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Abstract
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Strong models for explaining anomalous diffusion, non-local transport, and memory-dependent processes in physics, biology, engineering, and finance are the Caputo-Riesz time-space fractional diffusion equations. However, because the equations involve non-local operators in both time and space, their fractional nature presents significant analytical and numerical challenges. As a result, exact solutions are uncommon, and standard numerical techniques are either computationally expensive, low-order accurate, or unstable. In order to maintain accuracy, traditional finite difference or finite element methods frequently call for fine grids and small time steps, which significantly raises the computational cost for long-term or multidimensional simulations. High-order, reliable, and effective numerical techniques designed for these fractional models are therefore highly required. By fusing efficiency and accuracy, a fourth-order accurate hybrid approach fills this gap and allows for the precise resolution of fractional dynamics using less computing power. The significance of this work resides in developing a scalable and reliable computational framework that improves theoretical research and practical applications of anomalous transport phenomena by advancing the solution of intricate fractional diffusion problems.
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