Quasistationary problem of stefan type

An approximative method for solving the quasistationary thermophysical Stefan problem is presented.     

Monotonicity in a stefan problem for the heat equation

A theorem on differential inequalities related to a simple parabolic free boundary value problem is presented. The result can be used for the construction of pointwise upper and lower bounds of the free boundary curve.     

Numerical solution of a multidimensional two-phase stefan problem

This paper treats a multidimensional two-phase Stefan problem with variable coefficients and mixed type boundary conditions. A numerical method for solving the problem is of fixed domain type, based on a variational inequality formulation of the problem. Numerical solutions are obtained by using piecewise linear finite elements in space and finite difference in time, and by solving a strictly convex minimization problem at each time step. Some computational results are presented.     

Multidimensional two-phase quasistationary stefan problem

Existence and uniqueness of the classic solution to a two-dimensional quasistationary Stefan problem are considered. The family of model problems dependent on the parameter ε>0 which defines a heat conductivity of a matter in the direction of thex-axis is analysed. When ε→0 it is approximated by the approximate model problem having less dimensions. Analogous results are also valid for a three-dimensional problem.     

Time periodic solutions of a one-dimensional two-phase stefan problem

Summary This paper answers in the affirmative the question whether a stationary solution converts into a time periodic one if the constant boundary temperatures are slightly changed by small time periodic perturbations.     

Real-time control of the free boundary in a two-phase stefan problem

A two-phase Stefan-problem is considered where the progress of the free boundary is observed by fully automatic real-time controls (thermostats or photo-electric cells). The heat flux at both fixed boundaries can be determined by heaters which respond to the signals of the controls, possibly with a certain time lag. The corresponding mathematical model leads to a two-phase Stefan problem with nonlinear and discontinuous boundary conditions with delays at the fixed boundaries. The problem is transformed into a set-valued fixed point equation, and the existence of a solution is shown with the aid of a theorem due to Bohnenblust-Karlin. The consequence of this result is that a free boundary with the well-known smoothness properties develops under the impact of a fully automatic real-time control via thermostats or photo-electric cells. Some numerical experiments conclude the paper. They indicate that the model is realistic.     

Hyperbolic Yamabe problem

In this paper, we investigate the solutions of the hyperbolic Yamabe problem for the(1 + n)-dimensional Minkowski space-time. More precisely speaking, for the case of n = 1, we derive a general solution of the hyperbolic Yamabe problem; for the case of n = 2, 3, we study the global existence and blowup phenomena of smooth solutions of the hyperbolic Yamabe problem;while for general multi-dimensional case n ≥ 2, we discuss the global existence and non-existence for a kind of exact solutions of the hyperbolic Yamabe problem.     

Error estimates for a single phase quasilinear stefan problem with a forcing term

In this paper, we apply finite element Galerkin method to a singlephase quasi-linear Stefan problem with a forcing term. We consider the existence and uniqueness of a semidiscrete approximation and optimal error estimates inL ₂, L_∞,H ₁ andH ₂ norms for semidiscrete Galerkin approximations are derived.     

Quasilinear hyperbolic stefan problem with nonlocal boundary conditions

Using the method of contracting mappings, we prove, for small values of time, the existence and uniqueness of a generalized Lipschitz solution of a mixed problem with unknown boundaries for a hyperbolic quasilinear system of first-order equations represented in terms of Riemann invariants with nonlocal (nonseparated and integral) boundary conditions.     

On a stefan problem involving a spherical region: application of a Bergman-type expansion

A Bergman-type series expansion method is used in the analysis of a spherical reaction problem. The small-time solution so derived is employed as the starting point for the numerical solution, and the time for complete reaction is calculated.     

Iterative solution of the stefan problem with several boundaries

An iterative process is constructed for numerical solution of weakly stable boundary-value problems for parabolic equations with unknown moving boundaries. Special emphasis is placed on the choice of the initial approximation.Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 37–43, 1985.     

Periodic solutions of the stefan problem with hysteresis-type boundary conditions

We consider the Stefan problem with Dirichlet boundary conditions depending on a hysteresis functional where the free boundary is involved. We show existence of a positive valueT and existence of aT-periodic solution of the problem, provided the Stefan number is sufficiently small and the hysteresis functional is described by the elementary rectangular hysteresis loop. If in addition the Preisach hysteresis operator is Lipschitz-continuous we prove that every periodic solution must be stationary. Dedicated to Professor Avner Friedman on occasion of his 60th birthday supported by Humboldt Foundation Scholarship     

The bidmensional stefan problem with convection: the time dependent case

This paper considers the time dependent stefan problem with convection in the fluid phase governed by the Stokes equation, and with adherence of the fluid on the lateral boundaries: The existence of a weak solution is obtained via the introduction of a temperature dependent penalty term in the fluid flow equation, together with the application of various compactness arguments.     

The order of convergence in the stefan problem with vanishing specific heat

The paper is concerned with the two-phase Stefan problem with a small parameter ϵ, which coresponds to the specific heat of the material. It is assumed that the initial condition does not coincide with the solution for t = 0 of the limit problem related to ε = 0. To remove this discrepancy, an auxiliary boundary layer type function is introduced. It is proved that the solution to the two-phase Stefan problem with parameter ϵ differs from the sum of the solution to the limit Hele–Shaw problem and a boundary layer type function by quantities of order O(ϵ). The estimates are obtained in H?lder norms. Bibliography: 13 titles.