We discuss several outstanding theoretical problems in optical diffusion in random media. Specifically, we discuss which of several diffusion theories most closely approximates exact solutions of the equation of transfer. We consider a plane wave impinging upon a plane-parallel slab of a random medium as a model problem to compare the diffusion theories with a numerical solution of the equation...

We consider the oxygen diffusion equation. Oxygen diffusion in a sike cell with simultaneous absorption is an important problem. Oxygen diffusion has a wide range of medical applications. Numerical solutions of its partial differential equation are obtained by extrapolation method with the vector of values V approximating to C (diffusion) at the mesh points. And the results gave a good agreemen...

A framework that naturally unifies smoothing and enhancement processes is presented. We generalize the linear and nonlinear scale spaces in the complex domain, by combining the diffusion equation with the simplified Schrödinger equation. A fundamental solution for the linear case is developed. Preliminary analysis of the complex diffusion shows that the generalized diffusion has properties of b...

We consider singular-degenerate, multivalued stochastic fast diffusion equations with multiplicative Lipschitz continuous noise. In particular, this includes the stochastic sign fast diffusion equation arising from the BakTang-Wiesenfeld model for self-organized criticality. A well-posedness framework based on stochastic variational inequalities (SVI) is developed, characterizing solutions to t...

In cancer progress, the optical properties of tissues like absorption and scattering coefficient change, so by these changes, we can trace the progress of cancer, even it can be applied for pre-detection of cancer. In this paper, we investigate the effects of changes of optical properties on light penetrated into tissues. The diffusion equation is widely used to simulate light propagation into ...

We study the effect of transport processes (diffusion and free–streaming) on a collapsing spherically symmetric distribution of matter in a self–similar space– time. A very simple solution shows interesting features when it is matched with the Vaidya exterior solution. In the mixed case (diffusion and free– streaming), we find a barotropic equation of state in the stationary regime. In the diff...

The family of nonlinear weighted flux methods for solving the transport equation is derived for 2D Cartesian geometry. A linear polynomial weight is considered. An asymptotic diffusion limit analysis is performed on the discretized method. The analysis reveals conditions on the weight necessary for an accurate approximation of the diffusion equation. As a result, we developed a new weighted flu...

The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∈ (0, 1). In this work, an explicit finite-difference scheme for TFDE is presented. Discrete models of a non-Markovian random walk are generate for simulating random variables whose spatial probability density evolves...

Ehrenfest's urn model is modified by introducing nonlinear terms in the associated transition probabilities. It is shown that these modifications lead, in the continuous limit, to a Fokker-Planck equation characterized by two competing diffusion terms, namely, the usual linear one and a nonlinear diffusion term typical of anomalous diffusion. By considering a generalized H theorem, the associat...

This paper deals with the diffusion limit of a kinetic equation where the collisions are modeled by a Lorentz type operator. The main aim is to construct a discrete scheme to approximate this equation which gives for any value of the Knudsen number, and in particular at the diffusive limit, the right discrete diffusion equation with the same value of the diffusion coefficient as in the continuo...