One-Phase Stefan-Like Problems with Latent Heat Depending on the Position and Velocity of the Free Boundary and with Neumann or Robin Boundary Conditions at the Fixed Face

A one-phase Stefan-type problem for a semi-infinite material which has as its main feature a variable latent heat that depends on the power of the position and the velocity of the moving boundary is studied. Exact solutions of similarity type are obtained for the cases when Neumann or Robin boundary conditions are imposed at the fixed face. Required relationships between data are presented in order that these problems become equivalent to the problem where a Dirichlet condition at the fixed face is considered. Moreover, in the case where a Robin condition is prescribed, the limit behaviour is studied when the heat transfer coefficient at the fixed face goes to infinity.Fil: Bollati, Julieta. 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

Convergencia de una familia de problemas discretos de control óptimo elíptico frontera respecto de un parámetro

Se considera un dominio acotado D de Rn con una frontera regular compuesta de dos porciones de frontera F1 y F2. En Gariboldi – Tarzia, Adv. Diff. Eq. Control Processes, 1 (2008), 113-132, se considera la convergencia de una familia de problemas de controles óptimos frontera de tipo Neumann gobernados por ecuaciones variacionales elípticas cuando el parámetro alpha de la familia (el coeficiente de transferencia de calor sobre la porción de frontera F1) tiende a infinito. Se demuestra la convergencia del control óptimo, del estado del sistema y del estado adjunto de la familia de problemas de controles óptimos fronteras de tipo Neumann a los correspondientes de un problema de control óptimo frontera de tipo Neumann también gobernado por una ecuación variacional elíptica con condiciones de contorno de tipo Dirichlet sobre F1. Se consideran, tanto para la familia de problemas de controles óptimos frontera de tipo Neumann como para el problema de control óptimo frontera límite, las aproximaciones numéricas por el método de los elementos finitos con triángulos de Lagrange de tipo 1. Se discretizan las ecuaciones variacionales elípticas que definen el estado del sistema y de su estado adjunto y además las funciones de costo respectivas. El objetivo del presente trabajo es el de estudiar la convergencia de la familia de problemas de controles óptimos fronteras de tipo Neumann discretos cuando el parámetro alpha tiende a infinito. Se demuestra la convergencia del control óptimo discreto, del estado del sistema discreto y del estado adjunto discreto de la familia a los correspondientes del problema de control óptimo frontera de tipo Neumann límite discreto.Fil: Tarzia, Domingo Alberto. Universidad Austral. Facultad de Ciencias Empresariales; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Explicit solutions for the Solomon-Wilson-Alexiades's mushy zone model with convective or heat flux boundary conditions

We complete the Solomon-Wilson-Alexiades's mushy zone model (Letters Heat Mass Transfer, 9 (1982), 319-324) for the one-phase Lam\'e-Clapeyron-Stefan problem. We obtain explicit solutions when a convective or heat flux boundary condition is imposed on the fixed face for a semi-infinite material. We also obtain the necessary and sufficient condition on data in order to get these explicit solutions. Moreover, when these conditions are satisfied the two problems are equivalents to the same problem with a temperature boundary condition on the fixed face and therefore an inequality for the coefficient which characterized one of the two free interfaces is also obtained.Comment: 13 page

Determination of one unknown thermal coefficient through the one-phase fractional Lam\'e-Clapeyron-Stefan problem

We obtain explicit expressions for one unknown thermal coefficient (among the conductivity, mass density, specific heat and latent heat of fusion) of a semi-infinite material through the one-phase fractional Lam\'e-Clapeyron-Stefan problem with an over-specified boundary condition on the fixed face $x=0$. The partial differential equation and one of the conditions on the free boundary include a time Caputo's fractional derivative of order $\alpha \in (0,1)$. Moreover, we obtain the necessary and sufficient conditions on data in order to have a unique solution by using recent results obtained for the fractional diffusion equation exploiting the properties of the Wright and Mainardi functions, given in Roscani - Santillan Marcus, Fract. Calc. Appl. Anal., 16 (2013), 802-815, Roscani-Tarzia, Adv. Math. Sci. Appl., 24 (2014), 237-249, and Voller, Int. J. Heat Mass Transfer, 74 (2014), 269-277. This work generalizes the method developed for the determination of unknown thermal coefficients for the classical Lam\'e-Clapeyron-Stefan problem given in Tarzia, Adv. Appl. Math., 3 (1982), 74-82, which are recovered by taking the limit when the order $\alpha\nearrow 1$.Comment: 15 pages, 2 Table

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

Explicit Solution for the one-phase Stefan problem with latent heat depending on the position and a convective boundary condition at the fixed face

An explicit solution of a similarity type is obtained for a one-phase Stefan problem in a semi-infinite material using Kummer functions. Itis considered a phase-change problem with a latent heat defined as a powerfunction of the position with a non-negative real exponent and a convectiveboundary condition at the fixed face x = 0. Existence and uniqueness of thesolution is proved. Relationship between this problem and the problems withtemperature and flux boundary condition is also analysed. Furthermore it isstudied the limit behaviour of the solution when the coefficient which char-acterizes the heat transfer at the fixed boundary tends to infinity. Comput-ing this limit allows to demonstrate that the problem proposed in this paperwith a convective boundary condition generalizes the problem with Dirichletboundary condition. Numerical computation of the solution is done over cer-tain examples, with a view to comparing this results with those obtained bygeneral algorithms that solve Stefan problems.Fil: Bollati, Julieta. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Tarzia, Domingo Alberto. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Sufficient Conditions for Mixed Boundary Data to have a Steady-State two-Phase Stefan-Signorini Problem through Variational Inequalities

Projet PROMATHRésumé disponible dans le fichier PD

Numerical analysis of a mixed elliptic problem with flux and convective boundary conditions to obtain a discrete solution of non-constaint sign

Projet PROMATHRésumé disponibel dans le fichier PD