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Why nonlocal equations?
All partial differential equations are a limit case of nonlocal equations. A good understanding of nonlocal equations can ultimately provide a better understanding of their limit case: the PDEs. However, there are some cases in which a nonlocal equation gives a significantly better model than a PDE. Some of the most clear examples in which it is necessary to resort to nonlocal equations are
- Optimal control problems with Levy processes give rise to the Bellman equation, or in general any equation derived from jump processes will be some fully nonlinear integro-differential equation.
- In financial mathematics it is particularly important to study models involving jump processes. This can be considered a particular case of the item above (stochastic control), but it is a very relevant one. The pricing model for American options involves the obstacle problem.
- Nonlocal electrostatics is a very promising tool for drug design which could potentially have a strong impact in medicine in the future.
- The denoising algorithms in nonlocal image processing are able to detect patterns in a better way than the PDE based models. A simple model for denoising is the nonlocal mean curvature flow.
- The Boltzmann equation models the evolution of dilute gases and it is intrinsically an integral equation. In fact, simplified kinetic models can be used to derive the fractional heat equation without resorting to stochastic processes.
- In conformal geometry, the Paneitz operators $\mathcal{P}(s)$ encode information about the manifold, when the parameter is not an even integer they are fractional powers of the Laplacian, and therefore are nonlocal.
- In oceanography, the temperature on the surface may diffuse though the atmosphere giving rise to the surface quasi-geostrophic equation.
- Several stochastic models, in particular particle systems, can be used to derive nonlocal equations like the Nonlocal porous medium equation, the Hamilton-Jacobi equation with fractional diffusion, conservation laws with fractional diffusion, etc...
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