Time evolution of the reaction front in a subdiffusive system

Abstract

Using the quasistatic approximation, we show that in a subdiffusion-reaction system with arbitrary nonzero values of subdiffusion coefficients, the reaction front x_{f}(t) evolves in time as x_{f}(t)=Kt;{alpha2} , with alpha being the subdiffusion parameter and K being controlled by the subdiffusion coefficients. To check the correctness of our analysis, we compare approximate analytical solutions of the subdiffusion-reaction equations with the numerical ones.

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“…(1), (2) and (4) has its own structure where the depletion zone, the reaction region and diffusion region occur. As is shown in [3] within the reaction region the terms on the right hand side of Eq. (1) (or (2)) are comparable to each other and do not fulfil the assumptions of the perturbation method.…”

“…Such an example is nonlinear differential equation with a fractional time derivative which describes the subdiffusion-reaction symmetrical system with two initially separated diffusing particles of species A and B reacting according to the formula A + B → ∅(inert) [1,2,3] …”

“…Since the general method of solving fractional subdiffusion-reaction equations has not been found yet, one usually uses various approximations, such as the quasistationary approximation [3] or the scaling method [2]. In this paper, we present an idea to solve differential equations by means of the perturbation method.…”

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“…Frequently, the opinion is expressed that the definition of anomalous diffusion should be based upon the stochastic interpretation of this process. Let us note that in the system where the relation (1) is valid, other functions f (t) occur ensuring the relation f (t) ∼ t α which have the macroscopic interpretation and are experimentally measured, such as, for example, the time evolution of the near-membrane layer thickness [5], the time evolution of the reaction front in the subdiffusive system with chemical reactions [6] or the functions characterizing subdiffusive impedance [7]. Let us also note that there are models which do not have the stochastic interpretation (or such an interpretation has not been found yet) but in these models we can also find the important characteristics of the system, which satisfy the relation f (t) ∼ t α .…”

Conciliating the nonadditive entropy approach and the fractional model formulation when describing subdiffusion

“…where γ( ) denotes the special quantity characterizing the system which has a macroscopic interpretation and is experimentally measured, such as, for example, the time evolution of the near-membrane layer thickness [12,13], the time evolution of the reaction front in the subdiffusive system with chemical reactions [14,15] or the time evolution of the amount of a substance released from a thick membrane (or a thick slab) [16][17][18]. Let us note that in the cases we have mentioned, the coefficient A is controlled by subdiffusion parameters specific for the model under consideration and β is controlled only by the subdiffusion parameter α.…”