Artificial boundary conditions for the semi-discretized one-dimensional nonlocal Schrödinger equation

A general method is proposed to build exact artificial boundary conditions for the one-dimensional nonlocal Schrödinger equation. To this end, we first consider the spatial semi-discretization of the nonlocal equation, and then develop an accurate numerical method for computing the Green's function of the semi-discrete nonlocal Schrödinger equation. These Green's functions are next used to build the exact boundary conditions corresponding to the semi-discrete model. Numerical results illustrate the accuracy of the boundary conditions. The methodology can also be applied to other nonlocal models and could be extended to higher dimensions

On partial differential equations modified with fractional operators and integral transformations: Nonlocal and nonlinear PDF models

We explore nonlocal and pseudo-differential operators in the setting of partial differential equations (PDE). The two primary PDE in this work are the generalized heat equation and the nonlocal Burgers' type advection-diffusion equation. These nonlocal and nonlinear models arise in complex physical systems including material phase transition and fluid flow