One-dimensional Free Boundary Problem Research Articles

On a convective condition in the diffusion of a solvent into a polymer with non-constant conductivity coefficient

We studied a one-dimensional free boundary problem arising in the polymer industry. Imposing a convective boundary condition, the solution has an interesting asymptotic behavior. It is found that the free boundary is bounded by a constant which does not depend on the conductivity coefficient, which holds even if the diffusion process is non-linear. Numerical methods are presented to compute the solutions and to compare with the theoretical results.

On a convective condition in the diffusion of a solvent into a polymer

We studied a one-dimensional free boundary problem arising in the polymer industry, which solution has an interesting asymptotic behavior when a convective boundary condition is imposed. We show the asymptotic behavior of the free boundary and of the concentration of the solvent in the domain, for large t. Exact estimates and numerical results are obtained.

Exact solutions of one-dimensional two-phase free boundary problems for parabolic equations

We construct automodel solutions for the one-dimensional two-phase Stefan, Florin, and Verigin free boundary problems for parabolic equations in the case where the initial and boundary data are not adjusted. It is shown that in the Stefan problem with “supercooling,” the liquid temperature may be less than the temperature of the phase transition, i.e., the liquid may be “supercooled” while the solid may be “superheated.” Bibliography: 8 titles.

An one-dimensional two-phase free boundary problem in an angular domain

In this paper we consider an one-dimensional solidification problem of a pure substance which is initially in liquid state. The main feature is that the solid region is an angular domain. The existence and uniqueness of the solution are proved. The free boundary is monotonic increasing and disappears in finite time.

One-dimensional free boundary problem for actin-based propulsion of Listeria

Some bacteria move inside cells by recruiting the actin filaments of the host cells. The filaments are polymerized at the back surface of the bacteria, and they move away, forming a “comet” tail behind the bacterium, which consists of gel network. We develop a one-dimensional mathematical model of the gel based on partial differential equations which involve the number of filaments, the density and velocity of the gel, and the pressure. The two end-points of the gel form two free boundaries. The resulting free boundary problem is rather non-standard. We prove local existence and uniqueness.

On a parabolic free boundary problem arising from a Bingham-like flow model with a visco-elastic core

In this paper we study a two-phase one-dimensional free boundary problem for parabolic equation, arising from a mathematical model for Bingham-like fluids with visco-elastic core presented in [L. Fusi, A. Farina, A mathematical model for Bingham-like fluids with visco-elastic core, Z. Angew. Math. Phys. 55 (2004) 826–847]. The main feature of this problem consists in the very peculiar structure of the free boundary condition, not allowing to use classical tools to prove well posedness. Local existence is proved using a fixed point argument based on Schauder's theorem. Uniqueness is proved using a non-standard technique based on a weak formulation of the problem.

ONE DIMENSIONAL COMBUSTION FREE BOUNDARY PROBLEM The project supported by the National Natural Science Foundation of China (No. 10371045) and Guangdong Provincial Natural Science Foundation of China (No. 000671).

A free boundary problem for a parabolic system arising from the mathematical theory of combustion will be considered in the one dimensional case. The existence and uniqueness of the classical solution locally in time will be obtained by the use of a fixed point theorem. Also the existence of the classical solution globally in time and a convergence result with respect to a parameter will be proved under some reasonable assumptions.

Multiplicity and stability of travelling wave solutions in a free boundary combustion-radiation problem

We study travelling wave solutions of a one-dimensional two-phase Free Boundary Problem, which models premixed flames propagating in a gaseous mixture with dust. The model combines diffusion of mass and temperature with reaction at the flame front, the reaction rate being temperature dependent. The radiative effects due to the presence of dust account for the divergence of the radiative flux entering the equation for temperature. This flux is modelled by the Eddington equation. In an appropriate limit the divergence of the flux takes the form of a nonlinear heat loss term. The resulting reduced model is able to capture a hysteresis effect that appears if the amount of fuel in front of the flame, or equivalently, the adiabatic temperature is taken as a control parameter.

Global solutions to a one-dimensional hyperbolic free boundary problem which arises in peeling phenomena

This problem arises from the physical model “Peel a thin film from a domain”. The behavior of the peeling front is governed by a hyperbolic equation. As the first step, the one-dimensional case is treated. The local solutions have been already given. In this note, the global solution is constructed under the assumption the peeling speed is non negative.

Global existence of solutions to a one-dimensional free boundary problem of hyperbolic type

A one-dimensional free boundary problem of hyperbolic type will be analyzed. This problem arises from the physical model ``Peel a thin film from a domain''. The local solutions have been already given for our problem. However, it is not sure that the solutions can be extended globally. In this article, by applying an iteration method, it will be shown that time-global solutions exist.

FROM THE IMPLIED VOLATILITY SKEW TO A ROBUST CORRECTION TO BLACK-SCHOLES AMERICAN OPTION PRICES

We describe a robust correction to Black-Scholes American derivatives prices that accounts for uncertain and changing market volatility. It exploits the tendency of volatility to cluster, or fast mean-reversion, and is simply calibrated from the observed implied volatility skew. The two-dimensional free-boundary problem for the derivative pricing function under a stochastic volatility model is reduced to a one-dimensional free-boundary problem (the Black-Scholes price) plus the solution of a fixed boundary-value problem. The formal asymptotic calculation that achieves this is presented here. We discuss numerical implementation and analyze the effect of the volatility skew.

On the Solution of Inverse Problems for Generalized Oxygen Consumption

We present the solution of some inverse problems for one-dimensional free boundary problems of oxygen consumption type, with a semilinear convection-diffusion-reaction parabolic equation. Using a fixed domain transformation (Landau's transformation) the direct problem is reduced to a system of ODEs. To minimize the objective functionals in the inverse problems, we approximate the data by a finite number of parameters with respect to which automatic differentiation is applied.

A numerical approach to the asymptotic behavior of solutions of a one-dimensional free boundary problem of hyperbolic type

A free boundary problem which arises from the physical model called “peeling” will be analyzed numerically. To obtain an equation which describes the motion of the free boundary, the fixed domain method is applied. By using the equation, numerical computations are carried out. Our numerical computations suggest that the peeling speed plays an important role in the existence of global solutions.

A free boundary problem arising in the modelling of internal oxidation of binary alloys

A one-dimensional free boundary problem arising in the modelling of internal oxidation of binary alloys is studied in this paper. The free boundary of this problem is determined by the equation u = 0, where u is the solution of a parabolic partial differential equation with discontinuous coefficients across the free boundary. Local existence, uniqueness and the regularity of the free boundary are established. Global existence is also studied.

A Free Boundary Problem Arising from Swelling-Controlled Release Processes

In this paper a mathematical model describing the diffusion of a penetrant solvent in a polymer is studied. The model is formulated as a one-dimensional free boundary problem, which involves two free boundaries. Global existence, uniqueness, and regularity as well as asymptotic behaviors of the free boundaries are established.

A Free Boundary Focusing Problem

We consider a one-dimensional free boundary problem arising in combustion theory and establish that all solutions are asymptotically equal to a similarity solution which vanishes in a finite time.

A free boundary focusing problem

We consider a one-dimensional free boundary problem arising in combustion theory and establish that all solutions are asymptotically equal to a similarity solution which vanishes in a finite time.

A free boundary problem arising in some reacting–diffusing system

SynopsisWe prove the well-posedness for a one-dimensional free boundary problem arising from some reaction diffusion system. The interfacial point hits a boundary point in finite time or remains inside for all time. In the large diffusion limit, the system is reduced to ordinary differential equations of finite dimension.

Some Numerical Methods for the Computation of Capillary Free Boundaries Governed by the Navier–Stokes Equations

In §, the trial method, Newton’s method, and the total linearization method are applied to a simple one-dimensional free boundary (FB) problem for which the solution is known. Attention will be focused on convergence speed and computer implementation facilities of these methods. Section 2, discusses static FB problems in two- and three-dimensional containers. These FBs are governed by the Laplace–Young equation. In §3, two stationary FB problems governed by the Navier–Stokes equations are formulated and these problems are solved using the three methods introduced in §1. A thermocapillary FB problem governed by the Navier–Stokes equations coupled with the heat equation is studied in §4. In §5, a moving FB problem governed by the Navier–Stokes equations will be considered within the context of perturbation theory. Finally, in §6, some remarks are made on present and future research in the field of capillary FB problems governed by the Navier–Stokes equations.

On-line Identification of the State of the Surface of a Material Undergoing Thermal Processing

An inverse approach of a one-dimensional parabolic free boundary problem is presented. More precisely, the phase change interface location in a slab heated on one side is identified. The data consist of the temperature and the heat flux measured on the cold face of the domain. "Measurements" are produced by direct simulation and addition of noise. An objective function based on the difference between measured and simulated values of the heat flux is minimized. During the initial non-melting period, a reduced-order model, derived by the balanced state-space representation method, simulates the heat equation. The reduced model proves to be less time-consuming than a complete model and yet efficient.The importance of regularizing the problem is stressed.