Ivan Bazhlekov | Searcharxiv

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 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↗