In recent years, the study of fractional differential equations has become a hot spot. It is more difficult to solve fractional differential equations with nonlocal boundary conditions. In this article, we propose a multiscale orthonormal bases collocation method for linear fractional-order nonlocal boundary value problems. In algorithm construction, the solution is expanded by the multiscale orthonormal bases of a reproducing kernel space. The nonlocal boundary conditions are transformed into operator equations, which are involved in finding the collocation coefficients as constrain conditions. In theory, the convergent order and stability analysis of the proposed method are presented rigorously. Finally, numerical examples show the stability, accuracy and effectiveness of the method.

中文翻译：

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基于多尺度正交基方法在再生核空间中求解具有非局部边界条件的线性分数方程。

近年来，分数阶微分方程的研究已成为热点。用非局部边界条件求解分数阶微分方程更加困难。在本文中，我们提出了一种用于线性分数阶非局部边值问题的多尺度正交基搭配方法。在算法构造中，解决方案由可再生内核空间的多尺度正交基扩展。将非局部边界条件转换为算子方程，该算子方程涉及寻找搭配系数作为约束条件。在理论上，对提出的方法进行了收敛阶和稳定性分析。最后，数值算例表明了该方法的稳定性，准确性和有效性。