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Katarzyna D. Lewandowska

Publications and source records attributed to Katarzyna D. Lewandowska.

15 recordsLinked to original sources
arXiv2026

Subdiffusion equation with Cattaneo effect

The ordinary subdiffusion equation, with a fractional time derivative of at most first order, describes a process in which the propagation velocity of diffusing molecules is unlimited. To avoid this non-physical property different forms of the Cattaneo subdiffusion equation have been proposed. We define the Cattaneo effect as a delay of the ordinary subdiffusion flux activation by a random time. By incorporating this effect into the flux equation we get a Cattaneo--type subdiffusion equation (CTSE). We consider a subdiffusion process in which the Cattaneo effect is generated by the time-delay probability distribution controlled by the Mittag-Leffler function. Then, CTSE differs from the ordinary subdiffusion equation by a term with a fractional time derivative, whose order can be independent of the subdiffusion exponent. The influence of the Cattaneo effect on the solutions to the CTSE is discussed. We show that the process described by CTSE is subdiffusion in the entire time domain even though the temporal evolution of the mean square displacement of diffusing particle in the short-time limit is typical for superdiffusion. The delay in the flux activation in the subdiffusion equation should also cause a flux delay in a boundary condition. As an example, we study subdiffusion with the Cattaneo effect in a system with a partially absorbing wall at which the Robin boundary condition is assumed. We also propose a method for experimentally identifying the Cattaneo effect in a subdiffusive system.

cond-mat.stat-mech↗
arXiv2025

Cattaneo--type subdiffusion equation

The ordinary subdiffusion equation, with a fractional time derivative of at most first order, describes a process in which the propagation velocity of diffusing molecules is unlimited. To avoid this non-physical property different forms of the Cattaneo subdiffusion equation have been proposed. We define the Cattaneo effect as a delay of the ordinary subdiffusion flux activation by a random time. By incorporating this effect into the flux equation we get a Cattaneo--type subdiffusion equation (CTSE). We study the CTSE that differs from the ordinary subdiffusion equation by an additional integro--differential operator (AO) controlled by a time delay probability distribution. A method for deriving CTSE within the standard continuous time random walk model is also shown. As examples, we consider the CTSE with AO being the Caputo fractional time derivative of the order independent of the subdiffusion exponent and with the AO with a kernel that is a slowly varying function. In the first case the Cattaneo effect disappears over time much faster than in the second one. Based on the Green's functions, the time evolutions of the first passage time distribution and of the mean square displacement of a diffusing molecule, we discuss whether the influence of the Cattaneo effect is significant. In the considered examples, this influence seems to be small. However, a relative probability of finding a molecule at a long distance from the starting point for the CTSE equation with respect to the ordinary subdiffusion equation increases rapidly with distance. Even small changes caused by the Cattaneo effect can lead to different results in modeling processes where a faster appearance of a diffusing object changes the nature of the process. The effect may be important, for example, in modeling the spread of an epidemic when a diffusing object is a source of infection.

cond-mat.stat-mech↗
arXiv2022

Subdiffusion equation with Caputo fractional derivative with respect to another function in modelling diffusion in a complex system consisting of matrix and channels

Diffusion equation with a fractional Caputo time derivative with respect to another function $g$, which defines new time scale of the process, is applied to describe anomalous diffusion of antibiotic (colistin) in a system consisting of packed gel (alginate) beads immersed in water and impregnated with the antibiotic. We show that ultraslow diffusion of colistin in the system occurs. The equation, which is more general than the "ordinary" fractional subdiffusion equation, can be widely used in various fields of science to model diffusion in a matrix with channels system when subdiffusion parameters and even a type of diffusion evolves over time.

cond-mat.stat-mech↗
arXiv2017

How to determine a boundary condition at a thin membrane for diffusion from experimental data

We present a new method of deriving a boundary condition at a thin membrane for diffusion from experimental data. Based on experimental results obtained for normal diffusion of ethanol in water, we show that the derived boundary condition at a membrane contains a term with the Riemann--Liouville fractional time derivative of the $1/2$ order. Such a form of the boundary condition shows that a transfer of particles through a thin membrane is a `long memory process'. Presented method is an example that an important part of mathematical model of physical process may be derived directly from experimental data.

cond-mat.stat-mech↗
arXiv2014

Subdiffusion--reaction process with $A\longrightarrow B$ reactions versus subdiffusion--reaction process with $A+B\longrightarrow B$ reactions

We consider the subdiffusion-reaction process with reactions of a type A+B\arrow B (in which particles A are assumed to be mobile whereas B - static) in comparison to the subdiffusion-reaction process with A\rightarrow B reactions which was studied by I.M. Sokolov et al. in Phys. Rev. E 73, 031102 (2006). In both processes a rule that reactions can only occur between particles which continue to exist is taken into account. Although in both processes a probability of the vanishing of particle A due to a reaction is independent of both time and space variables (assuming that in the system with the A+B\arrow B reactions, particles B are distributed homogeneously) we show that subdiffusion-reaction equations describing these processes as well as their Greens' functions are qualitatively different. For the subdiffusion process with the A+B\arrow B reactions we consider three models. We base the method considered in this paper on a random walk model in a system with both discrete time and space variables. Then, the system with discrete variables is transformed into a system with continuous variables. Such a method seems to be convenient in analysing subdiffusion-reaction processes with partially absorbing or partially reflecting walls. The reason is that within this method we can determine Greens' functions without a necessity of solving a fractional differential subdiffusion-reaction equation with boundary conditions at the walls. As an example we use the model to find the Greens' functions for a subdiffusive-reaction system (with the reactions mentioned above), which is bounded by a partially absorbing wall. This example shows how the model can be used to analyze the subdiffusion-reaction process in a system with partially absorbing or reflecting thin membranes. Employing a simple phenomenological model, we derive equations related to the reaction parameters used in the considered models.

cond-mat.stat-mech↗
arXiv2012

First passage time for subdiffusion: The nonextensive entropy approach versus the fractional model

We study the similarities and differences between different models concerning subdiffusion. More particularly, we calculate first passage time (FPT) distributions for subdiffusion, derived from Greens' functions of nonlinear equations obtained from Sharma-Mittal's, Tsallis's and Gauss's nonadditive entropies. Then we compare these with FPT distributions calculated from a fractional model using a subdiffusion equation with a fractional time derivative. All of Greens' functions give us exactly the same standard relation $<(Δx)^2> =2 D_αt^α$ which characterizes subdiffusion ($0<α<1$), but generally FPT's are not equivalent to one another. We will show here that the FPT distribution for the fractional model is asymptotically equal to the Sharma--Mittal model over the long time limit only if in the latter case one of the three parameters describing Sharma--Mittal entropy $r$ depends on $α$, and satisfies the specific equation derived in this paper, whereas the other two models mentioned above give different FTPs with the fractional model. Greens' functions obtained from the Sharma-Mittal and fractional models - for $r$ obtained from this particular equation - are very similar to each other. We will also discuss the interpretation of subdiffusion models based on nonadditive entropies and the possibilities of experimental measurement of subdiffusion models parameters.

cond-mat.stat-mech↗
arXiv2012

The perturbation method to solve subdiffusion-reaction equations

We use the perturbation method to approximately solve subdiffusion-reaction equations. Within this method we obtain the solutions of the zeroth and the first order. The comparison our analytical solutions with the numerical results shown that the perturbation method can be useful to find approximate solutions of nonlinear subdiffusion--reaction equations.

cond-mat.stat-mech↗
arXiv2012

The nonextensive entropy approach versus the stochastic in describing subdiffusion

We have proposed a new stochastic interpretation of the sudiffusion described by the Sharma-Mittal entropy formalism which generates a nonlinear subdiffusion equation with natural order derivatives. We have shown that the solution to the diffusion equation generated by Gauss entropy (which is the particular case of Sharma-Mittal entropy) is the same as the solution of the Fokker-Planck (FP) equation generated by the Langevin generalised equation where the `long memory effect' is taken into account. The external noise which pertubates the subdiffusion coefficient (occuring in the solution of FP equation) according to the formula $D_α\rightarrow D_α/u$ where $u$ is a random variable described by the Gamma distribution, provides us with solutions of equations obtained from Sharma-Mittal entropy. We have also shown that the parameters $q$ and $r$ occuring in Sharma-Mittal entropy are controlled by the parameters $α$ and $ $, respectively.

cond-mat.stat-mech↗
arXiv2008

Subdiffusive model of substance releasing from a thick membrane

We study both theoretically and experimentally the process of subdiffusive substance releasing from a thick membrane. The theoretical model uses the subdiffusion equation with fractional time derivative and specific boundary conditions at the membrane surfaces. Using a special \textit{ansatz} we find analytical formulas describing the time evolution of concentration profiles and an amount of the substance remains in the membrane. Fitting the theoretical functions to the experimental results, we estimate the subdiffusion coefficient of polyethylene glycol 2000 in agarose hydrogel.

cond-mat.soft↗
arXiv2008

Hyperbolic subdiffusive impedance

We use the hyperbolic subdiffusion equation with fractional time derivatives (the generalized Cattaneo equation) to study the transport process of electrolytes in media where subdiffusion occurs. In this model the flux is delayed in a non-zero time with respect to the concentration gradient. In particular, we obtain the formula of electrochemical subdiffusive impedance of a spatially limited sample in the limit of large and of small pulsation of the electric field. The boundary condition at the external wall of the sample are taken in the general form as a linear combination of subdiffusive flux and concentration of the transported particles. We also discuss the influence of the equation parameters (the subdiffusion parameter and the delay time) on the Nyquist impedance plots.

cond-mat.stat-mech↗
arXiv2008

Hyperbolic versus parabolic equation with fractional derivative to describe subdiffusion in a membrane system

We use the parabolic and hyperbolic equation with fractional time derivative to describe the subdiffusion in a system with thin membrane. We find the Green's function and solutions of the equation for the system where the homogeneous solution is separated by a thin membrane from the pure solvent. The solutions were obtained for two boundary conditions where the ratio of the concentrations at the membrane surfaces does not change in time and where the flux flowing through the membrane is proportional to the concentration difference between membrane surfaces. We discuss the difference between the solutions for parabolic and hyperbolic subdiffusion equations obtained for both boundary conditions.

cond-mat.stat-mech↗
arXiv2008

Perturbation method to model enamel caries progress

We develop a theoretical model of the carious lesion progress caused by acids diffusing into the tooth enamel from the dental plaque. The acids react with static hydroxyapatite, which leads to demineralization of the enamel, and consequently to the development of the carious lesion. The model utilizes the diffusion-reaction equations with one static and one mobile reactant where the reaction term is proportional to the product of concentrations of acids and of mineral. The changes of concentrations are calculated approximately by means of a perturbation method. The analytical approximate solutions are compared with the numerical ones and experimental data.

cond-mat.stat-mech↗
arXiv2008

Time evolution of the reaction front in a subdiffusive system

Using the quasistatic approximation, we show that in a subdiffusion--reaction system the reaction front $x_{f}$ evolves in time according to the formula $x_{f} \sim t^{α/2}$, with $α$ being the subdiffusion parameter. The result is derived for the system where the subdiffusion coefficients of reactants differ from each other. It includes the case of one static reactant. As an application of our results, we compare the time evolution of reaction front extracted from experimental data with the theoretical formula and we find that the transport process of organic acid particles in the tooth enamel is subdiffusive.

cond-mat.stat-mech↗
arXiv2006

Numerical study of subdiffusion equation

We present a numerical procedure of solving the subdiffusion equation with Caputo fractional time derivative. On the basis of few examples we show that the subdiffusion is a 'long time memory' process and the short memory principle should not be used in this case.

cond-mat.other↗