The similarity method and explicit solutions for the fractional space one-phase Stefan problems

Roscani, Sabrina; Tarzia, Domingo A.; Lucas, Venturato,

doi:10.1007/s13540-022-00027-1

Cited by 3 publications

(2 citation statements)

References 25 publications

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“…The argument we use is based on the use of the maximum principle for (1.1). This idea was used first in the proof of a similar result in [19]. We provide our own and extended version of the argument.…”

Section: Remarkmentioning

confidence: 99%

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Special solutions to the space fractional diffusion problem

Namba

Rybka

Sato

2022

Fract Calc Appl Anal

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We derive a fundamental solution $${{\mathscr {E}}}$$ E to a space-fractional diffusion problem on the half-line. The equation involves the Caputo derivative. We establish properties of $${{\mathscr {E}}}$$ E as well as formulas for solutions to the Dirichlet and fixed slope problems in terms of convolution of $${{\mathscr {E}}}$$ E with data. We also study integrability of derivatives of solutions given in this way. We present conditions, which are sufficient for uniqueness of solutions. Finally, we show the infinite speed of signal propagation.

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Section: Remarkmentioning

confidence: 99%

“…We could justify the positivity of E on the grounds of the theory of Mittag-Leffler functions, (actually we do this in the Appendix). However, we would like to stress that our proof of positivity of E is entirely based on a PDE tool, which is the maximum principle, we use this idea after [19].…”

Section: Introductionmentioning

confidence: 99%

Special solutions to the space fractional diffusion problem

Namba

Rybka

Sato

2022

Fract Calc Appl Anal

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An alternative two-phase formulation and analytical solution for a Stefan problem in one-dimension accounting for volume change and sensible heat

Huynh,

Nguyen,

Gjerløw

2024

International Journal of Thermofluids

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Fractional Stefan Problem Solving by the Alternating Phase Truncation Method

Chmielowska

Słota

2022

Symmetry

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The aim of this paper is the adaptation of the alternating phase truncation (APT) method for solving the two-phase time-fractional Stefan problem. The aim was to determine the approximate temperature distribution in the domain with the moving boundary between the solid and the liquid phase. The adaptation of the APT method is a kind of method that allows us to consider the enthalpy distribution instead of the temperature distribution in the domain. The method consists of reducing the whole considered domain to liquid phase by adding sufficient heat at each point of the solid and then, after solving the heat equation transformed to the enthalpy form in the obtained region, subtracting the heat that has been added. Next the whole domain is reduced to the solid phase by subtracting the sufficient heat from each point of the liquid. The heat equation is solved in the obtained region and, after that, the heat that had been subtracted is added at the proper points. The steps of the APT method were adapted to solve the equations with the fractional derivatives. The paper includes numerical examples illustrating the application of the described method.

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The similarity method and explicit solutions for the fractional space one-phase Stefan problems

Cited by 3 publications

References 25 publications

Special solutions to the space fractional diffusion problem

Special solutions to the space fractional diffusion problem

An alternative two-phase formulation and analytical solution for a Stefan problem in one-dimension accounting for volume change and sensible heat

Fractional Stefan Problem Solving by the Alternating Phase Truncation Method

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