORDER DIFFERENTIAL EQUATIONS IN ANOMALOUS DIFFUSION PROCESSES - Badalov Tolib Tohir òģli (Prospects for innovative technologies in science and education, 2026)
The study focuses on the initial value problem (IVP) for fractional-order differential equations in anomalous diffusion processes. Unlike classical diffusion, anomalous diffusion exhibits nonlocal temporal behavior and memory effects, which can be accurately described using fractional derivatives. The paper reviews analytical and numerical methods for solving IVPs, including Laplace and Fourier transforms, finite difference schemes, and physics-informed neural networks (PINNs). Applications of fractional IVPs in physics, biology, hydrology, and finance are highlighted, demonstrating their ability to model subdiffusive and superdiffusive transport. The integration of stochastic models, computational techniques, and experimental validation provides a comprehensive framework for understanding complex transport phenomena in heterogeneous media.