In this paper, self-similarity is illustrated and compared in deterministic and stochastic dissipative systems. Examples are (1) deterministic self-similarity in reaction-diffusion system and Navier-Stokes equations, where solutions eventually decay to zero due to balance of diffusion (viscosity) and nonlinearity; (2) statistical self-similarity in randomly advected passive scalar model of Krai...

Numerous models of industrial processes, such as diffusion in glassy polymers or solidification phenomena, lead to general one phase free boundary value problemswith phase onset. The classicalwell-posedness of a fast diffusion approximation to the concerned free boundary value problems is proved. The analysis is performed via a singular change of variables leading to a singular system in a fixe...

Here, we adapt the concept of transformational thermodynamics, whereby the flux of temperature is controlled via anisotropic heterogeneous diffusivity, for the diffusion and transport of mass concentration. The n-dimensional, time-dependent, anisotropic heterogeneous Fick's equation is considered, which is a parabolic partial differential equation also applicable to heat diffusion, when convect...

where QT = R d×(0, T ) with T > 0 fixed, u : QT → R is the sough function, V : R → R is a (not necessarily divergence free) velocity field, f : R → R is the convective flux function, and A : R → R is the “diffusion” function. For the diffusion function the basic assumption is that A(·) is nonincreasing. This condition implies that (1.1) is a (strongly) degenerate parabolic problem. For example,...

Numerical methods are presented for simulating stochastic-in-time current pulses for an electrodiffusion model of the biological channel, with a fixed applied voltage across the channel. The electrodiffusion model consists of the parabolic advection-diffusion equation coupled either to Gauss' law or Poisson's equation, depending on the choice of boundary conditions. The TRBDF2 method is employe...

A novel anisotropic diffusion-based image denoising and restoration approach is proposed in this paper. A variational model for image restoration is introduced first, then the corresponding Euler-Lagrange equation being determined. A nonlinear parabolic PDE model is then obtained from this equation. It is based on a novel edge-stopping function and conductance parameter. A serious mathematical ...

In order to emphasize the possible relation between discontinuous and continuous approximations on different meshes, a two-grids method for the resolution of parabolic variational inequality problems is presented. The numerical methodology combines a time splitting algorithm to decouple a diffusion phenomenon from an obstacle problem. The diffusion problem is solved by using finite differences,...

In this work we propose a numerical scheme for a nonlinear and degenerate parabolic problem having application in petroleum reservoir and groundwater aquifer simulation. The degeneracy of the equation includes both locally fast and slow diffusion (i.e. the diffusion coefficients may explode or vanish in some point). The main difficulty is that the true solution is typically lacking in regularit...

We study solution techniques for parabolic equations with fractional diffusion and Caputo fractional time derivative, the latter being discretized and analyzed in a general Hilbert space setting. The spatial fractional diffusion is realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi-infinite cylinder in one more spatial dimension. We write our evolution ...

In this paper, we investigate the Cauchy problem for the generalized IBq equation with damping in one dimensional space. When σ = 1, the nonlinear approximation of the global solutions is established under small condition on the initial value. Moreover, we show that as time tends to infinity, the solution is asymptotic to the superposition of nonlinear diffusion waves which are given explicitly...