一类扩散非局部算子的数值方法
Numerical Methods for a Diffusive Class Nonlocal Operators
摘要：
在本文中，我们开发了一个数值方法的基础上求积近似解的积分微分方程的卷积核，$ν$，扩散型。特别地，我们假设$ν$是对称的并且在无穷远处指数衰减。我们考虑有界域和$\R$中的问题。在有界域的非局部Dirichlet边界条件的情况下，我们证明了该计划的收敛性的核有正的尾巴，但可以采取负值。当所有的$\R$上的方程，我们表明，我们的计划收敛的非负内核。由于非局部诺依曼边界条件导致一个等价的制定在无界的情况下，我们表明，这些最后的结果也适用于诺依曼问题。
英文摘要：
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.
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