A Caputo-based nonlocal arithmetic-mean discretization for solving nonlinear time-fractional diffusion equation using half-sweep KSOR

This paper introduces a novel numerical method for solving one-dimensional nonlinear time-fractional diffusion equations (1DNTFDEs), addressing computational challenges in modeling nonlinearity and fractional dynamics. The proposed method integrates the Half-sweep Kaudd Successive Over-Relaxation (HSKSOR) technique with a Caputo-based nonlocal arithmetic-mean discretization scheme. The Caputo fractional derivative is leveraged to model time-fractional dynamics, while the half-sweep Caputo-based nonlocal arithmetic-mean scheme efficiently handles nonlinear terms, transforming the nonlinear system into a linear one solved iteratively using HSKSOR. Numerical experiments on three benchmark examples demonstrate significant reductions in iteration counts and computational time. The HSKSOR method outperforms traditional iterative techniques such as Full-Sweep Gauss-Seidel (FSGS) and Full-Sweep Kaudd Successive Over-Relaxation (FSKSOR) methods, achieving superior computational efficiency without sacrificing accuracy. The proposed method provides an efficient and scalable computational framework for solving complex time-fractional models, offering high accuracy and substantial computational cost reductions. This advancement enhances the theoretical framework of nonlocal discretization and offers a powerful tool for applications in physics, engineering, and applied mathematics, where modeling fractional dynamics is critical.