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Homotopy Perturbation Method forSolving the Two-Phase Inverse StefanProblemDamian Słota a

a Institute of Mathematics, Silesian University of Technology ,Gliwice, PolandPublished online: 31 May 2011.

To cite this article: Damian Słota (2011) Homotopy Perturbation Method for Solving the Two-PhaseInverse Stefan Problem, Numerical Heat Transfer, Part A: Applications: An International Journal ofComputation and Methodology, 59:10, 755-768, DOI: 10.1080/10407782.2011.572763

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HOMOTOPY PERTURBATION METHOD FOR SOLVINGTHE TWO-PHASE INVERSE STEFAN PROBLEM

Damian SłotaInstitute of Mathematics, Silesian University of Technology, Gliwice, Poland

In this article, the possibility of the application of the homotopy perturbation method for

solving the two-phase inverse Stefan problem is presented. This problem consists in the cal-

culation of temperature distribution in the domain, as well as in the reconstruction of the

functions describing the temperature and the heat flux on the boundary when the position

of the moving interface is known. The validity of the approach is verified by comparing

the results obtained with the exact solution.

1. INTRODUCTION

The homotopy perturbation method was developed by He [1–5]. Using thismethod we are able to solve the operator equation.

LðuÞ þNðuÞ ¼ f ðzÞ; z 2 X ð1Þ

where L is the linear operator, N is the nonlinear operator, f is a known function,and u is a sought function. We define a homotopy map H as

Hðv; pÞ � ð1� pÞðLðvÞ � Lðu0ÞÞ þ pðLðuÞ þNðuÞ � f ðzÞÞ ð2Þ

where p2 [0, 1] is a homotopy parameter, v(z, p) :X� [0, 1]!R, and u0 is an initialapproximation of the solution of problem (1). The operator (2) can be written in thefollowing form.

Hðv; pÞ � LðvÞ � Lðu0Þ þ pLðu0Þ þ pðNðvÞ � f ðzÞÞ ð3Þ

As H(v, 0)¼L(v)�L(u0), so for p¼ 0 the solution of the operator equation H(v,0)¼ 0 is equivalent to the solution of a trivial problem L(v)�L(u0)¼ 0. However,for p¼ 1 the solution of the operator equation H(v, 1)¼ 0 is equivalent to the sol-ution of Eq. (1). Thus, a monotonous change of parameter p from zero to one cor-responds to a continuous change of the trivial problem L(v)�L(u0)¼ 0 to theoriginal problem (the same solution of v from u0 to u).

Received 9 September 2010; accepted 18 February 2011.

Address correspondence to Damian Słota, Institute of Mathematics, Silesian University of

Technology, Kaszubska 23, 44-100 Gliwice, Poland. E-mail: damian.slota@polsl.pl

Numerical Heat Transfer, Part A, 59: 755–768, 2011

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407782.2011.572763

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Next, we assume that the solution of equation H(v, p)¼ 0 can be written as apower series in homotopy parameter p.

v ¼X1j¼0

pjuj ð4Þ

Setting p¼ 1, we obtain the solution of problem (1) (if series is convergent).

u ¼ limp!1

v ¼X1j¼0

uj ð5Þ

The convergence of the method was discussed in references [1, 6]. Series (5) ishigh-rate convergent; thus, the limitation to the sum consisting of several initial com-ponents often provides a very good approximation of the sought solution. If weimpose the limitation to the first nþ 1 components, we obtain the so-called n-orderapproximate solution.

uun ¼Xnj¼0

uj ð6Þ

To find function uj, we substitute the relation (4) to equation H(v, p)¼ 0 and com-pare the terms of the same powers of parameter p. Thus, we obtain a series of oper-ator equations that make it possible to designate the successive functions uj.Accordingly, the solution of the initial problem was reduced to the solution of asequence of problems, which are easy to solve.

The homotopy perturbation method is efficient and effective for solving a wideclass of problems. He [7] applied the homotopy perturbation method to solve theboundary value problem (see also references [8–10]). The application of the homo-topy perturbation method to fractional partial differential equations in fluid mech-anics has been described by Yildirim [11]. For the time-fractional diffusionequation with a moving boundary condition, this method is applied in reference[12]. It is also applied for nonlinear wave equations [13] and nonlinear diffusion

NOMENCLATURE

Di domains of the problem

d length of domain

f function

H homotopy map

k thermal conductivity

L linear operator

N nonlinear operator

p homotopy parameter

q heat flux on the boundary

t time

t� end of process

u temperature

u� phase change temperature

x space variable

a thermal diffusivity

Ci boundary of domain

n phase change moving interface

d relative error

D absolute error

h temperature on the boundary

j latent heat of fusion per unit volume

u initial temperature

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equations [14]. Shakeri and Dehghan [15] applied the homotopy perturbationmethod for the inverse problem of the diffusion equation. Biazar and Ghazvini[16] used it for solving the hyperbolic partial differential equation. Numerous pub-lications [17–23] have focused on applying the homotopy perturbation method tosolve different problems related to heat transfer processes. For systems of partial dif-ferential equations, this method has been applied in reference [24].

The Stefan problem denotes mathematical models describing the thermal pro-cesses with phase change [25, 26]. Examples of such processes include: solidificationof metals, crystal growth, melting of ice, freezing of the ground and water, etc. TheStefan problem involves the designation of the temperature distribution in thedomain and the position of the phase change moving interface (freezing front). Inthis article, we solve the two-phase inverse Stefan problem, which consists in the cal-culation of temperature distribution in the domain, as well as in the reconstruction ofthe functions describing the temperature and the heat flux on the boundary when theposition of the moving interface is known. The proposed solution is based on thehomotopy perturbation method.

It is possible to find an exact analytical solution of the inverse Stefan problemin only a few simple cases [27]. In other cases, we are only left with approximate solu-tions [27–33]. In reference [27], the application of the homotopy perturbationmethod for solving the one-phase inverse Stefan problem was presented. The useof the Adomian decomposition method and variational iteration method for solvingthe same problem was discussed in references [34, 35]. However, in reference [29], theauthors compared some numerical methods to solve a one-phase inverse Stefanproblem.

Description of other methods finding applications in solving the various heatconduction problems with or without the change of phase can be found in references[36–44].

2. STATEMENT OF THE PROBLEM

Let us assume two domains (see Figure 1).

D1 ¼ fðx; tÞ; x 2 ½0; nðtÞ�; t 2 ½0; t��gD2 ¼ fðx; tÞ; x 2 ½nðtÞ; d�; t 2 ½0; t��g

and their boundaries,

C1 ¼ fðx; 0Þ; x 2 ð0; sÞ; s ¼ nð0ÞgC2 ¼ fðx; 0Þ; x 2 ðs; dÞ; s ¼ nð0ÞgC3 ¼ fð0; tÞ; t 2 ½0; t��gC4 ¼ fðd; tÞ; t 2 ½0; t��gC5 ¼ fðx; tÞ; t 2 ½0; t��; x ¼ nðtÞg

with an unknown function x¼ n(t).The two-phase Stefan problem [25, 26] consists in the determination of the

location of a phase change moving interface described by means of functionx¼ n(t), and in the determination of functions u1(x, t) and u2(x, t), defined in

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domains D1 and D2, respectively, which fulfill the following heat conductionequations.

qu1ðx; tÞqt

¼ a1q2u1ðx; tÞ

qx2in D1 ð7Þ

qu2ðx; tÞqt

¼ a2q2u2ðx; tÞ

qx2in D2 ð8Þ

where ai are the thermal diffusivity in the liquid phase (i¼ 1) and the solid phase(i¼ 2), and t and x refer to time and spatial location, respectively. On boundariesC1 and C2, they fulfill the initial conditions,

u1ðx; 0Þ ¼ u1ðxÞ on C1 ð9Þ

u2ðx; 0Þ ¼ u2ðxÞ on C2 ð10Þ

nð0Þ ¼ s ð11Þ

On boundary C3, they fulfill the Dirichlet boundary conditions.

u1ðx; tÞ ¼ h1ðtÞ on C3 ð12Þ

On boundary C4, they fulfill the Dirichlet or Neumann boundary conditions.

u2ðx; tÞ ¼ hðtÞ on C4 ð13Þ

�k2qu2ðx; tÞ

qx¼ qðtÞ on C4 ð14Þ

where ki are the thermal conductivity in the liquid phase (i¼ 1) and the solidphase (i¼ 2). On the phase change moving interface (C5), they fulfill the condition

Figure 1. Domain of the problem.

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of temperature continuity and the Stefan condition.

u1ðnðtÞ; tÞ ¼ u2ðnðtÞ; tÞ ¼ u� ð15Þ

jdnðtÞdt

¼ k2qu2ðx; tÞ

qx

����x¼nðtÞ

� k1qu1ðx; tÞ

qx

����x¼nðtÞ

ð16Þ

where u� is the phase change temperature and j is the latent heat of fusion per unitvolume.

The discussed inverse Stefan problem consists in finding the functions u1(x, t)and u2(x, t) describing the temperature distribution in domains D1, and D2, and thefunctions h(t) and q(t) describing the temperature and the heat flux on the boundaryC4, respectively, which should satisfy the heat conduction equations (7) and (8),initial conditions (9) and (10), boundary conditions (12) on the boundary C3, andfurthermore, the conditions (13) and (14) on the boundary C4 and conditions (15)and (16) on the phase change moving interface C5. All other functions (ui(x),h1(t), n(t)), and parameters (ai; ki; j; u�; s) are known.

3. SOLUTION OF THE PROBLEM

Let us construct the homotopy map for Eqs. (7) and (8), as follows (i¼ 1, 2).

Hiðvi; pÞ �q2viqx2

� q2ui;0qx2

þ pq2ui;0qx2

� 1

ai

qviqt

!ð17Þ

The solution to equations (i¼ 1, 2),

Hiðvi; pÞ ¼ 0 ð18Þ

are sought in the form of a power series in p (i¼ 1, 2).

vi ¼X1j¼0

pjui;j ð19Þ

By substituting dependence (19) into Eq. (18), we obtain (i¼ 1, 2)

X1j¼0

pjq2ui;jqx2

¼ q2ui;0qx2

� pq2ui;0qx2

þ 1

ai

X1j¼1

pjqui;j�1

qtð20Þ

Next, by comparing the above for the same powers of parameter p we derive

q2ui;1qx2

¼ 1

ai

qui;0qt

� q2ui;0qx2

ð21Þ

q2ui;jqx2

¼ 1

ai

qui;j�1

qtfor j � 2 ð22Þ

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The systems of partial differential equations (21) and (22) must be supplemented bythe boundary conditions securing their unique solution. So, for system (21) we set thefollowing conditions.

u1;0ð0; tÞ þ u1;1ð0; tÞ ¼ h1ðtÞ ð23Þ

u1;0ðnðtÞ; tÞ þ u1;1ðnðtÞ; tÞ ¼ u� ð24Þ

u2;0ðnðtÞ; tÞ þ u2;1ðnðtÞ; tÞ ¼ u� ð25Þ

k2qqx

ðu2;0ðx; tÞ þ u2;1ðx; tÞÞx¼nðtÞ � k1qqx

ðu1;0ðx; tÞ þ u1;1ðx; tÞÞx¼nðtÞ ¼ jdnðtÞdt

ð26Þ

Whereas, for successive systems (22), we set the conditions in the following form(j� 2).

u1;jð0; tÞ ¼ 0 ð27Þ

u1;jðnðtÞ; tÞ ¼ 0 ð28Þ

u2;jðnðtÞ; tÞ ¼ 0 ð29Þ

k2qqx

ðu2;jðx; tÞÞx¼nðtÞ � k1qqx

ðu1;jðx; tÞÞx¼nðtÞ ¼ 0 ð30Þ

The above conditions have been selected in such a way that the n-order approximatesolution

uu1;n ¼Xnj¼0

u1;j and uu2;n ¼Xnj¼0

u1;j ð31Þ

for n� 1 should fulfill the boundary condition (12) on the boundary C3, the con-dition of temperature continuity (15), and the Stefan condition (16) on the phasechange moving interface C5.

In this way, the solution of the initial problem is reduced to a sequence of easilysolved partial differential equations. While looking for the solution of the formu-lated problem by the homotopy perturbation method, we must determine the initialapproximation ui,0 (i¼ 1, 2), and, next, functions ui,1 will be found in the course ofsolving the system of equations (21) with conditions (23)–(26). Conversely, functionsui,j, j¼ 2, 3,. . ., which are the components of the series in Eq. (19), are determined bythe recurrent solution of the system of equations (22) with conditions (27)–(30).

4. EXAMPLE

The theoretical considerations introduced in the previous sections will be illu-strated with an example, where the approximate solution will be compared with anexact solution. We consider an example of the inverse Stefan problem, in which:

d ¼ 3; a1 ¼ 52 ; a2 ¼ 5

4 ; k1 ¼ 6; k2 ¼ 2; j ¼ 45 ; u

� ¼ 1; t� ¼ 5; s ¼ 32, and

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u1ðxÞ ¼ e3�2x10 u2ðxÞ ¼ e

3�2x5 h1ðtÞ ¼ e

tþ310 nðtÞ ¼ tþ 3

2

The exact solution of such a formulated inverse Stefan problem is determined by thefunctions (what can be easily verified by calculating the partial derivatives of func-tions u1, u2, and by substituting them into Eqs. (7), (8), and conditions (9)–(16)).

u1ðx; tÞ ¼ et�2xþ3

10 ð32Þ

u2ðx; tÞ ¼ et�2xþ3

5 ð33Þ

hðtÞ ¼ et�35 ð34Þ

qðtÞ ¼ 4

5et�35 ð35Þ

At the initial approximations, we assume the functions fulfilling the initial conditions(9) and (10)

u1;0ðx; tÞ ¼ e3�2x10 ð36Þ

u2;0ðx; tÞ ¼ e3�2x5 ð37Þ

By substituting the functions u1,0 and u2,0 into Eq. (21) and conditions (23)–(26) we receive the following system of equations.

q2u1;1ðx; tÞqx2

¼ � 1

25e3�2x10 ð38Þ

q2u2;1ðx; tÞqx2

¼ � 4

25e3�2x5 ð39Þ

with conditions

u1;1ð0; tÞ ¼ e310ðe t

10 � 1Þ ð40Þ

u1;1ðx; tÞjx¼tþ32¼ 1� e�

t10 ð41Þ

u2;1ðx; tÞjx¼tþ32¼ 1� e�

t5 ð42Þ

2qu2;1ðx; tÞ

qx x¼tþ32� 6

qu1;1ðx; tÞqx

��������x¼tþ3

2

¼ 2

5ð1þ 2e�

t5 � 3e�

t10Þ ð43Þ

which should be solved. Solution of this system is determined by the followingfunctions.

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u1;1ðx; tÞ ¼ �e310�x

5 þ e3þt10 ð3þ t� 2xÞ

3þ tþ 2x

3þ tð44Þ

u2;1ðx; tÞ ¼30e

310þ t

10 � t� 23

10� ð30e 3

10þ t10 � t� 33Þx5ð3þ tÞ � e

35�2x

5 ð45Þ

In the next step, functions ui,j(x, t), i¼ 1, 2, j� 2 are recurrently designated by solvingsystem (22),

q2u1;jqx2

¼ 2

5

qu1;j�1

qtð46Þ

q2u2;jqx2

¼ 4

5

qu2;j�1

qtð47Þ

with conditions (27)–(28).

u1;jð0; tÞ ¼ 0 ð48Þ

u1;jðx; tÞjx¼tþ32¼ 0 ð49Þ

u2;jðx; tÞjx¼tþ32¼ 0 ð50Þ

2qqx

ðu2;jðx; tÞÞx¼tþ32� 6

qqx

ðu1;jðx; tÞÞx¼tþ32¼ 0 ð51Þ

In particular, for j¼ 1 we obtain the system of the form,

q2u1;1ðx; tÞqx2

¼ e3þt10 ðt2 � 2tðx� 3Þ þ 14xþ 9Þ � 20x

25ð3þ tÞ2ð52Þ

q2u2;1ðx; tÞqx2

¼ 2ð3e3þt10 ðt2 � 2tðx� 3Þ þ 14xþ 9Þ � t2 � 6t� 60x� 9Þ

25ð3þ tÞ2ð53Þ

with the given above conditions (48)–(49), which should be solved. The solution ofthis system is determined by the functions

u1;2ðx; tÞ ¼ð2x� 3� tÞx150ð3þ tÞ2

ðe3þt10 ð24þ tð11þ t� xÞ þ 7xÞ

� 5ð3þ tþ 2xÞÞð54Þ

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u2;2ðx; tÞ ¼e3þt10

200ð3þ tÞ2ðð3þ tÞ3ð23þ tÞ � 10ð3þ tÞ2ð11þ tÞx

þ 24ð3þ tÞ2x2 � 16ð�7þ tÞx3Þ

þ 1

1004ð13þ tÞx� 4x2 � ð3þ tÞð13þ tÞ � 80x3

ð3þ tÞ2

!ð55Þ

Table 1 compiles the errors in the reconstruction of the functions describing theboundary conditions, whereas Table 2 compiles the errors in the reconstruction ofthe temperature distribution in the domain. As revealed by the above results,together with an increase of the number of components in sum (6), the errors quicklydecrease. Thus, the fifth-order approximate solution provides the approximation ofthe sought functions with the error not higher than 0.3%. Whereas, the eight-orderapproximate solution gives the error not higher than 0.09%. The error distributionsin the reconstruction of the boundary conditions are shown in Figures 2 and 3, wherethird-order and eight-order approximate solutions are presented.

The calculations were made for an accurate moving interface position, and fora position disturbed with an error with a size of 0.1%, 0.5%, 1.5%, and 3.0%. Table 3

Table 2. Values of errors in the reconstruction of the temperature distribution in domains D1 and D2

n Du1 du1 Du2 du2

1 0.051996 3.7358% 0.124086 28.5324%

2 0.010807 0.7764% 0.027865 6.4072%

3 0.003095 0.2223% 0.004007 0.9213%

4 0.001168 0.0839% 0.002564 0.5897%

5 0.000551 0.0396% 0.001292 0.2972%

6 0.000312 0.0224% 0.000747 0.1718%

7 0.000203 0.0146% 0.000500 0.1150%

8 0.000152 0.0109% 0.000380 0.0875%

D – absolute error, d – relative error.

Table 1. Values of errors in the reconstruction of the heat flux and temperature distribution on the

boundary C4

n Dq dq Dh dh

1 0.460733 58.7128% 0.160966 16.4100%

2 0.129175 16.4612% 0.038732 3.9486%

3 0.027348 3.4851% 0.008350 0.8513%

4 0.005715 0.7283% 0.002401 0.2448%

5 0.001969 0.2509% 0.001091 0.1112%

6 0.000988 0.1259% 0.000622 0.0634%

7 0.000612 0.0780% 0.000415 0.0423%

8 0.000442 0.0563% 0.000315 0.0321%

D – absolute error, d – relative error.

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presents the values of the absolute error and a relative error with which the heat fluxand temperature distribution were reconstructed for different perturbations and forthe eight-order approximate solution. Figures 4 and 5 present error distributions inthe reconstruction of the boundary conditions for perturbation equal to 1.5% and3.0%, and for the eight-order approximate solution. As shown in the results, the pre-sented algorithm is stable in terms of the input data errors. Each time when the inputdata were burdened with errors, the error of the boundary conditions reconstructiondid not exceed the error of the input data.

Figure 3. Error in the reconstruction of temperature on the boundary C4 for (a) n¼ 3 and for (b) n¼ 8.

Figure 2. Error in the reconstruction of the heat flux on the boundary C4 for (a) n¼ 3 and for (b) n¼ 8.

Table 3. Values of errors in the reconstruction of the heat flux and temperature distribution on the

boundary C4 for perturbed input data

Percent Dq dq Dh dh

0.1% 0.000349 0.0445% 0.000771 0.0786%

0.5% 0.003640 0.4639% 0.000929 0.0947%

1.5% 0.005831 0.7431% 0.009644 0.9832%

3.0% 0.011140 1.4195% 0.020161 2.0554%

Results for eight-order approximate solution: D – absolute error, d – relative error.

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5. CONCLUSION

In this article, the solution of the two-phase inverse Stefan problem is pre-sented. This problem consists in the calculation of temperature distribution in thedomain, as well as in the reconstruction of the functions describing the temperatureand the heat flux on the boundary when the position of the moving interface isknown. The proposed solution is based on the homotopy perturbation method.The calculations show that this method is effective for solving the problems underconsideration. Approximate solutions are convergent to the exact solution, andthe errors of approximation are small. The algorithm is stable in terms of the inputdata errors.

After the application of the homotopy perturbation method, we obtain the ser-ies convergent to the exact solution of the problem. In many cases, we are capable ofdesignating, in an analytical manner, the sum of the derived series, and consequently,the exact solution of the considered problem. As indicated by the example, if it isimpossible to predict a general form of function ui,j or the calculated sum of series(5), it is sufficient to make use of the sum of several first functions ui,j to provide avery good approximation of the solution.

Figure 4. Error in the reconstruction of the (a) heat flux and (b) temperature on the boundary C4 for

perturbation equal to 1.5%. (Results for eight-order approximate solution.)

Figure 5. Error in the reconstruction of the (a) heat flux and (b) temperature on the boundary C4 for

perturbation equal to 3.0%. (Results for eight-order approximate solution.)

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The applied method does not require the discretization of the region, as in thecase of classical methods based on the finite-difference method or the finite-elementmethod. The proposed method produces a wholly satisfactory result already in asmall number of iterations, whereas classical methods require a suitably dense latticein order to achieve similar accuracy, which considerably extends the time ofcalculations.

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3. J.-H. He, Homotopy Perturbation Method for Bifurcation of Nonlinear Problems, Int. J.Nonlin. Sci. Numer. Simulat., vol. 6, pp. 207–208, 2005.

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