A new mathematical formulation for a phase change problem with a memory flux

A mathematical formulation for a one-phase change problem in a form of Stefan problem with a memory flux is obtained. The hypothesis that the integral of weighted backward fluxes is proportional to the gradient of the temperature is considered. The model that arises involves fractional derivatives with respect to time both in the sense of Caputo and of Riemann–Liouville. An integral relation for the free boundary, which is equivalent to the “fractional Stefan condition”, is also obtained.Fil: Roscani, Sabrina Dina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; ArgentinaFil: Bollati, Julieta. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; ArgentinaFil: Tarzia, Domingo Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; Argentin

An integral relationship for a fractional one-phase Stefan problem

A one-dimensional fractional one-phase Stefan problem with a temperature boundary condition at the fixed face is considered by using the Riemann–Liouville derivative. This formulation is more convenient than the one given in Roscani and Santillan (Fract. Calc. Appl. Anal., 16, No 4 (2013), 802–815) and Tarzia and Ceretani (Fract. Calc. Appl. Anal., 20, No 2 (2017), 399–421), because it allows us to work with Green’s identities (which does not apply when Caputo derivatives are considered). As a main result, 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 also given.Fil: Roscani, Sabrina Dina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; ArgentinaFil: Tarzia, Domingo Alberto. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentin

A generalized neumann solution for the two-phase fractional lame-clapeyron-stefan problem

We obtain a generalized Neumann solution for the two-phase fractional Lam´eClapeyron-Stefan problem for a semi-infinite material with constant boundary and initial conditions. In this problem, the two governing equations and a governing condition for the free boundary include a fractional time derivative in the Caputo sense of order 0 < α ≤ 1. When α ↗ 1 we recover the classical Neumann solution for the two-phase Lam´eClapeyron-Stefan problem given through the error functionFil: Roscani, Sabrina Dina. Universidad Nacional de Rosario. Facultad de Cs.exactas Ingeniería y Agrimensura. Escuela de Cs.exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Tarzia, Domingo Alberto. Universidad Austral. Facultad de Cs.empresariales. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Global solution to a nonlinear fractional differential equation for the Caputo-Fabrizio derivative

In this article we prove existence and uniqueness of global solution to an initial value problem for a nonlinear fractional differential equation with a Caputo-Fabrizio (CF) derivative. We provide a new compact formula for the computation of the CF derivative to power functions (which is given in terms of Mittag-Leffler functions). We also give the convergence to classical derivatives for a regular class of functions when the order of the CF derivative tends to one, as well as some other useful properties.Fil: Roscani, Sabrina Dina. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; ArgentinaFil: Venturato, Lucas David. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; ArgentinaFil: Tarzia, Domingo Alberto. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentin

Hopf lemma for the fractional diffusion operator and its application to a fractional free-boundary problem

We consider a one-dimensional moving-boundary problem for the time-fractional diffusion equation, where the time-fractional derivative of order α. ∈. (0, 1) is taken in the Caputo sense. A generalization of the Hopf lemma is proved and then used to prove a monotonicity property for the free-boundary when a fractional free-boundary Stefan problem is investigated.Fil: Roscani, Sabrina Dina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentin

About Convergence and Order of Convergence of Some Fractional Derivatives

In this paper we obtain some convergence results for Riemann-Liouville, Caputo, and Caputo–Fabrizio fractional operators when the order of differentiation approaches one. We consider the errors given by D1−α f − f ′p for p=1 and p = ∞ and we prove that for bothm the Caputo and Caputo Fabrizio operators, the order of convergence is a positive real r ∈ (0, 1). Finally, we compare the speed of convergence between Caputo and Caputo–Fabrizio operators obtaining that they are related by the Digamma function

A One-Phase Space-Fractional Stefan Problem With No Liquid Initial Domain

We consider a phase-change problem for a one-dimensional material with a nonlocal 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 solution to an analogous problem with solid initial domain, when it is not possible to transform the domain into a cylinder.Fil: Roscani, Sabrina Dina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; ArgentinaFil: Ryszewska, Katarzyna. Warsaw University of Technology; PoloniaFil: Venturato, Lucas David. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; Argentin

About the convergence of a family of initial boundary value problems for a fractional diffusion equation of robin type

We consider a family of initial boundary value problems governed by a fractional diffusion equation with Caputo derivative in time, where the parameter is the Newton heat transfer coefficient linked to the Robin condition on the boundary. For each problem we prove existence and uniqueness of solution by a Fourier approach. This will enable us to also prove the convergence of the family of solutions to the solution of the limit problem, which is obtained by replacing the Robin boundary condition with a Dirichlet boundary condition.Fil: Cardoso, Isolda Eugenia. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; ArgentinaFil: Roscani, Sabrina Dina. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; ArgentinaFil: Tarzia, Domingo Alberto. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentin

Pressure driven lubrication flow of a Bingham fluid in a channel: A novel approach

In this paper we present a novel approach for modelling the lubrication flow of a Bingham fluid in a channel whose amplitude is non uniform. The novelty consists in deriving the rigid plug equation using an integral approach based on Newton's second law, where the unyielded part is treated as an evolving non material volume. Such an approach leads to an integro-differential equation for the pressure that can be solved with an iterative procedure. We prove that a true unyielded plug exists even when the maximum width variation is not "small" and we find constraints on the amplitude of the channel that prevent the plug from "breaking". We also extend our model to the case of a pressure-dependent viscosity.Fil: Fusi, Lorenzo. Dipartimento di Matematica e Informatica “Ulisse Dini”; ItaliaFil: Farina, Angiolo. Dipartimento di Matematica e Informatica “Ulisse Dini”; ItaliaFil: Rosso, Fabio. Dipartimento di Matematica e Informatica “Ulisse Dini”; ItaliaFil: Roscani, Sabrina Dina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentin