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5. Discrete operators to nonlocal operators

    1.     Consider the following nonlocal derivative, $$\mathcal{D}_\delta u(x) = \int_{-\delta}^\delta [u(x + y) - u(x)]K(x,y)dy.$$ If $K(x,y)$ is odd in $y$, then one can show that this converges to a derivative, $$\mathcal{D}_\delta u(x) \to u'(x).$$

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      Problem 5.1.

      [Petronela Radu] With the above notion of nonlocal derivative, characterize the behavior of the nonlocal transport equation $$\partial_t u + \mathcal{D}_\delta u = 0$$ as $\delta \to 0$.

        • Convergence of graph operators to nonlocal operators in $\R^n$


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          Problem 5.2.

          [Cynthia Flores] Show convergence of graph operators to their nonlocal counterparts in the limit as the $\#$ of vertices “fills up" $\R^n$.

              Cite this as: AimPL: Integro-differential equations in many-particle interacting systems, available at http://aimpl.org/manyparticle.