A generalized Stefan model accounting for system memory and non-locality

R. Garra, F. Falcini, V. R. Voller, G. Pagnini

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The Stefan problem, involving the tracking of an evolving phase-change front, is the prototypical example of a moving boundary problem. In basic one-dimensional problems it is well known that the front advances as the square root of time. When memory or non-locality are introduced into the system, however, this classic signal may be anomalous; replaced by a power-law advance with a time exponent that differs from n = 1/2. Up to now memory treatments in Stefan problem models have only been able to reproduce sub-diffusive front movements with exponents n < 1/2 and non-local treatments have only been able to reproduce super-diffusive behavior n > 1/2. In the present paper, by using a generalized Caputo fractional derivative operator, we introduce new memory and non-local treatment for Stefan problems. On considering a limit case Stefan problem, related to the melting problem, we are able to show that, this general treatment can not only produce arbitrary power-law in time predictions for the front movement but, in the case of memory treatments, can also produce non-power-law anomalous behaviors. Further, also in the context of the limit problem, we are able to establish an equivalence between non-locality and a space varying conductivity and memory and a time varying conductivity.

Original languageEnglish (US)
Article number104584
JournalInternational Communications in Heat and Mass Transfer
StatePublished - May 2020

Bibliographical note

Funding Information:
This research is supported by the Basque Government through the BERC 2014–2017 and BERC 2018–2021 programs, and by the Spanish Ministry of Economy and Competitiveness (MINECO) through BCAM Severo Ochoa excellence accreditations SEV-2013-0323 and SEV-2017-0718 . The research began and was primarily developed at BCAM - Basque Center for Applied Mathematics, Bilbao , during the visiting fellowship of Roberto Garra in August-October 2017.


  • Anomalous diffusion
  • Fractional moving boundary problems
  • Melting processes
  • Stefan problems


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