659 research outputs found
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 Lam\'e-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].Comment: 16 pages, 0 figures. arXiv admin note: text overlap with
arXiv:1610.0933
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
Existence, Uniqueness and Convergence of Simultaneous Distributed-Boundary Optimal Control Problems
We consider a steady-state heat conduction problem for the Poisson
equation with mixed boundary conditions in a bounded multidimensional domain
. We also consider a family of problems for the same
Poisson equation with mixed boundary conditions being the heat
transfer coefficient defined on a portion of the boundary. We
formulate simultaneous \emph{distributed and Neumann boundary} optimal control
problems on the internal energy within and the heat flux ,
defined on the complementary portion of the boundary of
for quadratic cost functional. Here the control variable is the vector .
We prove existence and uniqueness of the optimal control
for the system state of
, and
for the system state of , for each , 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 to the corresponding vectorial optimal
control, system and adjoint states governed by the problem , when the
parameter 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 (with boundary
optimal control ) and fixed (with distributed optimal control
), respectively, for both cases and .Comment: 14 page
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.Comment: 27 pages, 1 Table, 1 Appendi
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.Comment: 10 pages, 6 figures and 2 tables. Paper accepted, in press in Journal
of Biological Systems (2015
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.Comment: 23 page
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