A two-phase free boundary problem for system of reaction-diffusion equations

Purpose The main purpose of this paper is given below: To present a mathematical model of a two-phase Stefan problem including a moving phase change material and variable thermophysical properties. To find a numerical solution of the problem to discuss the dependence of considered phase change problem on variable thermal conductivity, variable specific heat and Peclet number. Design/methodology/approach In this paper, a numerical solution of the problem is obtained using the front-fixing method in tandem with the explicit finite difference scheme. The authors have also discussed the consistency and stability of proposed numerical scheme. Findings In this study, it is observed that the considered scheme is an efficient tool that provides sufficiently accurate results for exploring the behaviors of moving interface (free boundary) and temperature profile for a nonclassical two-phase free boundary problem. In this study, the authors have observed that the parameters α1 and α2 influence the temperature profiles of the liquid region and the solid region. It is also found that the free boundary propagates faster when the authors increase the parameter α1 or decrease the parameter α2. Originality/value From the literature, it is seen that most of the two-phase problems with free boundary in an infinite domain are considered by the authors with constant thermophysical properties. Because it is possible to establish an analytical solution of two-phase problems with free boundary in case of an infinite domain. Moreover, a two-phase problem in a finite domain involving moving phase change material with the unidirectional speed is not considered. Therefore, the authors have considered a two-phase free boundary problem with variable thermal coefficients.

A two-phase free boundary problem associated with a diffusion–convection equation is considered. The problem is reduced to a system of nonlinear integral equations, which admits a unique solution for small times. The system admits an explicit two-component solution corresponding to a two-component shock wave of the Burgers equation. The stability of such a solution is also discussed.

We study the existence and structure of branch points in two-phase free boundary problems. More precisely, we construct a family of minimizers to an Alt–Caffarelli–Friedman-type functional whose free boundaries contain branch points in the strict interior of the domain. We also give an example showing that branch points in the free boundary of almost-minimizers of the same functional can have very little structure. This last example stands in contrast with recent results of De Philippis, Spolaor and Velichkov on the structure of branch points in the free boundary of stationary solutions.

We provide an overview of some recent results about the regularity of the solution and the free boundary for so-called two-phase free boundary problems driven by uniformly elliptic equations.

Determination of one unknown coefficient in a two-phase free boundary problem in an angular domain with variable thermal conductivity and specific heat

In this paper we are interested in the study of a two-phase problem equipped with the $\Phi$-Laplacian operator $$ \Delta_\Phi u \coloneqq \mbox{div} \bigg(\varphi(|\nabla u|) \dfrac{\nabla u}{|\nabla u|}\bigg), $$% where $\Phi(s)=e^{s^2}-1$ and $\varphi=\Phi'$. We obtain the existence, boundedness, and Log-Lipschitz regularity of the minimizers of the energy functional associated to the two-phase problem. Furthermore, we also prove that the free boundaries of these minimizers have locally finite perimeter and Hausdorff dimension at most $(N-1)$.

Regularity for degenerate two-phase free boundary problems

A two-phase free boundary problem associated with nonlinear heat conduction is considered. The problem is mapped into two one-phase moving boundary problems for the linear heat equation, connected through a constraint on the relative motion of their moving boundaries. Existence and uniqueness of the solution is proved for small times and a particular exact solution is discussed.

Explicit solutions for one-dimensional two-phase free boundary problems with either shrinkage or expansion

A two-phase free boundary problem for coupled system including three elliptic equations is considered. The problem comes from the discussion of a growth model of self-maintaining protocell in multidimensional case. The global classical solution of the problem with free boundary Γ:y=g(x,t) between two domains is under search. The global existence and uniqueness of a quasi-stationary problem are proved in multidimensional case.

The paper is concerned with optimal control problems for systems governed by multidimensional two-phase parabolic free boundary value problems of Stefan type. In the case of nonlinear process equations and nonlinear boundary conditions existence of optimal controls enclosed in these conditions and convergence of approximating optimal control problems including smoothed coefficients are shown.

The field of the mathematical and numerical analysis of systems of nonlinear pdes involving interfaces and free boundaries is a burgeoning area of research. Many such systems arise from mathematical models in ma- terial science and fluid dynamics such as phase separation in alloys, crystal growth, dynamics of multiphase fluids and epitaxial growth. In applications of these mathematical models, suitable performance indices and appropriate control actions have to be specified. Mathematically this leads to optimiza- tion problems with pde constraints including free boundaries. It is now timely to consider such control problems because of the maturity of the field of com- putational free boundary problems. The aim of the mini-workshop was to bring together leading experts and young researchers from the separate fields of numerical free boundary problems and optimal control in order to estab- lish links and to identify suitable model problems to serve as paradigms for progressing knowledge of optimal control of free boundaries.

Classical solutions in the large of a two-phase free boundary problem. II

Inverse problem for a coupling model of reaction-diffusion and ordinary differential equations systems. Application to an epidemiological model

Static and dynamic analysis of two-phase peridynamic one-dimensional systems is thoroughly investigated in this paper for various boundary conditions by exact and numerical approaches. Such a nonlocal study concerns the axial or torsional responses of two-phase peridynamic rods or the behavior of a two-phase peridynamic string. For a normalized exponential kernel, the integro-differential equation of the two-phase peridynamic problem can be converted as a linear higher-order differential equation with constant coefficients. Exact static solutions of fixed–fixed and fixed–free two-phase peridynamic axial rod (TPPAR) under uniform distributed loading and concentrated force are analytically derived. It is shown that the pure peridynamic rod problem can be ill-posed due to a conflict between the natural boundary condition and the peridynamic constitutive boundary condition. The two-phase peridynamic problem may be justified from physical arguments, and is also seen as a regularized peridynamic problem. The exact eigen-frequencies of the two-phase peridynamic rods are also analytically obtained for various boundary conditions, including fixed–free, fixed–fixed or free–free boundary conditions. These theoretical approaches are corroborated by numerical investigations based on Finite Difference Method (FDM), Finite Element Method (FEM), exact FEM and variational iterative methods. Two FDM schemes are presented for the statics and the vibration of two-phase peridynamic rods, with convergence orders of 1 and 2, respectively. Moreover, three different FE procedures are proposed as: (1) FEM for direct solving the integro-differential equations of two-phase peridynamic one-dimensional system, (2) FEM for discretizing the pure differential equations, and (3) Exact FEM to obtain both the exact stiffness matrix and exact dynamic stiffness matrix of two-phase peridynamic one-dimensional system, based on Eisenberger’s methodology. The order of convergence for the first and the second proposed FEM is equal to 2 and 4, according to the type of selected finite elements. Moreover, the third proposed FEM based on the exact shape function of the peridynamic problem gives the exact solution from one single FEM element. In addition, the variational iterative method of Stodola–Vianello is successfully applied to the two-phase peridynamic eigenvalue problems.