We consider a subdiffusion problem described by a time fractional Riemann–Liouville derivative of order $$0<\alpha <1$$ 0 < α < 1 . The main purpose of this work is to show how we can apply fractional splines of order $$0<\beta \le 1$$ 0 < β ≤ 1 to approximate a fractional integral and hence how to solve the subdiffusion problem using this approach. To discuss the convergence of the numerical method we present the error bounds for the fractional splines and the fractional integral approximations and study the von Neumann stability analysis. We observe that, depending on the smoothness of the solution, the order of convergence will be affected by the values of $$\alpha $$ α and $$\beta $$ β . Numerical tests are presented along the work to highlight several properties of the fractional splines and the numerical tests in the end illustrate the performance of the numerical method.

中文翻译：

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用于子扩散问题的分数样条数值方法

我们考虑由阶 $$0<\alpha <1$$ 0 < α < 1 的时间分数 Riemann-Liouville 导数描述的子扩散问题。这项工作的主要目的是展示我们如何应用 $$0<\beta \le 1$$ 0 < β ≤ 1 阶分数样条来近似分数积分，从而展示如何使用这种方法解决子扩散问题。为了讨论数值方法的收敛性，我们提出了分数样条和分数积分近似的误差界限，并研究了冯诺依曼稳定性分析。我们观察到，根据解的平滑度，收敛的顺序将受到 $$\alpha $$ α 和 $$\beta $$ β 值的影响。