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Abstract
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Research into an effective finite-difference technique for solving fractional convection--diffusion problems is critically important because these equations model complex physical phenomena such as anomalous transport and memory-dependent processes that classical integer-order models cannot adequately capture. However, the nonlocal nature and weak singularities of fractional derivatives pose significant numerical challenges, including instability, reduced accuracy, and high computational cost. Developing specialized finite-difference schemes that ensure stability, convergence, computational efficiency, and physical fidelity is therefore essential not only for advancing numerical analysis of nonlocal operators but also for enabling reliable simulations in real-world applications across physics, engineering, finance, and biomedicine.
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