A Fully Discrete Numerical Scheme for Nonlinear Fractional PDEs with Caputo Derivatives and Fredholm Integral Terms

In this work, we propose a nonlinear fractional partial differential equation model incorporating a Caputo fractional derivative in time, a second-order spatial derivative, and a nonlinear Fredholm integral term. This model accounts for memory effects, anomalous diffusion, and nonlocal interactions, offering a more realistic description of complex transport phenomena compared to classical integer-order models. To solve the model numerically, we develop a fully discrete scheme that combines Lagrange interpolation-based approximation for the Caputo fractional derivative in time with central difference discretization for the spatial derivative. This approach ensures accuracy and flexibility in handling both the fractional derivative and the nonlinear integral term. A comprehensive convergence and stability analysis is conducted, establishing second-order accuracy in space and nearly second-order accuracy in time. Rigorous error estimates confirm the reliability and robustness of the proposed scheme for practical computations. Finally, a numerical example with a known exact solution is solved to validate the method. Errors are computed in both the L2 and maximum norms, and the temporal and spatial convergence orders are verified. The results, summarized in tables, demonstrate the effectiveness of the fully discrete scheme and underscore the practical utility of the proposed fractional model in complex physical and engineering systems.