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 Lam\'e--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 Lam\'e-Clapeyron-Stefan problem for the case \al=1.Comment: 19 pages, 1 figur

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 $\alpha \in (0,1)$ 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 $\alpha \nearrow 1$ recovering the heat equation with its respective Stefan condition.Comment: This paper was already accepted to be published in the in the journal "Fractional Calculus and Applied Analysis". arXiv admin note: substantial text overlap with arXiv:1306.175

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

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

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 $\alpha \in (0, 1)$ verifying that they coincide with the same classical Stefan problem at the limit case when $\alpha=1$. 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 $\alpha \nearrow 1$ 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.Comment: 21 page

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