A study of a Stefan problem governed with space–time fractional derivatives

The OHAM was developed by Marinca et al. [18], and it has been applied to solve a wide range of nonlinear differential equations [19-23].  Ghoreishi et. al. [24] presented the comparison between the homotopy analysis method and the OHAM for nonlinear age-structured population models. In 2013, Dinarvand and  Hosseini [25] also used the OHAM to investigate the temperature distribution equation in a convective straight fin with temperature-dependent thermal conductivity and the convective–radiative cooling of a lumped system with variable specific heat.

This paper presents a mathematical model for a Stefan problem [12] with a space–time fractional derivative. In this model, the OHAM is used to find the expression of the temperature distribution in a given domain and location of a moving interface with the help of the Taylor series [13]. The obtained results are compared with the existing exact solution for a standard case and are in good agreement. An approachable analysis for a fractional case also is discussed.

2. Mathematical formulation

In this section, a mathematical model of a one-dimensional Stefan problem with a position’s latent-heat power function [12] is considered. For this problem, we present a fractional model that involves space–time fractional derivatives, as given in [11]. The governing equations are as follow:

              (1)

                             (2)

                                           (3)

,                      (4)

where  is the temperature distribution, is the moving interface, is thermal conductivity, is the thermal diffusion coefficient, b is a constant ( for melting, for freezing),  is the variable latent heat per unit volume, and  is an non-negative integer. The operators  and  are the Caputo fractional derivatives [11,13], which are defined as

   (5)                                            ,                             (6)

where is the Gamma function.

In this paper, the following properties of fractional derivatives [13-14] are used:

(a)                                                     (7)

where is a constant.    

(b)     ,                           (8)

where and  is the Caputo fractional derivative of .            

3. Solution of the problem

First, Eqs. (1)–(4) are written in operator form as follows:                                                          

                           (9)                      ,                                                         (10)

where  is a linear operator,  is a nonlinear operator, and B is a boundary operator.

According to the OHAM [16, 21], we construct an optimal , which satisfies

   (11)

                              (12)

where  is an embedding parameter, is an unknown function, and is a nonzero auxiliary function for  and . Obviously, if ,

                ,                                             (13)

and when , then

.                                                (14)

Clearly, as p increases from 0 to 1, the unknown function  varies from to the solution .

Now, we choose the auxiliary function  in the following form:

,                          (15)

where  are constants to be determined later.

The solution to Eq. (11) is considered in the following series form:

   (16)

and

  ,                                              (17)

where  and .

Now, we expand the nonlinear term  into the following series form (as given in [24]):

 (18)

where .

Now, by substituting Eqs. (16) and (18) into Eq. (11) and equating the coefficients of like powers of , the following problems are obtained:

,                                                (19)

                (20)                                                                           

   (21)

and the general equation for  is given as   

  (22)

where  .

Substituting Eqs. (16) and (17) into the boundary conditions of (6) and (7) provides the following:

              (23)

and

,                                (24)

where .

The comparison of various powers of  can be shown by expending  in Taylor series form [13, 14] at a point, , as follows:

         (25)

where  and .

Eqs. (24) and (25) provide the following:

.           (26)

The interface condition (4) becomes

    (27)

By considering Eq. (19) and comparing the coefficients of the power of  from Eqs. (23), (26), and (27), the following system can be obtained:

           (28)

Taking Eq. (20) and comparing the coefficients of power for  from Eqs. (23), (26), and (27) provides the following:

             (29)

 Similarly, other systems can be found by comparing various powers of .

The solutions of the zeroth-order problem (28) are calculated as the following:

,                 (30)

and 

,                                                          (31)

Where ,       .

Substituting and into the first-order problem (29) and using the above process obtains the following expressions of : 

         (32)

where ,

,

and .

The expression of  can be calculated as:

,                                                         (33)

where ,

and .

The approximate solution of the temperature distribution can be determined as

, (34)

and an approximate solution of  is given as

 .                           (35)        

In order to get the constants involved in Eq. (34) for the expression of , the least square method is used [24]. For this purpose, residual is defined as:  

      (36)

If , then will be the exact solution. Generally, the OHAM gives an approximate solution. Therefore, in such a case, , but the function can be minimized as

                ,                             (37)

where R is the residual. The constants can be obtained optimally from the following conditions:

                                  (38)

4. Numerical results and discussion

In this section, numerical results for interface position  are obtained with the help of Wolfram Research (8.0.0) software by considering only , and the results are presented through tables and figures. 

Table 1. Comparison between exact and approximate              solution of at n = 0.

Table 2. Comparison between exact and approximate solution of at n = 1. 

Tables 1–2 represent comparisons between the exact and approximate values of the phase front ’s positions at particular times  for (standard motion). The tables clearly show that the approximate results are sufficiently accurate and in agreement with the existing exact solution [12] for standard motion.

Fig. 1  Plot of  vs.  at and n = 0

Figs. 1 and 2 represent the dependence of phase front ’s  movement trajectory on the thermal diffusion coefficient for n = 0at  and , respectively.

Fig. 2  Plot of  vs.  at and n = 0.

Fig. 3  Plot of  vs.  at  n = 1.

Fig. 4  Plot of  vs.  at and n = 1.

Figs. 3 and 4 also depict the dependence of phase front ’s path on the thermal diffusion coefficient for n = 1at  and , respectively. Figs. 1–4 portray that the interface’s movement increases with an increase in the value of the thermal diffusion coefficient for fractional cases (nonclassical or non-Fickian), which is similar to the case of standard motion [12].

Fig. 5  Plot of  vs.  at and n = 0.

Fig. 6  Plot of  vs.  at , and n = 0.

Figs. 5–6 show a variation in ’s path for a different value of b for a non-classical or non-Fickian case. From these figures, it is clear that the phase front’s movement increases with an increase in the value of the constant b; that is, the melting (or freezing) process becomes fast as the value of the constant b increases. 

5. Conclusion

In this work, we considered a mathematical model that contains space–time fractional derivatives and time-dependent surface-heat flux. An approximate solution of the model was obtained by the OHAM. It was observed that the interface movement increases with an increase in the value of the thermal diffusion coefficient v as well as the constant b for a nonclassical or non-Fickian case. Moreover, it can be seen that the proposed technique is sufficiently accurate and efficient for solving Stefan problems. It also was observed that it is convenient for controlling and adjusting the convergence of the series solution through the control parameters  in the OHAM.

Acknowledgements

The authors express their sincere thanks to the anonymous referees for their valuable suggestions for the improvement of the paper. 

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