A Finite Difference Method for Caputo Generalized Time Fractional Diffusion Equations

This paper presents a finite difference method for solving the Caputo generalized time fractional diffusion equation. The method extends the L1 scheme to discretize the time fractional derivative and employs the central difference for the spatial diffusion term. Theoretical analysis demonstrates that the proposed numerical scheme achieves a convergence rate of order 2−α in time and second order in space. These theoretical findings are further validated through numerical experiments. Compared to existing methods that only achieve a temporal convergence of order 1−α, the proposed approach offers improved accuracy and efficiency, particularly when the fractional order α is close to zero. This makes the method highly suitable for simulating transport processes with memory effects, such as oil pollution dispersion and biological population dynamics.