论文标题
扩散类非局部运算符的数值方法
Numerical Methods for a Diffusive Class Nonlocal Operators
论文作者
论文摘要
在本文中，我们开发了一个基于四元素的数值方案，以近似涉及卷积内核（$ν$）扩散类型的跨差异方程的解决方案。特别是，我们假设$ν$是对称的，在无穷大处呈指数衰减。我们考虑在有限域和$ \ r $中构成的问题。对于具有非本地dirichlet边界条件的有界域，我们显示了具有正尾巴的内核方案的收敛性，但可以承受负值。当方程式在所有$ \ r $上构成时，我们表明我们的方案会收敛于非负核。由于非局部边界条件与无界情况一样导致等效的配方，因此我们表明这些最后结果也适用于Neumann问题。
In this paper we develop a numerical scheme based on quadratures to approximate solutions of integro-differential equations involving convolution kernels, $ν$, of diffusive type. In particular, we assume $ν$ is symmetric and exponentially decaying at infinity. We consider problems posed in bounded domains and in $\R$. In the case of bounded domains with nonlocal Dirichlet boundary conditions, we show the convergence of the scheme for kernels that have positive tails, but that can take on negative values. When the equations are posed on all of $\R$, we show that our scheme converges for nonnegative kernels. Since nonlocal Neumann boundary conditions lead to an equivalent formulation as in the unbounded case, we show that these last results also apply to the Neumann problem.
