In the present study, we introduce a high-order non-polynomial spline method designed for non-linear time-fractional reaction–diffusion equations with an initial singularity. The method utilizes the L2-1 scheme on a graded mesh to approximate the Caputo fractional derivative and employs a parametric quintic spline for discretizing the spatial variable. Our approach successfully tackles the impact of the singularity. The obtained non-linear system of equations is solved using an iterative algorithm. We provide the solvability of the novel non-polynomial scheme and prove its stability utilizing the discrete energy method. Moreover, the convergence of the proposed scheme has been established using the discrete energy method in the norm. It is proven that the method is convergent of order in the temporal direction and 4.5 in the spatial direction, where denotes the order of the fractional derivative and the parameter is utilized in the construction of the graded mesh. Finally, we conduct numerical experiments to validate our theoretical findings and to illustrate how the mesh grading influences the convergence order when dealing with a non-smooth solution to the problem.

中文翻译：

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表现出弱初始奇异性的非线性时间分数反应扩散方程的高阶数值近似


在本研究中，我们引入了一种高阶非多项式样条方法，专为具有初始奇点的非线性时间分数反应扩散方程而设计。该方法利用分级网格上的 L2-1 方案来近似 Caputo 分数阶导数，并采用参数五次样条来离散空间变量。我们的方法成功地解决了奇点的影响。使用迭代算法求解所获得的非线性方程组。我们提供了新颖的非多项式格式的可解性，并利用离散能量方法证明了其稳定性。此外，所提出方案的收敛性是使用范数中的离散能量方法建立的。证明该方法在时间方向上阶数收敛，在空间方向上阶数收敛为4.5，其中表示分数阶导数的阶数，该参数用于构建分级网格。最后，我们进行数值实验来验证我们的理论发现，并说明在处理问题的非光滑解决方案时网格分级如何影响收敛阶数。