imported>Russell at 16:31, 17 November 2012

2012-11-17T16:31:32Z

← Older revision		Revision as of 11:31, 17 November 2012
Line 8:		Line 8:
	* Models for [[dislocation dynamics]] in crystals.		* Models for [[dislocation dynamics]] in crystals.
	* 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...		* 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...
			* In hydrology, the so called [[Fractional Advection Dispersion Equation]] involves fractional operators similar to sums of one-dimensional [[Fractional Laplacian \| fractional Laplacians]] to describe super-diffusive spreading rates of tracer particles flowing in underground aquifers. The more typically used second order (local) models cannot correctly capture this spreading phenomenon in a suitable way.

	{{stub}}		{{stub}}

imported>Luis: Created page with "* 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 in..."

2012-04-25T14:56:24Z

Created page with "* 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 in..."

New page

* 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 encode information about the manifold, they include fractional powers of the Laplacian, which are nonlocal operators.
* In oceanography, the temperature on the surface may diffuse though the atmosphere giving rise to the [[surface quasi-geostrophic equation]].
* Models for [[dislocation dynamics]] in crystals.
* 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...

{{stub}}

Applications - Revision history

imported>Russell at 16:31, 17 November 2012

imported>Luis: Created page with "* 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 in..."