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

Abstract

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

Explicit Solutions to Fractional Stefan-like problems for Caputo and Riemann-Liouville Derivatives

“…It is worth noting that this kind of problems are not deduced as in [17,27]. This paper is a continuation of a previous work [20], related to fractional one-phase change problems. In Section 2 some basic definitions and properties on fractional calculus are given.…”

“…From [23] we know that f 1 is a strictely decreassing function in R + . Taking a close interval [a, b] ⊂ R + such that η 1 ∈ [a, b], using the uniform convergence over compact sets of all the positive functions given in Proposition 6 and proceding like in [20,Theorem 2] we can state that lim α 1 η α = η 1 .…”

A self‐similar solution to time‐fractional Stefan problem

“…This result encouraged the researchers to investigate the time-fractional Stefan model more deeply. In papers, [6][7][8][9][10] the authors discussed other possible formulations of time-fractional Stefan problem and compare the formulas for special solutions. In paper, 11 there is shown that the time-fractional sharp-interphase model obtained in Falcini et al 1 is not a consequence of the assumption (1).…”

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

“…Different models are presented in [7], [18] and [26]. A rigorous existence analysis of self-similar solutions was done in [9], and results related to explicit solutions were established in [17,20,19] and references therein.…”