The Solution of One-Phase Inverse Stefan Problem. by Homotopy Analysis Method

Transcription

1 Applied Matheatical Sciences, Vol. 8, 214, no. 53, HIKARI Ltd, The Solution of One-Phase Inverse Stefan Proble by Hootopy Analysis Method O. N. Onyejekwe Eastern Florida State College, FL, USA Copyright 214 O. N. Onyejekwe. This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited. Abstract In this paper we obtain the solution of one-phase inverse Stefan proble by hootopy analysis ethod. The distribution of teperature on the boundary such that u (, = v(, along with the teperature distribution u are obtained. The oving front s ( is given as additional inforation. There are advantages to using the hootopy analysis ethod (HAM), firstly it is independent of sall/large physical paraeter, there is always a guarantee of convergence; there is flexibility on the choice of base function and the initial guess of solution and lastly there is great generality. The nuerical results obtained will show high accuracy and a strong rate of convergence. Keywords: Heat Equation, Hootopy Analysis Method, Inverse Stefan Proble. 1 Introduction The following proble is considered: () t, t, uxx ut =, < x < s > (1.1) ) ( x), < x s( ), (, = v(, t >, u =ϕ < (1.2) u (1.3) * u ( s() t, t ) = u, t >, (1.4) ( s() t, = s' (, t >, u x (1.5)

2 2636 O. N. Onyejekwe where x = s() t is the position of the freezing front, u * is teperature at the oving front, u(x,, x and t are the teperature, spatial location and tie, respectively. We assue that all the paraeters are diensionless. The inverse Stefan proble forulated above will be used in deterining the boundary value u (, of the teperature u(x, in the region < x < s(. Inverse Stefan probles are ill-posed, eaning that soe of the following, existence, uniqueness and continuous dependence on data are absent. Soe regularization ethods are required, and this is discussed by various authors [3], [9]. In recent years, nuerical ethods such as the finite difference ethod and finite eleent ethod [6] and approxiate analytical ethods such as the Adoain decoposition ethod, the hootopy perturbation ethod and the heat integral ethod have been used to solve the inverse Stefan proble [2-4], [7],[1]. We eploy the use of hootopy analysis ethod (HAM) proposed by Liao [11-13] to help us obtain exact and approxiate solutions to the inverse Stefan proble. The hootopy analysis ethod has been used in recent years to solve ordinary and partial differential equations [1] and parabolic inverse probles [8]. Unlike the ethods entioned above, HAM avoids discretization, provides us with efficient nuerical solution with high accuracy; inial calculation and the avoidance of physical unrealistic assuptions. The convergence region for the series solution obtained by HAM is deterined by the convergence-control paraeter h. This article contains the following sections: In section 2 we discuss the ethodology of the hootopy analysis ethod. We look at the solution of v () t in section 3. At the end of section 3, we discuss and analyze the results. 2 Hootopy Analysis Method To illustrate the basic idea of the HAM, we consider the following differential equation: N [ u ] = p, (2.1) Where N is a nonlinear operator, x and t denote independent and dependent variables respectively, u is an unknown function and p(x, is the nonhoogeneous ter. By eans of HAM, we first construct a zeroth-order deforation equation ( 1 q) L[ φ q) u ] = qhn[ φ q) p ], (2.2)

3 Solution of one-phase inverse Stefan proble 2637 where q [,1] is the ebedding paraeter, h is the convergence-control paraeter, L is an auxiliary linear operator, φ ( x, t ; q ) is an unknown factor, u is an initial guess of u(x,. It is obvious that when the ebedding paraeter q goes fro to 1, the values for φ ( x, q ) becoes φ ( x, ) = u, φ 1) = u, (2.3) Respectively. Thus as q increases fro to 1, the solution φ ( x, q) varies fro the initial guess u to the solution u(x,. Expanding φ ( x, q) in Taylor series with respect to q, we obtain φ (, q ) = u t ) + u t ) q, (2.4) where x + = 1 q) 1 φ u = q=, (2.5)! q The convergence of the series (2.4) depends upon the convergence-control paraeter h. With HAM, we have the freedo to choose the initial guess u t ), the auxiliary linear operator L and the nonzero convergence-control paraeter h. We assue that all of the are properly chosen so that: 1. The solution φ ( x, q ) of the zeroth-order deforation equation (2.2) exists for all q [,1]. 2. The hootopy analysis derivative D ( φ ( x, t ; q )) exists for = 1, 2, 3,, The power series (2.4) of φ ( x, q ) converges at q=1. Then fro Eqs. (2.3) and (2.4), we have under these assuptions the solution series u ( x, t ) = u ( x, t ) + u ( x, ), (2.6) + t = 1 which ust be one of the solutions of the original nonlinear equation, as proven by Liao [12]. Define the vectors u = n { u t ), u1 t ),..., u n t )}, (2.7) Differentiating the zeroth-order deforation equation (2.2) ties with respect to q and then dividing the by! and finally setting q = (Taking the th order hootopy derivative). Firstly, since L is a linear operator independent of q, it holds

4 2638 O. N. Onyejekwe D = D (( 1 q ) L[ φ ( x, q ) u ( x, t )]) L[ φ ( x, q ) q φ ( x, q ) + u ( x, t ) q u ( x, t )] ( ), and using Hootopy Properties [13] ( L[ φ q ) q φ q ) + u ( x, t ) q u t )]) D φ q ) D ( q φ q ) + u D ( q )) D = L[ ], and Hootopy Property [13] (2.8) (2.9) L = [ D ( φ ( x, q )) D ( q φ ( x, q )) + u ( x, t ) D ( q )] (2.1) L u ( x, t ) u ( x, t ) + u ( x, t ) D ( q ) [ ], 1 which is L [ x ] when =1, and L [ x x 1 ] when > 1, respectively. Thus, the th order deforation equation becoes where 1 L[ u χ u ] = hr u 1, (2.11) and R u = 1 1! ( ) 1 N [ φ q) g ] 1 1 q= q, (2.12), 1 χ =, (2.13) 1, > 1 It should be ephasized that u t ) for 1 is governed by the linear equation (2.11) with linear boundary conditions that coe fro the original proble. Therefore the solution to the differential equation obtained by HAM is a faily of solutions expressed using the convergence-control paraeter h. 3 Solution of Inverse Stefan Proble In this paper, we use HAM to obtain the approxiate analytical solution to the inverse Stefan Proble by finding both the teperature at the boundary u (, = v( and the teperature u(x, for (1.1)-(1.5).

5 Solution of one-phase inverse Stefan proble 2639 The following other conditions ϕ ( x), s (, * u are known. The existence and uniqueness of (1.1) (1.5) can be found in [5]. Using the solution procedure by HAM, we define a linear operator in the for. Solution procedure by HAM, we define a linear operator in the for [ ( )] q) L ϕ x, q = ϕ, (3.1) t with the property L [ c ( x) ], (3.2) 1 = where c 1 ( x ) is the integration constant. The nonlinear operator is N[ ϕ q) ] = u q) u q), (3.3) t + xx so we can define R as R u 2 u 1 u 1 1 = + (3.4) t x Using (2.11), (2.13) and (3.3), we can get 2 u, x ( x = χ u + h R u dt c ( ), (3.5) the paraeters used during the application of HAM are as follows: the initial guess is u = u), (3.6) the calculations for v () t and u are as follows: u ( x u = u + u + u +..., +, = 1 2 = + + = = = v () t v () t = u (, t ) = v + v + v +..., (3.7) 1 2

6 264 O. N. Onyejekwe Exaple 1. x ϕ ( x ) = e, (3.8) () t t, s = (3.9) u * = 1, (3.1) For which the exact solutions are t x u = e, (3.11) t v () t = e, (3.12) The results obtained for v () t and u(x, for calculated values of h using HAM are listed below Table 1. Results for v ( t h-values Exact v ( HAM Absolute Error

7 Solution of one-phase inverse Stefan proble 2641 Table 2. Results for u(x, when h = , t =.5 x Exact u(x, HAM Absolute Error Exaple ϕ ( x) = exp(1 2 ( 1 + x ) 1, (3.13) 1 2 s () t = 2 ( t ), (3.14) * = u, (3.15) For which the exact solutions are 2 u = exp(1 2 1 ( 1+ x) + 1 2) 1, (3.16) 2 v () t = exp( t 2) 1, (3.17) The results obtained for () t v and u(x, for calculated values of h using HAM are listed below

8 2642 O. N. Onyejekwe Table 3. Results for v ( t h-values Exact v ( HAM Absolute Error Table 4. Results for u(x, when h = , t = 1.5 x Exact u(x, HAM Absolute Error

9 Solution of one-phase inverse Stefan proble Conclusion We have shown that HAM can be used to accurately predict the results for both the teperature at a given boundary v ( and the teperature u(x,. We used the boundary condition defined in (1.4) to obtain the convergence-control paraeter h, instead of the h-curves proposed by Liao [11]. The freedo in choosing h enables us to adjust and control the convergence of the solution series and this differentiates the hootopy analysis ethod fro other existing ethods such as the hootopy perturbation ethod, adoain decoposition ethod and variational iteration ethod. MAPLE was used for the coputation presented in this paper. Acknowledgeents. The author would like to thank the referees for their constructive suggestions and valuable coents. References [1] A.K. Aloari, Modifications of Hootopy Analysis Method for Differential Equations: Modifications of Hootopy Analysis Method, Ordinary, Fractional, Delay, And Algebraic Equations, Labert Acadeic Publishing, Gerany, 212. [2] A.M.Shahrezaee, Solution of Soe Parabolic Inverse Probles by Adoian Decoposition Method, Applied Matheatical Sciences, 5 (211), [3] D.D.Ang, A.Pha Ngoc Dinh, D.N. Thanh, An inverse Stefan Proble: identification of boundary value, Journal of Coputational and Applied Matheatics 66(1996) [4] H.Ren, Application of the heat-balance integral to an inverse Stefan proble, International Journal of Theral Sciences, 46(27) [5] N. L Goldan, Inverse Stefan Proble, Kluwer Dordrecht, [6] N.Zabaras, Y. Ruan, A deforing finite eleent ethod analysis of inverse Stefan probles, International Journal for Nuerical Methods in Engineering, 28(1989)

10 2644 O. N. Onyejekwe [7] N.Zabaras, S. Mukherjee, O. Richond, An analysis of Inverse Heat Transfer Probles with Phase Changes Using an Integral Method, Journal of Heat Transfer,11(1988), [8] O.N.Onyejekwe, Solution of Soe Parabolic Inverse Probles by hootopy analysis ethod, International Journal of Applied Matheatical Research, 3 (214) [9] O.N. Onyejekwe, Nuerical Methods for Solving Inverse Free Boundary Probles, Dissertation, Florida Institute of Technology, ProQuest/UMI, 212, [1] R. Grzykowski, D. Slota, One- Phase Inverse Stefan Proble by Adoian Decoposition, International Journal of Coputer and Matheatics with Applications 51 (26) [11] S. Liao, Hootopy Analysis Method in Nonlinear Equations, Springer, New York, 212. [12] S. Liao, Beyond Perturbation: Introduction to the Hootopy Analysis Method, Chapan & Hall/CRC, 24. [13] S. Liao, Notes on the hootopy analysis ethod Soe definitions and theores, Coon, Nonlinear Sci. Nuer. Siulat, 14(29), Received: March 15, 214