Numerical Methods for a Diffusive Class Nonlocal Operators

TL;DR

This paper introduces a quadrature-based numerical scheme for solving integro-differential equations with symmetric, exponentially decaying diffusive kernels, demonstrating convergence in bounded and unbounded domains under various boundary conditions.

Contribution

The paper develops and proves convergence of a new numerical scheme for nonlocal operators with diffusive kernels, applicable to bounded and unbounded domains with different boundary conditions.

Findings

Convergence shown for kernels with positive tails and negative values in bounded domains.

Scheme converges for nonnegative kernels on the entire real line.

Results extend to nonlocal Neumann boundary conditions.

Abstract

In this paper we develop a numerical scheme based on quadratures to approximate solutions of integro-differential equations involving convolution kernels, $\nu$, of diffusive type. In particular, we assume $\nu$ 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.