arXiv subjects

Emilia Bazhlekova

Publications and source records attributed to Emilia Bazhlekova.

6 recordsLinked to original sources

arXiv2025

Fractional calculus approach to models of adsorption: Barrier-diffusion control

The mathematical model of surfactant adsorption under mixed barrier-diffusion control is analyzed using techniques from fractional calculus. The kinetic models of Henry, Langmuir, Frumkin, Volmer and van der Waals are considered. First, treating the Ward-Tordai integral equation as a fractional order one, the partial differential model is transformed into a single fractional ordinary differential equation for the adsorption. A transformation of the obtained equation is proposed that reduces the number of parameters to two dimensionless groups (at Frumkin and van der Waals models a third parameter appears). In the simplest case of Henry adsorption isotherm the fractional differential model depends on a single dimensionless group and an exact solution exists, represented in terms of Mittag-Leffler functions. Based on this solution, second order asymptotes (at small values of the adsorption) are derived for the other models. The asymptotes of the adsorption result in a higher order asymptotes for the surface pressure (surface tension). For small surface coverage, all considered models converge to the Henry model's predictions, making it a universal first-order approximation for the surface tension. Next, the fractional differential model is written as an integral equation %of fractional order that can be considered as a generalization of the well-known Ward-Tordai equation to the case of barrier-diffusion control. For computer simulation of the obtained integral equation a predictor-corrector numerical method is developed and numerical results are presented and discussed.

math.AP↗

arXiv2018

Subordination principle for space-time fractional evolution equations and some applications

The abstract Cauchy problem for the fractional evolution equation with the Caputo derivative of order $β\in(0,1)$ and operator $-A^α$, $α\in(0,1)$, is considered, where $-A$ generates a strongly continuous one-parameter semigroup on a Banach space. Subordination formulae for the solution operator are derived, which are integral representations containing a subordination kernel (a scalar probability density function) and a $C_0$-semigroup of operators. Some properties of the subordination kernel are established and representations in terms of Mainardi function and Lévy extremal stable densities are derived. Applications of the subordination formulae are given with a special focus on the multi-dimensional space-time fractional diffusion equation for some special values of the parameters.

math.AP↗

arXiv2018

Estimates for a general fractional relaxation equation and application to an inverse source problem

A general fractional relaxation equation is considered with a convolutional derivative in time introduced by A. Kochubei (Integr. Equ. Oper. Theory 71 (2011), 583-600). This equation generalizes the single-term, multi-term and distributed-order fractional relaxation equations. The fundamental and the impulse-response solutions are studied in detail. Properties such as analyticity and subordination identities are established and employed in the proof of an upper and a lower bound. The obtained results extend some known properties of the Mittag-Leffler functions. As an application of the estimates, uniqueness and conditional stability are established for an inverse source problem for the general time-fractional diffusion equation on a bounded domain.

math.AP↗

arXiv2018

Subordination approach to multi-term time-fractional diffusion-wave equations

This paper is concerned with the fractional evolution equation with a discrete distribution of Caputo time-derivatives such that the largest and the smallest orders, $α$ and $α_m$, satisfy the conditions $1<α\le 2$ and $α-α_m\le 1$. First, based on a study of the related propagation function, the nonnegativity of the fundamental solutions to the spatially one-dimensional Cauchy and signaling problems is proven and propagation speed of a disturbance is discussed. Next, we study the equation with a general linear spatial differential operator defined in a Banach space and suppose it generates a cosine family. A subordination principle is established, which implies the existence of a unique solution and gives an integral representation of the solution operator in terms of the corresponding cosine family and a probability density function. Explicit representation of the probability density function is derived. The subordination principle is applied for obtaining regularity results. The analytical findings are supported by numerical work.

math.AP↗

arXiv2015

Completely monotone functions and some classes of fractional evolution equations

The abstract Cauchy problem for the distributed order fractional evolution equation in the Caputo and in the Riemann-Liouville sense is studied for operators generating a strongly continuous one-parameter semigroup on a Banach space. Continuous as well as discrete distribution of fractional time-derivatives of order less than one are considered. The problem is reformulated as an abstract Volterra integral equation. It is proven that its kernel satisfy certain complete monotonicity properties. Based on these properties, the well-posedness of the problem is established and a series expansion of the solution is obtained. In case of ordered Banach space this representation implies positivity of the solution operator. In addition, a subordination formula is obtained.

math.AP↗

arXiv2015

An Analysis of the Rayleigh-Stokes problem for a Generalized Second-Grade Fluid

We study the Rayleigh-Stokes problem for a generalized second-grade fluid which involves a Riemann-Liouville fractional derivative in time, and present an analysis of the problem in the continuous, space semidiscrete and fully discrete formulations. We establish the Sobolev regularity of the homogeneous problem for both smooth and nonsmooth initial data $v$, including $v\in L^2(Ω)$. A space semidiscrete Galerkin scheme using continuous piecewise linear finite elements is developed, and optimal with respect to initial data regularity error estimates for the finite element approximations are derived. Further, two fully discrete schemes based on the backward Euler method and second-order backward difference method and the related convolution quadrature are developed, and optimal error estimates are derived for the fully discrete approximations for both smooth and nonsmooth initial data. Numerical results for one- and two-dimensional examples with smooth and nonsmooth initial data are presented to illustrate the efficiency of the method, and to verify the convergence theory.

math.NA↗