On a multiple time-scales perturbation analysis of a Stefan problem with a time-dependent Dirichlet boundary condition

In this paper, a classical Stefan problem with a prescribed and small time-dependent temperature at the boundary is studied. By using a multiple time-scales perturbation method, it is shown analytically how the moving boundary profile is influenced by the prescribed temperature at the boundary and the initial conditions. Only a few exact solutions are available for this type of problems and it turns out that the constructed approximations agree very well with these exact solutions. In particular, approximations of solutions for this type of problems, with periodic and decaying temperatures at the boundary, are constructed. Furthermore, these approximations are valid on a long time scale, and seems to be not available in the literature.     

On a two-phase Stefan problem with convective boundary condition including a density jump at the free boundary

We consider a two-phase Stefan problem for a semi-infinite body with a convective boundary condition including a density jump at the free boundary with a time-dependent heat transfer coefficient of the type $h/\sqrt{t}$, whose solution was given in D. A. Tarzia, PAMM. Proc. Appl. Math. Mech. 7, 1040307–1040308 (2007). We demonstrate that the solution to this problem converges to the solution to the analogous one with a temperature boundary condition when the heat transfer coefficient $h\to +\infty$. Moreover, we analyze the dependence of the free boundary respecting to the jump density.     

Optimal control for a parabolic problem in a domain with highly oscillating boundary

In this article we study the homogenization of an optimal control problem for a parabolic equation in a domain with highly oscillating boundary. We identify the limit problem, which is an optimal control problem for the homogenized equation and with a different cost functional.     

Asymptotics for eigenelements of Laplacian in domain with oscillating boundary: multiple eigenvalues

We study the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a domain with rapidly oscillating boundary. We consider the case where the eigenvalue of the limit problem is multiple. We construct the leading terms of the asymptotic expansions for the eigenelements and verify the asymptotics.     

Numerical bifurcation of a delayed diffusive food‐limited model with Dirichlet boundary condition

This paper studies the Neimark–Sacker bifurcation of a diffusive food‐limited model with a finite delay and Dirichlet boundary condition by the backward Euler difference scheme, Crank‐Nicolson difference scheme, and nonstandard finite‐difference scheme. The existence of Neimark‐Sacker bifurcation at the equilibrium is obtained. Our results show that Crank‐Nicolson and nonstandard finite‐difference schemes are superior to the backward Euler difference scheme under the means of describing approximately the dynamics of the original system. Finally, numerical examples are provided to illustrate the analytical results. Copyright © 2015 John Wiley & Sons, Ltd.     

Large time behavior for a viscous Hamilton-Jacobi equation with Neumann boundary condition

We prove the existence and the uniqueness of strong solutions for the viscous Hamilton-Jacobi equation: with Neumann boundary condition, and initial data μ₀, a continuous function. The domain Ω is a bounded and convex open set with smooth boundary, a∈R,a≠0 and p>0. Then, we study the large time behavior of the solution and we show that for p∈(0,1), the extinction in finite time of the gradient of the solution occurs, while for p?1 the solution converges uniformly to a constant, as t→∞.     

Numerical investigation of quenching for a nonlinear diffusion equation with a singular Neumann boundary condition

For a nonlinear diffusion equation with a singular Neumann boundary condition, we devise a difference scheme which represents faithfully the properties of the original continuous boundary value problem. We use non‐uniform mesh in order to adequately represent the spatial behavior of the quenching solution near the boundary. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 429–440, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10013     

双曲-抛物型偏微分方程奇摄动混合问题的数值解法

构造了二阶双曲—抛物型方程奇摄动混合问题的差分格式，给出了差分解的能量不等式，并证明了差分解在离散范数下关于小参数一致收敛于摄动问题的解。     

Dynamics of a free boundary in a binary medium with variable thermal conductivity

We construct an asymptotic solution of the phase field system with variable thermal conductivity different in domains occupied by different phases. We show that, depending on relations between parameters characterizing the substance, the dynamics of the free interface between the phases is determined by solutions of the classical or modified Stefan problems. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 231–241, August, 1999.     

一类带记忆边界条件波动系统解的长时间性态

研究一类带记忆边界条件波动系统解的长时间性态.在初边值满足一定条件时,利用Faedo-Galerkin近似方法得到了系统解的存在唯一性.使用扰动能量方法,证明了系统解的一致衰减性.     

Stability of difference schemes for two-dimensional parabolic equations with non-local boundary conditions

The stability of difference schemes for one-dimensional and two-dimensional parabolic equations, subject to non-local (Bitsadze-Samarskii type) boundary conditions is dealt with. To analyze the stability of difference schemes, the structure of the spectrum of the matrix that defines the linear system of difference equations for a respective stationary problem is studied. Depending on the values of parameters in non-local conditions, this matrix can have one zero, one negative or complex eigenvalues. The stepwise stability is proved and the domain of stability of difference schemes is found.     

Stability of Rothe difference scheme for the reverse parabolic problem with integral boundary condition

In this paper, we study the approximation of reverse parabolic problem with integral boundary condition. The Rothe difference scheme for an approximate solution of reverse problem is discussed. We establish stability and coercive stability estimates for the solution of the Rothe difference scheme. In sequel, we investigate the first order of accuracy difference scheme for approximation of boundary value problem for multidimensional reverse parabolic equation and obtain stability estimates for its solution. Finally, we give numerical results together with an explanation on the realization in one- and two-dimensional test examples.     

An inverse coefficient problem for a quasilinear parabolic equation with periodic boundary and integral overdetermination condition

In this paper, the inverse problem of finding the time‐dependent coefficient of heat capacity together with the solution periodic boundary and integral overdetermination conditions is considered. Under some natural regularity and consistency conditions on the input data, the existence, uniqueness, and continuous dependence upon the data of the solution are shown. Some considerations on the numerical solution for this inverse problem are presented with an example. Copyright © 2014 John Wiley & Sons, Ltd.     

Homogenization of the boundary value problem for the laplace operator in a perforated domain with a rapidly oscillating nonhomogeneous Robin-type condition on the boundary of holes in the critical case

The asymptotic behavior of solutions to a boundary value problem in a domain periodically perforated by small holes with a rapidly oscillating nonhomogeneous Robin-type condition on their boundaries is investigated in the case of critical parameter values.     

Error estimates of the finite element method for the exterior Helmholtz problem with a modified DtN boundary condition

A priori error estimates in the H¹- and L²-norms are established for the finite element method applied to the exterior Helmholtz problem, with modified Dirichlet-to-Neumann (MDtN) boundary condition. The error estimates include the effect of truncation of the MDtN boundary condition as well as that of discretization of the finite element method. The error estimate in the L²-norm is sharper than that obtained by the author [D. Koyama, Error estimates of the DtN finite element method for the exterior Helmholtz problem, J. Comput. Appl. Math. 200 (1) (2007) 21-31] for the truncated DtN boundary condition.     

Mathematical modeling of coupled turbulent flow and solidification in a single belt caster with electromagnetic brake

A mathematical model has been developed to simulate turbulent fluid flow and solidification in the presence of a DC magnetic field in an extended nozzle for metal delivery to a single belt caster. This paper reports on predicted effects of DC magnetic field conditions in modifying flows and solidification behavior in the metal delivery system. It is shown that the application of a DC magnetic brake to the proposed system can result in a reasonably uniform feeding of melt onto the cooled moving belt. This, in turn, optimises the rate of even shell growth along the chilled substrate. In order to account for the effects of turbulence, a revised low-Reynolds k–ε turbulent model was employed. A Darcy-porosity approach was used to simulate fluid flow within the mushy solidification region. Simulations were carried out for plain carbon steel strip casting. The fully coupled transport equations were numerically solved using the finite volume method. The computed flow patterns were compared with those reported in the literature. The performance of the magnetic flow control device proposed in this work is evaluated and compared with flow modifications obtained by inserting a ceramic filter within the reservoir.     

Skin effect at a finite temperature with a diffusive condition on the boundary of the half-space of the conducting medium

An analytical solution of the skin effect problem at a finite temperature with diffusive scattering of electrons on the boundary of the half-space filled by a conducting medium is obtained.