Two equivalent Stefan’s problems for the time fractional diffusion equation

Roscani, Sabrina; Marcus, Eduardo A. Santillan

doi:10.2478/s13540-013-0050-7

Cited by 30 publications

(61 citation statements)

References 10 publications

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“…In Section 5, an inequality for the coefficient which characterizes the free boundary of the two-phase fractional Lamé-Clapeyron-Stefan problem with a fractional temperature boundary condition given recently in [26], is also obtained. In Section 6, we recover the results obtained in [25] for the one-phase fractional Lamé-Clapeyron-Stefan problem as a particular case of the present work (see Sections 3 and 4). order 0 < α < 1 for the solid phase in the first quadrant with an initial constant temperature and a heat flux boundary condition at x = 0:…”

Section: [38]supporting

confidence: 73%

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Explicit solution for a two-phase fractional Stefan problem with a heat flux condition at the fixed face

Roscani

Tarzia

2018

Comp. Appl. Math.

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A generalized Neumann solution for the two-phase fractional Lamé-Clapeyron-Stefan problem for a semi-infinite material with constant initial temperature and a particular heat flux condition at the fixed face is obtained, when a restriction on data is satisfied. The fractional derivative in the Caputo sense of order α ∈ (0, 1) respect on the temporal variable is considered in two governing heat equations and in one of the conditions for the free boundary. Furthermore, we find a relationship between this fractional free boundary problem and another one with a constant temperature condition at the fixed face and based on that fact, we obtain an inequality for the coefficient which characterizes the fractional phasechange interface obtained in Roscani-Tarzia, Adv. Math. Sci. Appl., 24 (2014), 237-249. We also recover the restriction on data and the classical Neumann solution, through the error function, for the classical two-phase Lamé-Clapeyron-Stefan problem for the case α = 1.

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Section: [38]supporting

confidence: 73%

“…Taking into account that Γ(3/2) = √ π 2 , M 1/2 (x) = e −(x/2) 2 and that W −x, − 1 2 , 1 = erfc x 2 (see [25]), it results that…”

Section: [38]mentioning

confidence: 99%

Explicit solution for a two-phase fractional Stefan problem with a heat flux condition at the fixed face

Roscani

Tarzia

2018

Comp. Appl. Math.

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“…This choice is justified by the fact that the location of the moving boundary is described by power function S. / D p ˛=2 as was showed in papers [10,20]. Assuming a priori the correct value of parameter p, transformation, (15) fixes the dissolution front at u D 1 for all .…”

Section: Front Fixing Methodsmentioning

confidence: 99%

“…We integrate both sides of equation (20) applying the left-sided Riemann-Liouville integral of order˛2 .0, 1/, and we get the following equation:…”

Section: Front Fixing Methodsmentioning

confidence: 99%

Numerical solution of the one phase 1D fractional Stefan problem using the front fixing method

Błasik

Klimek

2014

Math Methods in App Sciences

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Communicated by M. KiraneIn this paper, we present a novel numerical scheme to solve a 1D, one-phase extended Stefan problem with fractional Caputo derivative with respect to time. The proposed method is based on a suitable choice of the new space coordinate for the subdiffusion equation and extends the front-fixing method to the subdiffusion case. In the final part, examples of numerical results are discussed.

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“…Some works [5][6][7][8][9][10][11] focused in problems like (3). Let us aboard now the physical approach.…”

Section: Introductionmentioning

confidence: 99%

Two different fractional Stefan problems that are convergent to the same classical Stefan problem

Roscani

Tarzia

2018

Math Methods in App Sciences

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Two fractional Stefan problems are considered by using Riemann-Liouville and Caputo derivatives of order ∈ (0, 1) such that, in the limit case ( = 1), both problems coincide with the same classical Stefan problem. For the one and the other problem, explicit solutions in terms of the Wright functions are presented.We prove that these solutions are different even though they converge, when ↗ 1, to the same classical solution. This result also shows that some limits are not commutative when fractional derivatives are used. KEYWORDSCaputo derivative, explicit solutions, fractional Stefan problem, Riemann-Liouville derivative, Wright functions t 0 t u(x, ) (t − ) d ,6842

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Two equivalent Stefan’s problems for the time fractional diffusion equation

Cited by 30 publications

References 10 publications

Explicit solution for a two-phase fractional Stefan problem with a heat flux condition at the fixed face

Explicit solution for a two-phase fractional Stefan problem with a heat flux condition at the fixed face

Numerical solution of the one phase 1D fractional Stefan problem using the front fixing method

Two different fractional Stefan problems that are convergent to the same classical Stefan problem

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