Two methods to solve a fractional single phase moving boundary problem

Li, Xicheng; Wang, Shaowei; Zhao, Moli

doi:10.2478/s11534-013-0227-z

Cited by 4 publications

(6 citation statements)

References 17 publications

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“…Fractional moving boundary problems have gained in recent years great interest for the applications and these contributions have a potential significant impact because exact solutions and equivalence of different problems are provided. To have a complete review of the results in this field, as a reference for further studies to know the mathematical results obtained in literature and their applications, see [1], [10], [11], [12], [18], [19]. An interesting physical meaning of the fractional Stefan's problems is discussed in [4].…”

Section: Introductionmentioning

confidence: 99%

A new equivalence of Stefan’s problems for the time fractional diffusion equation

Roscani

Marcus

2014

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A fractional Stefan's problem with a boundary convective condition is solved, where the fractional derivative of order α ∈ (0, 1) is taken in the Caputo sense. Then an equivalence with other two fractional Stefan's problems (the first one with a constant condition on x = 0 and the second with a flux condition) is proved and the convergence to the classical solutions is analyzed when α 1 recovering the heat equation with its respective Stefan's condition.MSC 2010 : Primary 26A33; Secondary 33E12, 35R11, 35R35, 80A22

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Section: Introductionmentioning

confidence: 99%

A new equivalence of Stefan’s problems for the time fractional diffusion equation

Roscani

Marcus

2014

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“…Some advances in this direction were recently given in [13]. Looking for approximate solutions (both numerical and analytical) is an active area of research, [3,6,7,16,17,25,33,38,40].…”

Section: Brief Review On Time-fractional Stefan Problemsmentioning

confidence: 99%

“…Timefractional conservation equations were also considered in [5], where the relation with non-local transport theory with memory effects is discussed. In this way, we provide a new approach to derive some fractional problems that are of interest in pure and applied fields, [2,3,6,7,13,16,17,19,25,33,[38][39][40].…”

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confidence: 99%

A Note on Models for Anomalous Phase-Change Processes

Ceretani

2020

Fract Calc Appl Anal

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We review some fractional free boundary problems that were recently considered for modeling anomalous phase-transitions. All problems are of Stefan type and involve fractional derivatives in time according to Caputo's definition. We survey the assumptions from which they are obtained and observe that the problems are nonequivalent though all of them reduce to a classical Stefan problem when the order of the fractional derivatives is replaced by one. We further show that a simple heuristic approach built upon a fractional version of the energy balance and the classical Fourier's law leads to a natural generalization of the classical Stefan problem in which time derivatives are replaced by fractional ones.

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“…A particular approach of promise, that has a theoretical bases in generalized Brownian motion (Metzler and Klafter, 2000;Schumer et al, 2009), is the use of fractional calculus (Podlubny, 1999;Meerschaert and Sikorskii, 2012); essentially the transient and flux components in the governing transport equations are molded in terms of time and space convolutions representing memory and non-locality, respectively. In this way, a number of authors (Voller, 2010(Voller, , 2012(Voller, , 2014Voller et al, 2013;Liu and Xu, 2009;Li et al, 2013;Atkinson, 2012) have investigated the predictive consequences of replacing the transient and flux terms in standard heat conduction phase change models with fractional derivative counterparts. And indeed, it has been shown that such replacements do lead to anomalous behavior.…”

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confidence: 99%

Computations of anomalous phase change

Voller

2016

International Journal of Numerical Methods for Heat &Amp; Fluid Flow

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Purpose – The purpose of this paper is to demonstrate how anomalous diffusion behaviors can be manifest in physically realizable phase change systems. Design/methodology/approach – In the presence of heterogeneity the exponent in the diffusion time scale can become anomalous, exhibiting values that differ from the expected value of 1/2. Here the author investigates, through directed numerical simulation, the two-dimensional melting of a phase change material (PCM) contained in a pattern of cavities separated by a non-PCM matrix. Under normal circumstances we would expect that the progress of melting F(t) would exhibit the normal diffusion time exponent, i.e., F∼t1/2. The author’s intention is to investigate what features of the PCM cavity pattern might induce anomalous phase change, where the progress of melting has a time exponent different from n=1/2. Findings – When the PCM cavity pattern has an internal length scale, i.e., when there is a sub-domain pattern which, when reproduced, gives us the full domain pattern, the direct simulation recovers the normal ∼t1/2 phase change behavior. When, however, there is no internal length scale, e.g., the pattern is a truncated fractal, an anomalous super diffusive behavior results with melting going as t n; n > 1/2. By studying a range of related fractal patterns, the author is able to relate the observed sub-diffusive exponent to the cavity pattern’s fractal dimension. The author also shows, how the observed behavior can be modeled with a non-local fractional diffusion treatment and how sub-diffusion phase change behavior (F∼t n; n < 1/2) results when the phase change nature of the materials in the cavity and matrix are inverted. Research limitations/implications – Although the results clearly demonstrate under what circumstances anomalous phase change behavior can be practically produced, the question of an exact theoretical relationship between the cavity pattern geometry and the observed anomalous time exponent is not known. Practical implications – The clear role of the influence of heterogeneity on heat flow behavior is illustrated. Suggesting that modeling heat and fluid flow in heterogeneous systems requires careful consideration. Originality/value – The novel direct simulation of melting in a two-dimensional PCM cavity pattern provides a clear illustration of anomalous behavior in a classic heat and fluid flow system and by extension provides motivation to continue the numerical investigation of anomalous and non-local behaviors and fractional calculus tools.

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Two methods to solve a fractional single phase moving boundary problem

Cited by 4 publications

References 17 publications

A new equivalence of Stefan’s problems for the time fractional diffusion equation

A new equivalence of Stefan’s problems for the time fractional diffusion equation

A Note on Models for Anomalous Phase-Change Processes

Computations of anomalous phase change

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