Perturbative estimates for the one-phase Stefan problem

AbstractThe classical Stefan problem is one of the most studied free boundary problems of evolution type. Recently, there has been interest in treating the corresponding free boundary problem with nonlocal diffusion. We start the paper by reviewing the main properties of the classical problem that are of interest to us. Then we introduce the fractional Stefan problem and develop the basic theory. After that we center our attention on selfsimilar solutions, their properties and consequences. We first discuss the results of the one-phase fractional Stefan problem, which have recently been studied by the authors. Finally, we address the theory of the two-phase fractional Stefan problem, which contains the main original contributions of this paper. Rigorous numerical studies support our results and claims.

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On the one-phase reduction of the Stefan problem with a variable phase change temperature

International Communications in Heat and Mass Transfer ◽

10.1016/j.icheatmasstransfer.2014.11.008 ◽

2015 ◽

Vol 61 ◽

pp. 37-41 ◽

Cited By ~ 10

Author(s):

T.G. Myers ◽

F. Font

Keyword(s):

Phase Change ◽

Stefan Problem ◽

Variable Phase ◽

Phase Change Temperature ◽

Phase Reduction ◽

Change Temperature ◽

The One

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Global Lipschitz continuity of free boundaries in the one-phase Stefan problem

Annali di Matematica Pura ed Applicata (1923 -) ◽

10.1007/bf01766594 ◽

1985 ◽

Vol 142 (1) ◽

pp. 197-213 ◽

Cited By ~ 1

Author(s):

Chen Ya-zhe

Keyword(s):

Stefan Problem ◽

Lipschitz Continuity ◽

Free Boundaries ◽

The One

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An integral relationship for a fractional one-phase Stefan problem

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2018-0049 ◽

2018 ◽

Vol 21 (4) ◽

pp. 901-918 ◽

Cited By ~ 3

Author(s):

Sabrina Roscani ◽

Domingo Tarzia

Keyword(s):

Boundary Condition ◽

Stefan Problem ◽

One Dimensional ◽

Similarity Type ◽

Integral Relationship ◽

Temperature Boundary ◽

Liouville Derivative ◽

The One ◽

Stefan Condition ◽

Temperature Boundary Condition

Abstract A one-dimensional fractional one-phase Stefan problem with a temperature boundary condition at the fixed face is considered by using the Riemann–Liouville derivative. This formulation is more convenient than the one given in Roscani and Santillan (Fract. Calc. Appl. Anal., 16, No 4 (2013), 802–815) and Tarzia and Ceretani (Fract. Calc. Appl. Anal., 20, No 2 (2017), 399–421), because it allows us to work with Green’s identities (which does not apply when Caputo derivatives are considered). As a main result, an integral relationship between the temperature and the free boundary is obtained which is equivalent to the fractional Stefan condition. Moreover, an exact solution of similarity type expressed in terms of Wright functions is also given.

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Oscillatory instabilities and dynamics of multi-spike patterns for the one-dimensional Gray-Scott model

European Journal of Applied Mathematics ◽

10.1017/s0956792508007766 ◽

2009 ◽

Vol 20 (2) ◽

pp. 187-214 ◽

Cited By ~ 31

Author(s):

WAN CHEN ◽

MICHAEL J. WARD

Keyword(s):

Hopf Bifurcation ◽

Stefan Problem ◽

Differential Algebraic Equations ◽

Finite Domain ◽

Quasi Equilibrium ◽

Algebraic Equations ◽

Source Terms ◽

One Dimensional ◽

The One ◽

Oscillatory Instabilities

The dynamics and oscillatory instabilities of multi-spike solutions to the one-dimensional Gray-Scott reaction–diffusion system on a finite domain are studied in a particular parameter regime. In this parameter regime, a formal singular perturbation method is used to derive a novel ODE–PDE Stefan problem, which determines the dynamics of a collection of spikes for a multi-spike pattern. This Stefan problem has moving Dirac source terms concentrated at the spike locations. For a certain subrange of the parameters, this Stefan problem is quasi-steady and an explicit set of differential-algebraic equations characterizing the spike dynamics is derived. By analysing a nonlocal eigenvalue problem, it is found that this multi-spike quasi-equilibrium solution can undergo a Hopf bifurcation leading to oscillations in the spike amplitudes on an O(1) time scale. In another subrange of the parameters, the spike motion is not quasi-steady and the full Stefan problem is solved numerically by using an appropriate discretization of the Dirac source terms. These numerical computations, together with a linearization of the Stefan problem, show that the spike layers can undergo a drift instability arising from a Hopf bifurcation. This instability leads to a time-dependent oscillatory behaviour in the spike locations.

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