Fractional Stefan problems

Abstract

The solution of the classical one-dimensional Stefan problem predicts that in time t the melt front goes as s(t)∼t12. In the presence of heterogeneity, however, anomalous behavior can be observed where the time exponent n≠12. In such a case, it may be appropriate to write down the governing equations of the Stefan problem in terms of fractional order time (1⩾β>0) and space (1⩾α>0) derivatives. Here, we present sharp and diffuse interface models of fractional Stefan problems and discuss available analytical solutions. We illustrate that in the fractional time case (β<1), a solution of the diffuse interface model in the sharp interface limit will not coincide with the solution of the sharp interface counterpart; negating a well know result of integer derivative Stefan problems. The paper concludes with the development of an implicit time stepping numerical solution for the diffuse interface fractional Stefan model. Results from this solution are verified with available analytical solutions.