Some mathematical models of applied problems lead to the need of solving boundary value problems with a fractional power of an elliptic operator. In a number of works, approximations of such a nonlocal operator are constructed on the basis of an integral representation with a singular integrand. In the present article, new integral representations are proposed for operators with fractional powers. Approximations are based on the classical quadrature formulas. The results of numerical experiments on the accuracy of quadrature formulas are presented. The proposed approximations are used for numerical solving a model two‐dimensional boundary value problem for fractional diffusion.

中文翻译：

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椭圆算子的分数幂的逼近

一些应用问题的数学模型导致需要用椭圆算子的分数幂来解决边值问题。在许多工作中，这种非局部算子的近似值是基于带有奇异被积数的整数表示的。在本文中，为具有小数幂的运算符提出了新的积分表示形式。近似值基于经典的正交公式。给出了关于正交公式精度的数值实验结果。所提出的近似值用于数值求解分数扩散的模型二维边界值问题。