Untitled

Abstract

The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order β ∈ (0, 1). From a physical view-point this generalized diffusion equation is obtained from a fractional Fick law which describes transport processes with long memory. The fundamental solution for the Cauchy problem is interpreted as a probability density of a selfsimilar non-Markovian stochastic process related to a phenomenon of slow anomalous diffusion. By adopting a suitable finite-difference scheme of solution, we generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation.

Time Evolution Of The Reaction Front In A Subdiffusive System

“…3 and 4 there are presented the concentration profiles C A and C B obtained numerically according to the formula (75) and the functions given by Eqs. (30) and (32) with ρ A = 0.40 and ρ B = 3.64, respectively. We observe a quite good agreement of the analytical and numerical functions in the diffusion region.…”

“…Starting with the above assumptions, we show at first the following (a) The concentration profiles (30) and (32) extended to the reaction region vanish at the points which are identified with the point x z defined in Fig. (1) and by Eq.…”

“…In the papers [30,31] there were presented the procedures of the numerical solving of the subdiffusion equation without chemical reaction, when one can use the equation with Riemann-Liouville as well as Caputo fractional time derivative. The situation is different in the case of the subdiffusion-reaction equations.…”

“…For this reason we can compare the functions (30) and (32) with experimental data obtained for a subdiffusive system without chemical reactions. Our theoretical and the experimental functions presented in [33] are qualitatively similar to each other if we take the units commonly used in real systems where x is given in 10 −2 m, t in sec, D A and D B are of the order 10 −8 m 2 /s α .…”

Fractional thermoelasticity applications for porous asphaltic materials

“…Fractional order models often work well, particularly for dielectrics and viscoelastic materials over extended ranges of time and frequency (Lakes 1999;Grimnes and Martinsen 2000). In heat transfer and electrochemistry, for example, the half-order fractional integral is the natural integral operator connecting the applied gradients (thermal or material) with the diffusion of ions with heat (Gorenflo et al 2002). One can refer to Podlubny (1999) for a survey of applications of fractional calculus.…”