On Computation of Prefactor of Free Boundary in One Dimensional One-Phase Fractional Stefan Problems

Abstract

We consider the melting of a one-dimensional domain (x≥0), initially at the melting temperature u=0, by fixing the boundary temperature to a value u(0,t)=U0>0—the so called Stefan melting problem. The governing transient heat-conduction equation involves a time derivative and the spatial derivative of the temperature gradient. In the general case the order of the time derivative and the gradient can take values in the range (0,1]. In these problems it is known that the advance of the melt front s(t) can be uniquely determined by a specified prefactor multiplying a power of time related to the order of the fractional derivatives in the governing equation. For given fractional orders the value of the prefactor is the unique solution to a transcendental equation formed in terms of special functions. Here, our main purpose is to provide efficient numerical schemes with low computational complexity to compute these prefactors. The values of the prefactors are obtained through a dimensionalization that allows the recovery of the solution for the quasi-stationary case when the Stefan number approaches zero. The mathematical analysis of this convergence is given and provides consistency to the numerical results obtained.