arXiv subjects

Domingo A. Tarzia

Publications and source records attributed to Domingo A. Tarzia.

At least 37 records · Page 2Linked to original sources

arXiv2020

Convergence Results for Optimal Control Problems Governed by Elliptic Quasivariational Inequalities

We consider an optimal control problem $\cQ$ governed by an elliptic quasivariational inequality with unilateral constraints. The existence of optimal pairs of the problem is a well known result, see \cite{SS}, for instance. We associate to $\cQ$ a new optimal control problem $\wQ$, obtained by perturbing the state inequality (including the set of constraints and the nonlinear operator) and the cost functional, as well. Then, we provide sufficient conditions which guarantee the convergence of solutions of Problem $\wQ$ to a solution of Problem $\cQ$. The proofs are based on convergence results for elliptic quasivariational inequalities, obtained by using arguments of compactness, lower semicontinuity, monotonicity, penalty and various estimates. Finally, we illustrate the use of the abstract convergence results in the study of optimal control associated with two boundary value problems. The first one describes the equilibrium of an elastic body in frictional contact with an obstacle, the so-called foundation. The process is static and the contact is modeled with normal compliance and unilateral constraint, associated to a version of Coulomb's law of dry friction. The second one describes a stationary heat transfer problem with unilateral constraints. For the two problems we prove existence, uniqueness and convergence results together with the corresponding physical interpretation.

math.OC↗

arXiv2020

Explicit Solutions for Distributed, Boundary and Distributed-Boundary Elliptic Optimal Control Problems

We consider a steady-state heat conduction problem in a multidimensional bounded domain Omega for the Poisson equation with constant internal energy g and mixed boundary conditions given by a constant temperature b in the portion Gamma_1 of the boundary and a constant heat flux q in the remaining portion Gamma_2 of the boundary. Moreover, we consider a family of steady-state heat conduction problems with a convective condition on the boundary Gamma_1 with heat transfer coefficient alpha and external temperature b. We obtain explicitly, for a rectangular domain in R^2, an annulus in R^2 and a spherical shell in R^3, the optimal controls, the system states and adjoint states for the following optimal control problems: a distributed control problem on the internal energy g, a boundary optimal control problem on the heat flux q, a boundary optimal control problem on the external temperature b and a distributed-boundary simultaneous optimal control problem on the source g and the flux q. These explicit solutions can be used for testing new numerical methods as a benchmark test. In agreement with theory, it is proved that the system state, adjoint state, optimal controls and optimal values corresponding to the problem with a convective condition on Gamma_1 converge, when alpha\to\infty, to the corresponding system state, adjoint state, optimal controls and optimal values that arise from the problem with a temperature condition on Gamma_1. Also, we analyze the order of convergence in each case, which turns out to be 1/alpha being new for these kind of elliptic optimal control problems.

math.OC↗

arXiv2019

Auxiliary functions in the study of Stefan-like problems with variable thermal properties

We address the existence and uniqueness of the so-called modified error function that arises in the study of phase-change problems with specific heat and thermal conductivity given by linear functions of the material temperature. This function is defined from a differential problem that depends on two parameters which are closely related with the slopes of the specific heat and the thermal conductivity. We identify conditions on these parameters which allow us to prove the existence of the modified error function. In addition, we show its uniqueness in the space of non-negative bounded analytic functions for parameters that can be negative and different from each other. This extends known results from the literature and enlarges the class of associated phase-change problems for which exact similarity solutions can be obtained. In addition, we provide some properties of the modified error function considered here.

math.CA↗

arXiv2019

A free boundary problem for a diffusion-convection equation

One-dimensional free boundary problem for a nonlinear diffusion - convection equation with a Dirichlet condition at fixed face $x=0$, variable in time, is considered. Throught several transformations the problem is reduced to a free boundary problem for a diffusion equation and the integral formulation is obtained. By using fixed point theorems, the existence of at least a solution, for small time, to a system of coupled nonlinear integral equations is obtained.

math.AP↗

arXiv2019

Convergence of simultaneous distributed-boundary parabolic optimal control problems

We consider a heat conduction problem $S$ with mixed boundary conditions in a n-dimensional domain $Ω$ with regular boundary $Γ$ and a family of problems $S_α$, where the parameter $α>0$ is the heat transfer coefficient on the portion of the boundary $Γ_{1}$. In relation to these state systems, we formulate simultaneous \emph{distributed-boundary} optimal control problems on the internal energy $g$ and the heat flux $q$ on the complementary portion of the boundary $Γ_{2}$. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system and the adjoint states when the heat transfer coefficient $α$ goes to infinity. Finally, we prove estimations between the simultaneous distributed-boundary optimal control and the distributed optimal control problem studied in a previous paper of the first author.

math.OC↗

arXiv2019

A Heat Conduction Problem with Sources Depending on the Average of the Heat Flux on the Boundary

Motivated by the modeling of temperature regulation in some mediums, we consider the non-classical heat conduction equation in the domain $D=\mathbb{R}^{n-1}\times\br^{+}$ for which the internal energy supply depends on an average in the time variable of the heat flux $(y, s)\mapsto V(y,s)= u_{x}(0 , y , s)$ on the boundary $S=\partial D$. The solution to the problem is found for an integral representation depending on the heat flux on $S$ which is an additional unknown of the considered problem. We obtain that the heat flux $V$ must satisfy a Volterra integral equation of second kind in the time variable $t$ with a parameter in $\mathbb{R}^{n-1}$. Under some conditions on data, we show that a unique local solution exists, which can be extended globally in time. Finally in the one-dimensional case, we obtain the explicit solution by using the Laplace transform and the Adomian decomposition method.

math-ph↗

arXiv2019

Approximate solutions to the one-phase Stefan problem with non-linear temperature-dependent thermal conductivity

In this chapter we consider different approximations for the one-dimensional one-phase Stefan problem corresponding to the fusion process of a semi-infinite material with a temperature boundary condition at the fixed face and non-linear temperature-dependent thermal conductivity. The knowledge of the exact solution of this problem, allows to compare it directly with the approximate solutions obtained by applying the heat balance integral method, an alternative form to it and the refined balance integral method, assuming a quadratic temperature profile in space. In all cases, the analysis is carried out in a dimensionless way by the Stefan number (Ste) parameter.

cond-mat.stat-mech↗

arXiv2018

Two different fractional Stefan problems which are convergent to the same classical Stefan problem

Two fractional Stefan problems are considered by using Riemann-Liouville and Caputo derivatives of order $α\in (0,1)$ such that in the limit case ($α=1$) both problems coincide with the same classical Stefan problem. For the one and the other problem, explicit solutions in terms of the Wright functions are presented. We prove that these solutions are different even though they converge, when $α\nearrow 1$, to the same classical solution. This result also shows that some limits are not commutative when fractional derivatives are used.

math.AP↗

arXiv2018

Integral balance methods applied to a non-classical Stefan problem

In this paper we consider two different Stefan problems for a semi-infinite material for the non classical heat equation with a source which depends on the heat flux at the fixed face x = 0. One of them (with constant temperature on x = 0) was studied in [4] where it was found a unique exact solution of similarity type and the other (with a convective boundary condition at the fixed face) is presented in this work. Due to the complexity of the exact solution it is of interest to obtain some kind of approximate solution. For the above reason, the exact solution of each problem is compared with approximate solutions obtained by applying the heat balance integral method and the refined heat balance integral method, assuming a quadratic temperature profile in space. In all cases, a dimensionless analysis is carried out by using the parameters: Stefan number (Ste) and the generalized Biot number (Bi). In addition it is studied the case when Bi goes to infinity, recovering the approximate solutions when a Dirichlet condition is imposed at the fixed face. Some numerical simulations are provided in order to verify the accuracy of the approximate methods

physics.class-ph↗

arXiv2018

One-phase Stefan-like problems with a latent heat depending on the position and velocity of the free boundary, and with Neumann or Robin boundary conditions at the fixed face

In this paper, 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.

math.AP↗

arXiv2018

Exact solution for a two-phase Stefan problem with variable latent heat and a convective boundary condition at the fixed face

Recently it was obtained in [Tarzia, Thermal Sci. 21A (2017) 1-11] for the classical two-phase Lamé-Clapeyron-Stefan problem an equivalence between the temperature and convective boundary conditions at the fixed face under a certain restriction. Motivated by this article we study the two-phase Stefan problem for a semi-infinite material with a latent heat defined as a power function of the position and a convective boundary condition at the fixed face. An exact solution is constructed using Kummer functions in case that an inequality for the convective transfer coefficient is satisfied generalizing recent works for the corresponding one-phase free boundary problem. We also consider the limit to our problem when that coefficient goes to infinity obtaining a new free boundary problem, which has been recently studied in [Zhou-Shi-Zhou, J. Engng. Math. (2017) DOI 10.1007/s10665-017-9921-y].

math-ph↗

arXiv2018

Heat balance integral methods applied to the one-phase Stefan problem with a convective boundary condition at the fixed face

In this paper we consider a one-dimensional one-phase Stefan problem corresponding to the solidification process of a semi-infinite material with a convective boundary condition at the fixed face. The exact solution of this problem, available recently in the literature, enable us to test the accuracy of the approximate solutions obtained by applying the classical technique of the heat balance integral method and the refined integral method, assuming a quadratic temperature profile in space. We develop variations of these methods which turn out to be optimal in some cases. Throughout this paper, a dimensionless analysis is carried out by using the parameters: Stefan number (Ste) and the generalized Biot number (Bi). In addition it is studied the case when Bi goes to infinity, recovering the approximate solutions when a Dirichlet condition is imposed at the fixed face. Some numerical simulations are provided in order to estimate the errors committed by each approach for the corresponding free boundary and temperature profiles.

math.AP↗

arXiv2018

Approximation of the modified error function

In this article, we obtain explicit approximations of the modified error function introduced in Cho, Sunderland. Journal of Heat Transfer 96-2 (1974), 214-217, as part of a Stefan problem with a temperature-dependent thermal conductivity. This function depends on a parameter $δ$, which is related to the thermal conductivity in the original phase-change process. We propose a method to obtain approximations, which is based on the assumption that the modified error function admits a power series representation in $δ$. Accurate approximations are obtained through functions involving error and exponential functions only. For the special case in which $δ$ assumes small positive values, we show that the modified error function presents some characteristic features of the classical error function, such as monotony, concavity, and boundedness. Moreover, we prove that the modified error function converges to the classical one when $δ$ goes to zero.

math.CA↗

arXiv2017

An exact solution to a Stefan problem with variable thermal conductivity and a Robin boundary condition

In this article it is proved the existence of similarity solutions for a one-phase Stefan problem with temperature-dependent thermal conductivity and a Robin condition at the fixed face. The temperature distribution is obtained through a generalized modified error function which is defined as the solution to a nonlinear ordinary differential problem of second order. It is proved that the latter has a unique non-negative bounded analytic solution when the parameter on which it depends assumes small positive values. Moreover, it is shown that the generalized modified error function is concave and increasing, and explicit approximations are proposed for it. Relation between the Stefan problem considered in this article with those with either constant thermal conductivity or a temperature boundary condition is also analysed.

math.AP↗

arXiv2017

A family of singular ordinary differential equations of third order with an integral boundary condition

We establish in this paper the equivalence between a Volterra integral equation of second kind and a singular ordinary differential equation of third order with two initial conditions and an integral boundary condition, with a real parameter. This equivalence allow us to obtain the solution to some problems for nonclassical heat equation, the continuous dependence of the solution with respect to the parameter and the corresponding explicit solution to the considered problem.

math.CA↗

arXiv2017

One-phase Stefan problem with a latent heat depending on the position of the free boundary and its rate of change

From the one-dimensional consolidation of fine-grained soils with threshold gradient, it can be derived a special type of Stefan problems where the seepage front, due to the presence of this threshold gradient, exhibits the features of a moving boundary. In this kind of problems, in contrast with the classical Stefan problem, the latent heat is considered to depend inversely with the rate of change of the seepage front. In this paper a one-phase Stefan problem with a latent heat that not only depends on the rate of change of the free boundary but also on its position is studied. The aim of this analysis is to extend prior results, finding an analytical solution that recovers, by specifying some parameters, the solutions that have already been examined in the literature regarding Stefan problems with variable latent heat. Computational examples will be presented in order to examine the effect of this parameters on the free boundary.

math.AP↗

arXiv2016

Existence and uniqueness of the modified error function

This article is devoted to prove the existence and uniqueness of solution to the non-linear second order differential problem through which is defined the modified error function introduced in Cho-Sunderland, J. Heat Transfer, 96-2:214-217, 1974. We prove here that there exists a unique non-negative analytic solution for small positive values of the parameter on which the problem depends.

math.CA↗

arXiv2016

Non-classical heat conduction problem with non local source

We consider the non-classical heat conduction equation, in the domain $D=\br^{n-1}\times\br^{+}$, for which the internal energy supply depends on an integral function in the time variable of % $(y , t)\mapsto \int_{0}^{t} u_{x}(0 , y , s) ds$, %where $u_{x}(0 , y , s)$ is the heat flux on the boundary $S=\partial D$, with homogeneous Dirichlet boundary condition and an initial condition. The problem is motivated by the modeling of temperature regulation in the medium. The solution to the problem is found using a Volterra integral equation of second kind in the time variable $t$ with a parameter in $\br^{n-1}$. The solution to this Volterra equation is the heat flux $(y, s)\mapsto V(y , t)= u_{x}(0 , y , t)$ on $S$, which is an additional unknown of the considered problem. We show that a unique local solution exists, which can be extended globally in time. Finally a one-dimensional case is studied with some simplifications, we obtain the solution explicitly by using the Adomian method and we derive its properties.

math.AP↗