We analyze operator splitting methods applied to scalar equations with a nonlinear advection operator, and a linear (local or nonlocal) diffusion operator or a linear dispersion operator. The advection velocity is determined from the scalar unknown itself and hence the equations are so-called active scalar equations. Examples are provided by the surface quasi-geostrophic and aggregation equatio...

We review some results about nonlocal advection-diffusion equations based on lower bounds for the fractional Laplacian. To Haim, with respect and admiration.

The present notes contain the material presented at the XLI Summer School on Mathematical Physics (Ravello September 05 10, 2016). The notes aim in introducing a physical way to look at well-known functional inequalities in sharp form. Among them, the classical Young’s inequality and its converse in sharp form, BrascampLieb type inequalities, Babenko’s inequality and Prékopa-Leindler inequality...

The basic governing equations in anisotropic elastic material under the effect of porous piezothermoelastic are presented. Biot [1], Lord & Shulman [4] and Sherief et al. [5] theories are used to develop the basic equations for porous piezothermoelastic with mass diffusion material. The variational principle, uniqueness theorem and theorem of reciprocity in this model are established under the ...

In this paper, we show that the calibration to an implied volatility surface and the pricing of contingent claims can be as simple in a jump-diffusion framework as in a diffusion one. Indeed, after defining the jump densities as those of diffusions sampled at independent and exponentially distributed random times, we show that the forward and backward Kolmogorov equations can be transformed int...

the kinetics of zn sorption by ten calcareous soils of iran were measured and fitted to several equations (zero-, first-, second-, third-order, parabolic diffusion, simple elovich, and exponential equations). two initial concentrations of zinc i.e. 20 and 80 mg zn l-1 were used and observations were carried out for 100 hours. the time required to reach the near equilibrium state was about 24 ho...

Fractional diffusion equations are useful for applications where a cloud of particles spreads faster than the classical equation predicts. In a fractional diffusion equation, the second derivative in the spatial variable is replaced by a fractional derivative of order less than two. The resulting solutions spread faster than the classical solutions and may exhibit asymmetry, depending on the fr...

Recently, we have proposed a method for solving steady-state convection-diffusion equations, including the full compressible Navier-Stokes equations [17]. The method is a combination of a mixed Finite Element method for the diffusion terms, and a Discontinuous Galerkin method for the convection term. The method is fully implicit, and the globally coupled unknowns are the hybrid variables, i.e.,...