On the Solutions of a Stefan Problem with Variable Latent Heat

On comparison of this condition with the one-phase Stefan condition, we observe that the latent heat term s is not a constant but, rather, a linear function of position [1]. Problems such as this have been treated under the name of Stefan problems with variable latent heat [1–3]. An analytical solution for this Stefan problem is presented in [1], by introducing the similarity variables ξ = x/√t and seeking the solution in the form u(t, x) = F(ξ) with F being an unknown function. Accordingly, it is natural that the interface location s(t) should be proportional to √t; that is, s(t) = A√t, where A is a constant. The purpose of this paper is to apply the Adomian decomposition method [4–25] to find the solution of (1), (2), and (5), that is, the temperature distribution of the water u(t, x), and then use a direct method to determine the position of the ice-water interface as a function of time. Also, an analytical solution based on a similarity variable is presented in the case when the Dirichlet boundary condition at the water-ice interface depends on time of the form u(t) = α√t, where α is a constant.


Introduction
The solution of the Stefan problem with variable latent heat consists of finding  and the moving melt interface  such that which is the governing equation, subject to the following Neumann boundary condition at the left end of the domain describing the inlet heat flux: the Dirichlet boundary condition at the water-ice interface: the Stefan condition: and the initial temperature distribution: (0, ) =  () , 0 <  <  (0) = .
On comparison of this condition with the one-phase Stefan condition, we observe that the latent heat term  is not a constant but, rather, a linear function of position [1].
Problems such as this have been treated under the name of Stefan problems with variable latent heat [1][2][3].
An analytical solution for this Stefan problem is presented in [1], by introducing the similarity variables  = /√ and seeking the solution in the form (, ) = () with  being an unknown function.Accordingly, it is natural that the interface location () should be proportional to √; that is, () = √, where  is a constant.
The purpose of this paper is to apply the Adomian decomposition method  to find the solution of (1), (2), and (5), that is, the temperature distribution of the water (, ), and then use a direct method to determine the position of the ice-water interface as a function of time.Also, an analytical solution based on a similarity variable is presented in the case when the Dirichlet boundary condition at the water-ice interface depends on time of the form  * () = √, where  is a constant.

The Adomian Decomposition Method
Based on the Adomian decomposition method, we write (1) in Adomian's operator-theoretic notation as where 2

Mathematical Problems in Engineering
We conveniently define the inverse linear operator as Applying the inverse linear operator  −1  to ( 6) and taking into account that (/)(, 0) = −ℎ(), we obtain where the unknown boundary condition () = (, 0) will be determined.Define the solution (, ) by an infinite series of components in the form Consequently, the components   can be elegantly determined by setting the recursion scheme for the complete determination of these components.In view of ( 11), the components  0 (, ),  1 (, ),  2 (, ), . . .are immediately determined as Consequently, the solution is readily found to be obtained by substituting ( 12) into (10).
We remark here that the unknown boundary condition () can be easily determined by using the initial condition equation (5).Substituting  = 0 into (13) and using the Taylor expansion of () lead to Equating the coefficients of like power of  in both sides of ( 14) and taking into account that the compatibility conditions yield Thus Accordingly, the solution equation ( 13) is completely determined by defining the function ().
Once the function (, ) is obtained, we can rewrite the Stefan condition equation ( 4) in terms of the known function (, ) including the Dirichlet and Neumann boundary conditions.For this, integrating (1) with respect to  from 0 to  and taking into account that −(/)(, 0) = ℎ(), we obtain Thus ( 4) can be replaced by Using the following Leibniz rule for differentiation under the integral sign: and taking into account that (, ()) =  * , we obtain Substituting ( 21) into (19), we obtain    13) of the heat equation into (25), we obtain where Thus where We now can determine the shoreline () with time by solving the nonlinear equation (28).In order to demonstrate the feasibility and efficiency of this method, we consider the following case: If we choose () =   , then a simple calculation leads to () =   , and (28) becomes where (, ()) =  + .Let us write () and () in series forms To compute [    ]  we need the following theorem [4].
Using this theorem with the given formula   = 1/!, we see that Consequently, we obtain the recurrence relations for the coefficients where (37)