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

Abstract

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.

A variant of Adams — Bashforth — Moulton method to solve fractional ordinary differential equation

“…The developed method, applied here for equation of fractional order in (0, 1), will work for differential equations of arbitrary fractional order as they can also be transformed into equivalent integral ones and the numerical integration method described here works for arbitrary order. We note that lately similar approach was applied for partial fractional differential equations [6]. Namely, fractional Stefan problem was solved numerically by transforming the subdiffusion equation into an equivalent integral one.…”

“…The developed method overcomes this difficulty and can be successfully applied in all cases when the fractional differential equation is converted into an equivalent fractional integral one. The future applications will include sequential fractional differential equations considered in [3]- [5] and fractional Stefan problem investigated in [6], [7].…”

A Note on Models for Anomalous Phase-Change Processes

“…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].…”

“…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].…”

Modeling of Heat Distribution in Porous Aluminum Using Fractional Differential Equation

“…Various phenomena in nature can be modeled using fractional derivatives [8][9][10][11][12][13][14][15][16]-for example, in [9,11], the authors surveyed fractional-order electric circuit models, Reference [12] shows applications of fractional derivatives in control theory, and, in [13,14,16,17], we can find information about application fractional derivatives in heat conduction problems. In [16], the authors present an algorithm to solve the fractional heat conduction equation.…”