标题： 关于通过PDE系统近似空间卷积的研究 显示英文标题

标题： On the approximation of spatial convolutions by PDE systems

摘要： 本文考虑了给定积分核的空间卷积的逼近问题。 通过偏微分方程系统对空间卷积进行逼近的方法已被提出，以消除一维空间中积分引起的解析困难。 在本文中，我们建立了高维空间中空间卷积的偏微分方程系统逼近方法。 我们通过Green函数的线性组合推导出给定任意径向积分核的逼近函数。 在该方法的证明中，我们引入适当的积分变换，以证明由Green函数构造的基的完备性。 从这一理论出发，可以通过偏微分方程组解的线性组合来逼近具有任意径向积分核的非局部卷积型算子。 最后，使用此方法展示了逼近的数值示例。 显示英文摘要

摘要： This paper considers the approximation problem for the spatial convolution with a given integral kernel. The approximation to the spatial convolution by PDE systems has been proposed to eliminate the analytical difficulties caused by the integrals for the one-dimensional space. In this paper we establish a PDE system approximation for the spatial convolutions in higher spatial dimensions. We derive an approximation function for given arbitrary radial integral kernels by linear sums of the Green functions. In the proof of this methodology we introduce an appropriate integral transformation to show the completeness of the basis constructed by the Green functions. From this theory, it is possible to approximate the nonlocal operator of the convolution type with any radial integral kernels by the linear sum of the solutions to the PDE system. Finally, numerical examples for the approximation are demonstrated using this method.