arXiv subjects

Domingo A. Tarzia

Publications and source records attributed to Domingo A. Tarzia.

54 records · Page 3Linked to original sources

arXiv2016

Similarity solution for a two-phase one-dimensional Stefan problem with a convective boundary condition and a mushy zone model

A two-phase solidification process for a one-dimensional semi-infinite material is considered. It is assumed that it is ensued from a constant bulk temperature present in the vicinity of the fixed boundary, which it is modelled through a convective condition (Robin condition). The interface between the two phases is idealized as a mushy region and it is represented following the model of Solomon, Wilson and Alexiades. An exact similarity solution is obtained when a restriction on data is verified, and it is analysed the relation between the problem considered here and the problem with a temperature condition at the fixed boundary. Moreover, it is proved that the solution to the problem with the convective boundary condition converges to the solution to a problem with a temperature condition when the heat transfer coefficient at the fixed boundary goes to infinity, and it is given an estimation of the difference between these two solutions. Results in this article complete and improve the ones obtained in Tarzia, Compt. Appl. Math., 9 (1990), 201-211.

math.AP↗

arXiv2016

Determination of two unknown thermal coefficients through an inverse one-phase fractional Stefan problem

We consider a semi-infinite one-dimensional phase-change material with two unknown constant thermal coefficients among the latent heat per unit mass, the specific heat, the mass density and the thermal conductivity. Aiming at the determination of them, we consider an inverse one-phase Stefan problem with an over-specified condition at the fixed boundary and a known evolution for the moving boundary. We assume that the phase-change process presents latent-heat memory effects by considering a fractional time derivative of order $α$ ($0<α<1$) in the Caputo sense and a sharp front model for the interface. According to the choice of the unknown thermal coefficients, six inverse fractional Stefan problems arise. For each of them, we determine necessary and sufficient conditions on data to obtain the existence and uniqueness of a solution of similarity type. Moreover, we present explicit expressions for the temperature and the unknown thermal coefficients. Finally, we show that the results for the classical statement of this problem, associated with $α=1$, are obtained through the fractional model when $α\to 1^-$.

math-ph↗

arXiv2016

Properties of the financial break-even point in a simple investment project as a function of the discount rate

We consider a simple investment project with the following parameters: I>0: Initial investment which is amortizable in n years; n: Number of years the investment allows production with constant output per year; A>0: Annual amortization (A=I/n); Q>0: Quantity of products sold per year; Cv>0: Variable cost per unit; p>0: Price of the product with p>Cv; Cf>0: Annual fixed costs; te: Tax of earnings; r: Annual discount rate. We also assume inflation is negligible. We derive a closed expression of the financial break-even point Qf (i.e. the value of Q for which the net present value (NPV) is zero) as a function of the parameters I, n, Cv, Cf, te, r, p. We study the behavior of Qf as a function of the discount rate r and we prove that: (i) For r negligible Qf equals the accounting break-even point Qc (i.e. the earnings before taxes (EBT) is null) ; (ii) When r is large the graph of the function Qf=Qf(r) has an asymptotic straight line with positive slope. Moreover, Qf(r) is an strictly increasing and convex function of the variable r; (iii) From a sensitivity analysis we conclude that, while the influence of p and Cv on Qf is strong, the influence of Cf on Qf is weak.

q-fin.GN↗

arXiv2015

Numerical analysis of a family of optimal distributed control problems governed by an elliptic variational inequality

The numerical analysis of a family of distributed mixed optimal control problems governed by elliptic variational inequalities (with parameter $α>0$) is obtained through the finite element method when its parameter $h\rightarrow 0$. We also obtain the limit of the discrete optimal control and the associated state system solutions when $α\rightarrow \infty$ (for each $h>0$) and a commutative diagram for two continuous and two discrete optimal control and its associated state system solutions is obtained when $h\rightarrow 0$ and $α\rightarrow \infty$. Moreover, the double convergence is also obtained when $(h, α)\rightarrow(0, \infty)$.

math.NA↗

arXiv2015

Double convergence of a family of discrete distributed mixed elliptic optimal control problems with a parameter

We consider a bounded domain $Ω$ in $\mathbb{R}^{n}$ whose regular boundary $\partialΩ$ consists of the union of two disjoint portions $Γ_{1}$ and $Γ_{2}$ with $meas(Γ_{1})>0$. The convergence of a family of continuous distributed mixed elliptic optimal control problems (DMEOCPs) $P_α$, governed by elliptic variational equalities (EVE), when the parameter $α$ goes to infinity was studied in Gariboldi-Tarzia, Appl. Math. Optim. (2003). It has been proved that the optimal control (OC), and their corresponding system and adjoint system states (SASSs) are strongly convergent, in adequate functional spaces, to the OC, and the SASSs of another CDMEOPC $P$ governed also by an EVE with a different boundary condition on $Γ_{1}$. We consider the discrete approximations $P_{hα}$ and $P_{h}$ of the OCPs $P_α$ and $P$ respectively, for each $h>0$ and $α>0$, through the finite element method with parameter $h$. We also discrete the EVEs which define the SASSs, and the corresponding cost functional of the DMEOCPs $P_α$ and $P$. The goal is to study the double convergence of this family of discrete DMEOCPs $P_{hα}$ when $α\to +\infty$ and $h\to 0$ simultaneously. We prove the convergence of the discrete OCs, the discrete SASSs of the family $P_{hα}$ to the corresponding to the discrete DMEOCP $P_{h}$ when $α\to +\infty$, for each $h>0$,. We study the convergence of the discrete OCPs $P_{hα}$ and $P_{h}$ when $h\to 0$ obtaining a commutative diagram which relates the continuous and discrete DMEOCPs $P_{hα}$, $P_{h}$, $P_α$ and $P$ by taking the limits $h\to 0$ and $α\to +\infty$ respectively. We also study the double convergence of $P_{hα}$ to $P$ when $(h,α)\to (0,+\infty)$.

math.NA↗

arXiv2015

Simultaneous determination of two unknown thermal coefficients through a mushy zone model with an overspecified convective boundary condition

The simultaneous determination of two unknown thermal coefficients for a semi-infinite material under a phase-change process with a mushy zone according to the Solomon-Wilson-Alexiades model is considered. The material is assumed to be initially liquid at its melting temperature and it is considered that the solidification process begins due to a heat flux imposed at the fixed face. The associated free boundary value problem is overspecified with a convective boundary condition with the aim of the simultaneous determination of the temperature of the solid region, one of the two free boundaries of the mushy zone and two thermal coefficients among the latent heat by unit mass, the thermal conductivity, the mass density, the specific heat and the two coefficients that characterize the mushy zone. The another free boundary of the mushy zone, the bulk temperature and the heat flux and heat transfer coefficients at the fixed face are assumed to be known. According to the choice of the unknown thermal coefficients, fifteen phase-change problems arise. The study of all of them is presented and explicit formulae for the unknowns are given, beside necessary and sufficient conditions on data in order to obtain them. Formulae for the unknown thermal coefficients, with their corresponding restrictions on data, are summarized in a table.

math.AP↗

arXiv2015

Numerical analysis of distributed optimal control problems governed by elliptic variational inequalities

A continuous optimal control problem governed by an elliptic variational inequality was considered in Boukrouche-Tarzia, Comput. Optim. Appl., 53 (2012), 375-392 where the control variable is the internal energy $g$. It was proved the existence and uniqueness of the optimal control and its associated state system. The objective of this work is to make the numerical analysis of the above optimal control problem, through the finite element method with Lagrange's triangles of type 1. We discretize the elliptic variational inequality which define the state system and the corresponding cost functional, and we prove that there exists a discrete optimal control and its associated discrete state system for each positive $h$ (the parameter of the finite element method approximation). Finally, we show that the discrete optimal control and its associated state system converge to the continuous optimal control and its associated state system when the parameter $h$ goes to zero.

math.NA↗

arXiv2015

Existence, Uniqueness and Convergence of Simultaneous Distributed-Boundary Optimal Control Problems

We consider a steady-state heat conduction problem $P$ for the Poisson equation with mixed boundary conditions in a bounded multidimensional domain $Ω$. We also consider a family of problems $P_α$ for the same Poisson equation with mixed boundary conditions being $α>0$ the heat transfer coefficient defined on a portion $Γ_{1}$ of the boundary. We formulate simultaneous \emph{distributed and Neumann boundary} optimal control problems on the internal energy $g$ within $Ω$ and the heat flux $q$, defined on the complementary portion $Γ_{2}$ of the boundary of $Ω$ for quadratic cost functional. Here the control variable is the vector $(g,q)$. We prove existence and uniqueness of the optimal control $(\overline{\overline{g}},\overline{\overline{q}})$ for the system state of $P$, and $(\overline{\overline{g}}_α,\overline{\overline{q}}_α)$ for the system state of $P_α$, for each $α>0$, and we give the corresponding optimality conditions. We prove strong convergence, in suitable Sobolev spaces, of the vectorial optimal controls, system and adjoint states governed by the problems $P_α$ to the corresponding vectorial optimal control, system and adjoint states governed by the problem $P$, when the parameter $α$ goes to infinity. We also obtain estimations between the solutions of these vectorial optimal control problems and the solution of two scalar optimal control problems characterized by fixed $g$ (with boundary optimal control $\overline{q}$) and fixed $q$ (with distributed optimal control $\overline{g}$), respectively, for both cases $α>0$ and $α=\infty$.

math.OC↗

arXiv2015

Determination of one unknown thermal coefficient through a mushy zone model with a convective overspecified boundary condition

A semi-infinite material under a solidification process with the Solomon-Wilson- Alexiades's mushy zone model with a heat flux condition at the fixed boundary is considered. The associated free boundary problem is overspecified through a convective boundary condition with the aim of the simultaneous determination of the temperature, the two free boundaries of the mushy zone and one thermal coefficient among the latent heat by unit mass, the thermal conductivity, the mass density, the specific heat and the two coefficients that characterize the mushy zone. Bulk temperature and coefficients which characterize the heat flux and the heat transfer at the boundary are assumed to be determined experimentally. Explicit formulae for the unknowns are given for the resulting six phase-change problems, beside necessary and sufficient conditions on data in order to obtain them. In addition, relationship between the phase-change process solved in this paper with an analogous process overspecified by a temperature boundary condition is presented, and this second problem is solved by considering a large heat transfer coefficient at the boundary in the problem with the convective boundary condition. Formulae for the unknown thermal coefficients corresponding to both problems are summarized in two tables.

math.AP↗

arXiv2015

A free boundary model for oxygen diffusion in a spherical medium

The goal of this article is to find a correct approximated solution using a polynomial of sixth degree for the free boundary problem corresponding to the diffusion of oxygen in a spherical medium with simultaneous absorption at a constant rate, and to show some mistakes in previously published solutions.

math.AP↗

arXiv2014

Analysis of the nutrient uptake by roots in fixed volume of soil as predicted by fixed boundary, moving boundary and architectural models

This work examines the relevance of the one-dimensional models used to study the influx and the cumulative uptake of nutrient by roots. The physical models studied are the fixed boundary model (Barber and Cushman 1981) and an improved version of our moving boundary model (Reginato et al. 2000). A weight averaged expression to compute influx on root surface and a generalized formula to estimate the cumulative nutrient uptake are used. The moving boundary model problem is solved by the adaptive finite element method. For comparison of simulations of influx and cumulative uptake versus observed results six set of data extracted from literature are used. For ions without limitations of availability fixed and moving boundary models produces similar results with small errors. Instead, to low concentrations, the fixed boundary model over predicts while the moving boundary model always produces better results mainly for K. For the P uptake the moving boundary model produces better results only when the concentrations are very low and their predictions are comparable to the obtained by a 3D-architectural model. The obtained improvements would explain any failures of previous models for ions of low availability. Therefore, our model could be a simpler alternative due to its low computational burden.

q-bio.TO↗

arXiv2014

A commutative diagram among discrete and continuous Neumann boundary optimal control problems

We consider a bounded domain D whose regular boundary consists of the union of two portions F1 and F2. The convergence of a family of continuous Neumann boundary mixed elliptic optimal control problems (Pa), governed by elliptic variational equalities, when the parameter a of the family goes to infinity was studied in Gariboldi - Tarzia, Adv. Diff. Eq. Control Processes, 1 (2008), 113-132, being the control variable the heat flux on the boundary F2. It has been proved that the optimal control problem (Pa) are strongly convergent to another optimal control (P) governed also by an elliptic variational equality with a different boundary condition on the portion of the boundary F1. We consider the discrete approximations (Pha) and (Ph) of the optimal control problems (Pa) and (P) respectively, for each h>0, a>0, through the finite element method with Lagrange's triangles of type 1 with parameter h (the longest side of the triangles). We also discrete the elliptic variational equalities which define the system and their adjoint system states, and the corresponding cost functional of the Neumann boundary optimal control problems (Pa) and (P). The goal of this paper is to study the convergence of this family of discrete Neumann boundary mixed elliptic optimal control problems (Pha) when the parameter a goes to infinity. We prove the convergence of the discrete optimal controls, the discrete system and adjoint system states of the family (Pha) to the corresponding to the discrete Neumann boundary mixed elliptic optimal control problem (Ph) when a goes to infinity, for each h>0, in adequate functional spaces. We also study the convergence when h goes to zero and we obtain a commutative diagram which relates the continuous and discrete Neumann boundary mixed elliptic optimal control problems (Pha), (Pa), (Ph) and (P) by taking the limits h goes to zero and a goes to infinity respectively.

math.NA↗

arXiv2014

Explicit solutions for a non-classical heat conduction problem for a semi-infinite strip with a non-uniform heat source

A non-classical initial and boundary value problem for a non-homogeneous one-dimensional heat equation for a semi-infinite material with a zero temperature boundary condition at the face $x=0$ is studied with the aim of finding explicit solutions. It is not a standard heat conduction problem because a heat source $-Φ(x)F(V(t),t)$ is considered, where $V$ represents the heat flux at $x=0$. Explicit solutions independents of the space or temporal variables are given. Solutions with separated variables when the data functions are defined from the solution $X=X(x)$ of a linear initial value problem of second order and the solution $T=T(t)$ of a non-linear (in general) initial value problem of first order which involves the function $F$, are also given and explicit solutions corresponding to different definitions of $F$ are obtained. A solution by an integral representation depending on the heat flux at $x=0$ for the case in which $F=F(V,t)=νV$, $ν>0$, is obtained and explicit expressions for the heat flux at $x=0$ and for its corresponding solution are calculated when $h=h(x)$ is a potential function and $Φ=Φ(x)$ is given by $Φ(x)=λx$, $Φ(x)=-μ\sinh{(λx)}$ or $Φ(x)=-μ\sin{(λx)}$, $λ>0$ and $μ>0$. The limit when the temporal variable $t$ tends to $+\infty$ of each explicit solution obtained in this paper is studied and the "controlling" effects of the source term $-ΦF$ are analysed by comparing the asymptotic behavior of each solution with the asymptotic behavior of the solution to the same problem but in absence of source term. Finally, a relationship between this problem with another non-classical initial and boundary value problem for the heat equation is established and explicit solutions for this second problem are also obtained.

math.AP↗

arXiv2014

A Generalized Neumann Solution for the Two-Phase Fractional Lamé-Clapeyron-Stefan Problem

We obtain a generalized Neumann solution for the two-phase fractional Lamé-Clapeyron-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 $0<\al\leq 1$. When $ \al \nearrow $ 1 we recover the classical Neumann solution for the two-phase Lamé-Clapeyron-Stefan problem given through the error function.

math.AP↗

arXiv2014

Similarity solutions for thawing processes with a convective boundary condition

Similarity solutions for a one-dimensional mathematical model for thawing in a saturated semi-infinite porous media is considered when change of phase induces a density jump and a convective boundary condition is imposed at the fixed face $x=0$. Different cases depending on physical parameters are analysed and an explicit solution of a similarity type is obtained if and only if an inequality for data is verified. Moreover, a monotony property respect to the coefficient which characterizes the heat transfer at the fixed face $x=0$ is obtained for the coefficient involved in the definition of the free boundary. Relationship between the Stefan problem with convective condition at $x=0$ considered in this paper and the Stefan problem with temperature condition at the same face studied in (Fasano-Primicerio-Tarzia, Math. Models Meth. Appl. Sci., 9 (1999), 1-10) is analized and conditions for physical parameters under which both problems became equivalents are obtained. Furthermore, an inequality to be satisfied for the coefficient which characterizes the free boundary of the problem with a temperature or a convective boundary condition at the fixed face $x=0$ is also obtained.

math.AP↗

arXiv2013

Existence, Uniqueness, and Convergence of optimal control problems associated with Parabolic variational inequalities of the second kind

Let $u_{g}$ the unique solution of a parabolic variational inequality of second kind, with a given $g$. Using a regularization method, we prove, for all $g_{1}$ and $g_{2}$, a monotony property between $μu_{g_{1}} + (1-μ)u_{g_{2}}$ and $u_{μg_{1} + (1-μ)g_{2}}$ for $μ\in [0, 1]$. This allowed us to prove the existence and uniqueness results to a family of optimal control problems over $g$ for each heat transfer coefficient $h>0$, associated to the Newton law, and of another optimal control problem associated to a Dirichlet boundary condition. We prove also, when $h\to +\infty$, the strong convergence of the optimal controls and states associated to this family of optimal control problems with the Newton law to that of the optimal control problem associated to a Dirichlet boundary condition.

math.AP↗

arXiv2013

Convergence of Distributed Optimal Control Problems Governed by Elliptic Variational Inequalities

First, let $u_{g}$ be the unique solution of an elliptic variational inequality with source term $g$. We establish, in the general case, the error estimate between $u_{3}(μ)=μu_{g_{1}}+ (1-μ)u_{g_{2}}$ %(the convex combination of two solutions) and $u_{4}(μ)=u_{μg_{1}+ (1-μ) g_{2}}$ %(the solution corresponding to the convex combination of two data) for $μ\in [0, 1]$. Secondly, we consider a family of distributed optimal control problems governed by elliptic variational inequalities over the internal energy $g$ for each positive heat transfer coefficient $h$ given on a part of the boundary of the domain. For a given cost functional and using some monotony property between $u_{3}(μ)$ and $u_{4}(μ)$ given in F. Mignot, J. Funct. Anal., 22 (1976), 130-185, we prove the strong convergence of the optimal controls and states associated to this family of distributed optimal control problems governed by elliptic variational inequalities to a limit Dirichlet distributed optimal control problem, governed also by an elliptic variational inequality, when the parameter $h$ goes to infinity. We obtain this convergence without using the adjoint state problem (or the Mignot's conical differentiability) which is a great advantage with respect to the proof given in C.M. Gariboldi - D.A. Tarzia, Appl. Math. Optim., 47 (2003), 213-230, for optimal control problems governed by elliptic variational equalities.

math.AP↗

arXiv2013

Convergence of optimal control problems governed by second kind parabolic variational inequalities

We consider a family of optimal control problems where the control variable is given by a boundary condition of Neumann type. This family is governed by parabolic variational inequalities of the second kind. We prove the strong convergence of the optimal controls and state systems associated to this family to a similar optimal control problem. This work solves the open problem left by the authors in IFIP TC7 CSMO2011.

math.AP↗