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Sabrina Roscani

Publications and source records attributed to Sabrina Roscani.

12 recordsLinked to original sources
arXiv2026

On the closed solution of a problem coupling fluid infiltration with a hydration reaction

We present a one-dimensional model for water infiltration coupled with a hydration reaction, relevant to coupled transport and chemical processes in the Earth's subsurface. In this model the sharp interface separating the saturated and dry regions evolves over time, leading to a moving free boundary problem. Consistent with recently presented numerical treatments in the literature, this solution indicates an interesting dynamic for the free boundary. At early time, for a given domain porosity, the infiltration front advances with a specific square-root-in-time behavior. At later time, depending on the consumption of the hydration reaction, the front advance can exhibit square-root, linear, or exponential forms. The presented closed solution provides an analytical tool that can be used to quantify important behavior in coupled transport and reaction systems.

math.AP↗
arXiv2021

A one-phase space -- fractional Stefan problem with no liquid initial domain

Taking into account the recent works \cite{RoTaVe:2020} and \cite{Rys:2020}, we consider a phase-change problem for a one dimensional material with a non-local flux, expressed in terms of the Caputo derivative, which derives in a space-fractional Stefan problem. We prove existence of a unique solution to a phase-change problem with the fractional Neumann boundary condition at the fixed face $x=0$, where the domain, at the initial time, consists of liquid and solid. Then we use this result to prove the existence of a limit solution to an analogous problem with solid initial domain, when it is not possible to transform the domain into a cylinder.

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arXiv2020

About Convergence and Order of Convergence of some Fractional Derivatives

In this paper we establish some convergence results for Riemann-Liouville, Caputo, and Caputo-Fabrizio fractional operators when the order of differentiation approaches one. We consider some errors given by $\left|\left| D^{1-\al}f -f'\right|\right|_p$ for p=1 and $p=\infty$ and we prove that for both Caputo and Caputo Fabrizio operators the order of convergence is a positive real r, 0<r<1. Finally, we compare the speed of convergence between Caputo and Caputo-Fabrizio operators obtaining that they a related by the Digamma function.

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arXiv2020

Explicit Solutions to Fractional Stefan-like problems for Caputo and Riemann-Liouville Derivatives

Two fractional two-phase Stefan-like problems are considered by using Riemann-Liouville and Caputo derivatives of order $α\in (0, 1)$ verifying that they coincide with the same classical Stefan problem at the limit case when $α=1$. For both problems, explicit solutions in terms of the Wright functions are presented. Even though the similarity of the two solutions, a proof that they are different is also given. The convergence when $α\nearrow 1$ of the one and the other solutions to the same classical solution is given. Numerical examples for the dimensionless version of the problem are also presented and analyzed.

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arXiv2018

An integral Relationship for a new Fractional One-phase Stefan Problem

A one-dimensional fractional one-phase Stefan problem with a temperature boundary condition at the fixed face is considered. An integral relationship between the temperature and the free boundary is obtained which is equivalent to the fractional Stefan condition. Moreover, an exact solution of similarity type expressed in terms of Wright functions is given.

math.AP↗
arXiv2018

A New Mathematical Formulation for a Phase Change Problem with a Memory Flux

A mathematical model for a one-phase change problem (particularly a Stefan problem) with a memory flux, is obtained. The hypothesis that the weighted sum of fluxes back in time is proportional to the gradient of temperature is considered. The model obtained involves fractional derivatives with respect on time in the sense of Caputo and in the sense of Riemann--Liouville. An integral relationship for the free boundary which is equivalent to the `fractional Stefan condition' is also obtained.

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arXiv2018

Global Solution to a Nonlinear Fractional Differential Equation for the Caputo-Fabrizio Derivative

This paper deals with the fractional Caputo--Fabrizio derivative and some basic properties related. A computation of this fractional derivative to power functions is given in terms of Mittag--Lefler functions. The inverse operator named the fractional Integral of Caputo--Fabrizio is also analyzed. The main result consists in the proof of existence and uniqueness of a global solution to a nonlinear fractional differential equation, which has been solved previously for short times by Lozada and Nieto (Progr. Fract. Differ. Appl., 1(2):87--92, 2015). The effects of memory as well as the convergence of the obtained results when $\al \nearrow 1$ (and the classical first derivative is recovered) are analyzed throughout the paper.

math.AP↗
arXiv2018

Explicit solution for a two--phase fractional Stefan problem with a heat flux condition at the fixed face

A generalized Neumann solution for the two-phase fractional Lamé--Clapeyron--Stefan problem for a semi--infinite material with constant initial temperature and a particular heat flux condition at the fixed face is obtained, when a restriction on data is satisfied. The fractional derivative in the Caputo sense of order $\al \in (0,1)$ respect on the temporal variable is considered in two governing heat equations and in one of the conditions for the free boundary. Furthermore, we find a relationship between this fractional free boundary problem and another one with a constant temperature condition at the fixed face and based on that fact, we obtain an inequality for the coefficient which characterizes the fractional phase-change interface obtained in Roscani--Tarzia, Adv. Math. Sci. Appl., 24 (2014), 237-249. We also recover the restriction on data and the classical Neumann solution, through the error function, for the classical two-phase Lamé-Clapeyron-Stefan problem for the case $\al=1$.

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arXiv2015

On the Initial-Boundary Problem for the Time-Fractional Diffusion Equation in the Quarter Plane

Taking into account the asymptotic behavior of some Wright functions and the existence of bounds for the Mainardi and the Wright function $W(-x,\fracα{2}, 1)$ in $\mathbb{R}^+$ , three different initial-boundary-value problems for the time-fractional diffusion equation in the quarter plane, where the time-fractional derivative is taken in the Caputo sense of order $α$ $\in (0,1)$ are solved. Moreover, the limit when $α\nearrow 1$ of the respective solutions are analyzed, recovering the respective solutions of the classical boundary-value problems when $α=1$ and the fractional diffusion equation becomes the heat equation.

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arXiv2014

A new equivalence of Stefan's problems for the Time-Fractional-Diffusion Equation

A fractional Stefan problem with a boundary convective condition is solved, where the fractional derivative of order $ α\in (0,1) $ is taken in the Caputo sense. Then an equivalence with other two fractional Stefan problems (the first one with a constant condition on $ x = 0 $ and the second with a flux condition)is proved and the convergence to the classical solutions is analyzed when $ α\nearrow 1$ recovering the heat equation with its respective Stefan condition.

math.AP↗
arXiv2013

Two equivalent Stefan's problems for the Time Fractional Diffusion Equation

Two Stefan's problems for the diffusion fractional equation are solved, where the fractional derivative of order $ \al \in (0,1) $ is taken in the Caputo's sense. The first one has a constant condition on $ x = 0 $ and the second presents a flux condition $ T_x (0, t) = \frac {q} {t ^ {\al/2}} $. An equivalence between these problems is proved and the convergence to the classical solutions is analysed when $ \al \nearrow $ 1 recovering the heat equation with its respective Stefan's condition.

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