Document Type : Full Lenght Research Article
Authors
^{1} Indian Institute of Technology(BHU)
^{2} IIT (BHU), Varanasi
^{3} IIT (BHU), VARANASI
Abstract
Keywords
Main Subjects
History: Submitted 2 June 2015 Revised 21 July 2015 Accepted 27 July 2016
Keywords:
Optimal homotopy asymptotic method Stefan problem moving interface fractional derivatives 
A B S T R A C T This paper presents a fractional mathematical model of a onedimensional phasechange problem (Stefan problem) with a position’s latentheat power function. This model includes space–time fractional derivatives in the Caputo sense and timedependent surfaceheat flux. An approximate solution of this model is obtained using the optimal homotopy asymptotic method to find approximate solutions of temperature distribution in the domain and using the interface’s tracking or location. The results thus obtained are compared with existing exact solutions for the case of the integer order derivative at some particular values of the governing parameters. A study also is performed of the interface movement’s dependence on certain parameters.
© 2016 Published by Semnan University Press. All rights reserved.

A Study of a Stefan Problem Governed With Space–Time Fractional Derivatives
Rajeev^{*}, M.S. Kushwaha, A.K. Singh
Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi221005, India


Journal of Heat and Mass Transfer Research
Journal homepage: http://jhmtr.journals.semnan.ac.ir 

Corresponding Author: Rajeev, Department of Mathematical Sciences,Indian Institute of Technology (Banaras Hindu University),Varanasi221005, India Email: rajeevbhu.mac@gmail.com

The OHAM was developed by Marinca et al. [18], and it has been applied to solve a wide range of nonlinear differential equations [1923]. Ghoreishi et. al. [24] presented the comparison between the homotopy analysis method and the OHAM for nonlinear agestructured population models. In 2013, Dinarvand and Hosseini [25] also used the OHAM to investigate the temperature distribution equation in a convective straight fin with temperaturedependent 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 onedimensional Stefan problem with a position’s latentheat 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 nonnegative 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 [1314] 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 zerothorder problem (28) are calculated as the following:
, (30)
and
, (31)
Where , .
Substituting and into the firstorder 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.
b 
t 
Exact value of S(t) 
Approximate value of S(t) by OHAM 
Error (%) 
0.1 
0 5 10 15 20 25 
0.0 0.4428497 0.6262841 0.7668507 0.8856994 0.9902421 
0.0 0.4449844 0.6293030 0.7707735 0.8899689 0.9950155 
0.0 0.4820 0.4820 0.5112 0.4820 0.4820 
0.25 
0 1 2 3 4 5 
0.0 0.4728215 0.6686706 0.8189509 0.9456430 1.0030059 
0.0 0.4847701 0.6854703 0.8395262 0.9694014 1.0282054 
0.0 2.5271 2.5124 2.5124 2.5124 2.5124 
0.5 
0 0.25 0.50 0.75 1.00 1.30 
0.0 0.4193648 0.5930714 0.7263611 0.8387296 0.9562989 
0.0 0.4439988 0.6279091 0.7690284 0.8879975 1.0124729 
0.0 5.8741 5.8741 5.8741 5.8741 5.8741 
Table 2. Comparison between exact and approximate solution of at n = 1.
b 
t 
Exact value of S(t) 
Approximate value of S(t) by OHAM 
Error (%) 
0.1 
0 1 2 3 4 5 
0.0 0.4275253 0.6046120 0.7404955 0.8550505 0.9559756 
0.0 0.4332152 0.6126588 0.7503507 0.8664304 0.9686986 
0.0 1.3309 1.3309 1.3309 1.3309 1.3308 
0.25 
0 0.5 1.0 1.5 2.0 2.5 
0.0 0.4527556 0.6402931 0.7841957 0.9055112 1.0123923 
0.0 0.4719994 0.6675079 0.8175269 0.9439988 1.0554227 
0.0 4.2504 4.2504 4.2504 4.2504 4.2504 
0.5 
0.0 0.25 0.5 0.75 1.0 1.25 
0.0 0.4222513 0.5971534 0.7313607 0.8445026 0.9441826 
0.0 0.4536318 0.6415322 0.7857133 0.9072635 1.0143515 
0.0 7.4317 7.4317 7.4317 7.4317 7.4317 
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.
v = 1.0 v = 0.1 v = 0.05 
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.
v = 1.0 v = 0.1 v = 0.05 
Fig. 2 Plot of vs. at and n = 0.
v = 1.0 v = 0.1 v = 0.05 
Fig. 3 Plot of vs. at n = 1.
v = 1.0 v = 0.1 v = 0.05 
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 nonFickian), which is similar to the case of standard motion [12].
b =0.15 b=0.1 b=0.05 
Fig. 5 Plot of vs. at and n = 0.
b =0.15
b=0.10
b=0.05 
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 nonclassical or nonFickian 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 timedependent surfaceheat 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 nonFickian 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.
References
[1] A.S. Chaves, A fractional diffusion equation to describe Lévy flights. Phys. Lett. A, 239, 13–16 (1998).
[2] D.A. Benson, S.W. Wheatcraft, M. M. Meerschaert, The fractionalorder governing equation of Lévy motion, Water Resources Res., 36, 1413–1423 (2000).
[3] Y. Aoki, M. Sen, S. Paolucci, Approximation of transient temperatures in complex geometries using fractional derivatives, Heat Transfer, 44, 771–777 (2008).
[4] H. Jiang, F. Liu, I. Turner, K. Burrage, Analytical solutions for the multiterm timespace CaputoRiesz fractional advectiondiffusion equations on a finite domain, Journal of Mathematical Analysis and Applications, 389, 11171127 (2012).
[5] Zˇ. Tomovskia, T. Sandev, R. Metzler, J. Dubbeldam, Generalized space–time fractional diffusion equation with composite fractional time derivative, Physica A, 391, 2527–42 (2012).
[6] X.C. Li, M. Y. Xu, S.W. Wang, Analytical solutions to the moving boundary problems with time–space fractional derivatives in drug release devices, J Phys A: Math. Theor., 40, 12131–12141 (2007).
[7] X.C. Li, M.Y. Xu, S.W. Wang, Scaleinvariant solutions to partial differential equations of fractional order with a moving boundary condition, J Phys A: Math. Theor., 41, 155202 (2008).
[8] J. Liu, M. Xu, Some exact solutions to Stefan problems with fractional differential equations, J. Math Anal. Appl., 351, 536542 (2009).
[9] C. J. Vogl, M. J. Miksis, S. H. Davis, Moving boundary problems governed by anomalous diffusion.Proc. R. Soc. A,468, 33483369 (2012).
[10] S. Das, R. Kumar, P.K. Gupta, Analytical approximate solution of space–time fractional diffusion equation with a moving boundary condition, Z. Naturforsch. A 66 a, 281–288 (2011).
[11] V.R. Voller, An exact solution of a limit case Stefan problem governed by a fractional diffusion equation, Int. J. Heat Mass Transfer., 53, 56225625 (2010).
[12] Y. Zhou, Y. Wang, W. Bu, Exact solution for a Stefan problem with latent heat a power function of position, International Journal of Heat and Mass Transfer, 69, 451–454 (2014).
[13] L. Xicheng, M. Xu, X. Jiang, Homotopy perturbation method to timefractional diffusion equation with a moving boundary condition, Applied Mathematics and Computation, 208, 434–439 (2009).
[14] Rajeev, M.S. Kushwaha, Homotopy perturbation method for a limit case Stefan problem governed by fractional diffusion equation, Appl. Math Model, 37, 35893599 (2013).
[15] S. Das, Rajeev, Solution of fractional diffusion equation with a moving boundary condition by variational iteration method and Adomian decomposition method, Z Naturforsch . 65a, 793799 (2010).
[16] R. Grzymkowski, D. Słota, Onephase inverse Stefan problem solved by Adomian decomposition method, Comput Math Appl., 51, 3340 (2006).
[17] Rajeev, M. S. Kushwaha, A. Kumar, An approximate solution to a moving boundary problem with space–time fractional derivative in fluviodeltaic sedimentation process, Ain Shams Engineering Journal, 4, 889–895 (2013).
[18] V. Marinca, N. Herisanu, Application of homotopy Asymptotic method for solving nonlinear equations arising in heat transfer, Int. Comm. Heat Mass Transfer, 35 , 710–715 (2008).
[19] N. Herisanu, V. Marinca, Accurate analytical solutions to oscillators with discontinuities and fractionalpower restoring force by means of the optimal homotopy asymptotic method. Comput. Math. Appl. 60, 1607–1615 (2010).
[20] V. Marinca, N. Herisanu, Determination of periodic solutions for the motion of a particle on a rotating parabola by means of the optimal homotopy asymptotic method, J. Sound Vib. 329, 1450–1459 (2010).
[21] S. Iqbal, M. Idrees, A.M. Siddiqui, A.R. Ansari, Some solutions of the linear and nonlinear Klein–Gordon equations using the optimal homotopy asymptotic method, Appl. Math. Comput., 216, 2898–2909 (2010).
[22] S. Iqbal, A. Javed, Application of optimal homotopy asymptotic method for the analytic solution of singular Lane–Emden type equation, Appl. Math. Comput., 217, 7753–7761 (2011).
[23] M. S. Hashmi, N. Khan, S. Iqbal, Optimal homotopy asymptotic method for solving nonlinear Fredholm integral equations of second kind, Applied Mathematics and Computation, 218, 10982–10989 (2012).
[24] M. Ghoreishi , A.I.B. Md. Ismail , A.K. Alomari ,A.S.Bataineh, The comparison between Homotopy Analysis Method and Optimal Homotopy Asymptotic Method for nonlinear agestructured population Models, Commun Nonlinear Sci. Numer. Simulat., 17, 1163–1177 (2012).
[25] S.Dinarvand, R.Hosseini, Optimal homotopy asymptotic method for convective–radiative cooling of a lumped system, and convective straight fin with temperaturedependent thermal conductivity, AfrikaMatematika., 24, 103116 (2013).
