Homotopy Perturbation Method for Solving Partial Differential Equations with Variable Coefficients

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 28, Homotopy Perturbation Method for Solving Partial Differential Equations with Variable Coefficients Lin Jin Modern Educational Technology Center Zhejiang Gongshang University Hangzhou, , P. R. China Abstract In this paper, we present the exact solutions of the partial differential equations in different dimensions with variable coefficients by using the homotopy perturbation method. The feature of this method is its flexibility and ability to solve parabolic-like equations and hyperbolic-like equations without the calculation of complicated Adomian polynomials or unrealistic nonlinear assumptions. The numerical results show that this method is a promising and powerful tool for solving the partial differential equations with variable coefficients. Mathematics Subject Classification: 35K15, 35C05, 65D99, 65M99 Keywords: homotopy perturbation method, parabolic-like equation, hyperbolic-like equation, exact solution 1 Introduction Many physical problems can be described by partial differential equations with variable coefficients in mathematical physics, and other areas of science and engineering. The investigation of exact or approximate solutions to these partial differential equations will help us to understand these physical phenomena better. There are some valuable efforts that focus on solving the partial differential equations arising in engineering and scientific applications [2, 6, 7, 8, 12, 13, 24, 28, 36]. Reviewing these improvements, the Adomian decomposition method [2, 12, 24, 36], the tanh method [6, 28], the extended tanh function method [7, 8] and other methods [13] are proposed to solve the partial differential equations. Among these solution methods, Adomian decomposition method is the most transparent method for solutions of the

2 1396 Lin Jin partial differential equations, however, this method is involved in the calculation of complicated Adomian polynomials which narrows down its applications. To overcome this disadvantage of the Adomian decomposition method, we consider the homotopy perturbation method to solve the partial differential equations with variable coefficients. The homotopy perturbation method proposed by J.H. He [14, 15] has been the subject of extensive studies, and applied to various linear and nonlinear problems [1, 3, 4, 5, 9, 10, 11, 16, 17, 18, 19, 20, 21, 26, 27, 29, 30, 31, 32, 33, 34, 35]. Unlike analytical perturbation methods, the significant feature of this method which doesn t depend on a small parameter is that it provides the exact or approximate solutions to a wide range of nonlinear problems without unrealistic assumptions, linearization, discretization and the computation of the Adomian polynomials. The goal of this work is to extend the homotopy perturbation method to solve the partial differential equations with variable coefficients. Based on the homotopy perturbation method, we present exact solutions of some parabolic-like equations and hyperbolic-like equations. It s easy to see that this method gives the exact solutions without the calculation of Adomian polynomials which is inevitable to solve the partial differential equations with variable coefficients in [36], therefore, this method provides efficient approach to solve the partial differential equations. The rest of this paper is organized as follows. In section 2, we give the analysis of the homotopy perturbation method. The parabolic-like equations and hyperbolic-like equations are introduced in sections 3 and 4. In section 5, we present numerical results to demonstrate the efficiency of the homotopy perturbation method for some partial differential equations with variable coefficients. Finally, we give the conclusion in section 6. 2 Analysis of the homotopy perturbation method To clarify the basic ideas of the homotopy perturbation method [14, 15], we consider the following nonlinear differential equation with boundary conditions A(u) f(r) =0,r Ω, (1) B(u, du )=0,r Γ, (2) dn where A is a general differential operator, B is a boundary operator, u is a known analytical function, and Γ is the boundary of the domain Ω.

3 HPM for solving partial differential equations with variable coefficients 1397 The operator A can be divided into two parts L and N, where L is linear, while N is nonlinear. Therefore (1) can be rewritten as follows L(u)+N(u) f(r) =0. (3) By the homotopy technique proposed by Liao [23], we can construct a homotopy v(r, p) :Ω [0, 1] R which satisfies or H(v, p) = (1 p)[l(v) L(u 0 )] + p[a(v) f(r)] = 0, (4) H(v, p) = L(v) L(u 0 )+pl(u 0 )+p[n(v) f(r)] = 0, (5) where r Γ and p [0, 1] is an embedding parameter, u 0 is an initial approximation of (1), which satisfies the boundary conditions. By (4), it easily follows that H(v, 0) = L(v) L(u 0 )=0, (6) H(v, 1) = A(v) f(r) =0, (7) and the changing process of p from zero to unity is just that of H(v, p) from L(v) L(u 0 )toa(v) f(r). In topology, this is called deformation, L(v) L(u 0 ) and A(v) f(r) are called homotopic. The embedding parameter p is introduced much more naturally, unaffected by artificial factors. Furthermore, it can be considered as a small parameter for 0 <p 1. By applying the perturbation technique used in [25, 22], we assume that the solution of (4) can be expressed as v = v 0 + pv 1 + p 2 v 2 +. (8) Therefore, the approximate solution of (1) can be readily obtained as follows: u = lim p 1 v = v 0 + v 1 + v 2 +. (9) The convergence of this method has been proved by J.H. He in [14, 15]. 3 Parabolic-like equation Consider the parabolic-like equation in three dimensions of the form u t + f 1 (x, y, z)u xx + f 2 (x, y, z)u yy + f 3 (x, y, z)u zz =0, (10)

4 1398 Lin Jin with the initial condition u(x, y, z, 0) = f 4 (x, y, z). (11) According to the homotopy perturbation method, we construct the homotopy Ω [0, 1] R which satisfies u t y 0t (x, y, z)+py 0t (x, y, z)+p(f 1 (x, y, z)u xx (12) +f 2 (x, y, z)u yy + f 3 (x, y, z)u zz )=0, with the initial approximation y 0 (x, y, z) =u(x, y, z, 0) = f 4 (x, y, z). Assume that the form of the solution to (10) is as follows u = u 0 + pu 1 + p 2 u 2 +. (13) Substituting the solution (13) into Eq. (12), and equating the terms of the same power of p, we have p 0 : p 1 : u 0t y 0t (x, y, z) =0, y 0 (x, y, z) = f 4 (x, y, z), u 1t + f 1 (x, y, z)u 0xx + f 2 (x, y, z)u 0yy + f 3 (x, y, z)u 0zz +y 0t (x, y, z) =0,u 1 (x, y, z, 0) = 0, p 2 : u 2t + f 1 (x, y, z)u 1xx + f 2 (x, y, z)u 1yy + f 3 (x, y, z)u 1zz =0, u 2 (x, y, z, 0) = 0, p n : u nt + f 1 (x, y, z)u n 1xx + f 2 (x, y, z)u n 1yy + f 3 (x, y, z)u n 1zz =0, u n (x, y, z, 0) = 0. Letting u 0 = y 0 (x, y, z), and solving the above equations results in the approximate solution u = lim n (u 0 + u 1 + u u n ). (14) which converges to the exact solution if such a solution exists. 4 Hyperbolic-like equation Consider the three dimensional hyperbolic-like equation of the form u tt + g 1 (x, y, z)u xx + g 2 (x, y, z)u yy + g 3 (x, y, z)u zz =0, (15)

5 HPM for solving partial differential equations with variable coefficients 1399 subject to the initial condition u(x, y, z, 0) = g 4 (x, y, z), u t (x, y, z, 0) = g 5 (x, y, z). (16) Similarly, we can construct the homotopy Ω [0, 1] R which satisfies u tt y 0tt (x, y, z, t)+py 0tt (x, y, z, t)+p(g 1 (x, y, z)u xx (17) +g 2 (x, y, z)u yy + g 3 (x, y, z)u zz )=0, with the initial approximation y 0 (x, y, z, t) =g 4 (x, y, z)+tg 5 (x, y, z). Suppose that the solution of (15) can be expressed as u = u 0 + pu 1 + p 2 u 2 +. (18) Substituting the solution (18) into Eq. (17), and equating the terms of the identical power of p gives the following results p 0 : p 1 : u 0tt y 0tt (x, y, z) =0,y 0 (x, y, z, t) = g 4 (x, y, z)+tg 5 (x, y, z), u 1tt + g 1 (x, y, z)u 0xx + g 2 (x, y, z)u 0yy + g 3 (x, y, z)u 0zz +y 0tt (x, y, z, t) =0,u 1 (x, y, z, 0) = 0, p 2 : u 2tt + g 1 (x, y, z)u 1xx + g 2 (x, y, z)u 1yy + g 3 (x, y, z)u 1zz =0, u 2 (x, y, z, 0) = 0, p n : u ntt + g 1 (x, y, z)u n 1xx + g 2 (x, y, z)u n 1yy + g 3 (x, y, z)u n 1zz =0, u n (x, y, z, 0) = 0. Solving the above equations results in the approximate solution u = lim n (u 0 + u 1 + u u n ). (19) Note that the approximate solution converges to the exact solution if such an exact solution exists. 5 Numerical examples In this section, we will present the exact solutions of the parabolic-like equations and hyperbolic-like equations investigated by A.M. Wazwaz [36] to assess the efficiency of the homotopy perturbation method.

6 1400 Lin Jin Example 1. Consider the following one-dimensional parabolic-like equation with variable coefficients in the form [36] u t (x, t) x2 2 u xx(x, t) =0, (20) subject to the initial condition u(x, 0) = x 2. According to the homotopy perturbation method, we can construct the homotopy Ω [0, 1] R which satisfies u t (x, t) y 0t + py 0t p( x2 2 u xx(x, t)) = 0 (21) with the initial approximation y 0 = u(x, 0) = x 2. Suppose that the solution of (20) can be represented as u = u 0 + pu 1 + p 2 u 2 +. (22) Substituting (22) into (21), and equating the terms of the same power of p, it follows that p 0 : u 0t (x, t) y 0t =0,y 0 = x 2, p 1 : p 2 : u 1t (x, t) x2 2 u 0xx(x, t)+y 0t =0,u 1 (x, 0) = 0, u 2t (x, t) x2 2 u 1xx(x, t) =0,u 2 (x, 0) = 0, p n : u nt (x, t) x2 2 u n 1xx(x, t) =0,u n (x, 0) = 0. By choosing u 0 (x, t) =y 0, and solving the above equations, we obtain the following approximations u 1 (x, t) =tx 2, u 2 (x, t) = t2 2! x2,,u n (x, t) = tn n! x2. (23) Then the exact solution of (20) is given by u(x, t) = lim (1 + t + t2 tn + + n 2! n! )x2 = e t x 2, (24) which is exactly the same solution obtained in [36] by using the Adomian decomposition method, however, the homotopy perturbation method provides the exact solution without the computation of the complicated Adomian polynomials, therefore, this method can be seen as a more readily implemented and promising method for solving the partial differential equations with variable coefficients.

7 HPM for solving partial differential equations with variable coefficients 1401 Example 2. Consider the two-dimensional parabolic-like equation with variable coefficients [36] u t (x, y, t) y2 2 u xx(x, y, t) x2 2 u yy(x, y, t) =0, (25) subject to the initial condition u(x, y, 0) = y 2. Similarly, by using the homotopy perturbation method, we have p 0 : u 0t (x, y, t) y 0t =0, y 0 = y 2, p 1 : p 2 : p n : u 1t (x, y, t) y2 2 u 0xx(x, y, t) x2 2 u 0yy(x, y, t)+y 0t =0, u 1 (x, y, 0) = 0, u 2t (x, y, t) y2 2 u 1xx(x, y, t) x2 2 u 1yy(x, y, t) =0, u 2 (x, y, 0) = 0, u nt (x, y, t) y2 2 u n 1xx(x, y, t) x2 2 u n 1yy(x, y, t) =0, u n (x, y, 0) = 0. With the initial approximation u 0 (x, y, t) =y 0, by solving the above equations, we have u 1 (x, y, t) =tx 2, u 2 (x, y, t) = t2 2! y2, u 3 (x, y, t) = t3 3! x2,, (26) u 2n (x, y, t) = t2n (2n)! y2,u 2n+1 (x, y, t) = Then the exact solution of (25) is given by t2n+1 (2n + 1)! x2. u(x, y, t) = lim (t + t3 t2n n 3! (2n + 1)! )x2 +(1+ t2 t2n + + 2! (2n)! )y2 = x 2 sinh t + y 2 cosh t. Example 3. Consider the three-dimensional parabolic-like equation with variable coefficients [36] u t (xyz) (x2 u xx + y 2 u yy + z 2 u zz )=0, (27) subject to the initial condition u(x, y, z, 0) = 0.

8 1402 Lin Jin Similar to the manipulation of the previous examples, we obtain p 0 : u 0t (x, y, z, t) y 0t =0, y 0 =0, p 1 : u 1t (x, y, z, t) (xyz) (x2 u 0xx (x, y, z, t)+y 2 u 0yy (x, y, z, t) +z 2 u 0zz (x, y, z, t)) + y 0t =0,u 1 (x, y, z, 0) = 0, p 2 : u 2t (x, y, z, t) 1 36 (x2 u 1xx (x, y, z, t)+y 2 u 1yy (x, y, z, t) +z 2 u 1zz (x, y, z, t)) = 0, u 2 (x, y, z, 0) = 0, p n : u nt (x, y, z, t) 1 36 (x2 u n 1xx (x, y, z, t)+y 2 u n 1yy (x, y, z, t) +z 2 u n 1zz (x, y, z, t)) = 0, u n (x, y, z, 0) = 0. Letting u 0 (x, y, z, t) =y 0, and solving the above equations, it leads to that the exact solution of (27) can be expressed as u(x, y, z, t) = lim (t + t2 tn + + n 2! n! )x4 y 4 z 4 = x 4 y 4 z 4 (e t 1). Example 4. Consider the one-dimensional hyperbolic-like equation with variable coefficients as [36] u tt (x, t) x2 2 u xx(x, t) =0, (28) with the initial condition u(x, 0) = x, u t (x, 0) = x 2. Similarly, we have p 0 : u 0tt (x, t) y 0tt =0,y 0 = x + tx 2, p 1 : p 2 : p n : u 1tt (x, t) x2 2 u 0xx(x, t)+y 0tt =0,u 1 (x, 0) = 0, u 2tt (x, t) x2 2 u 1xx(x, t) =0,u 2 (x, 0) = 0, u ntt (x, t) x2 2 u n 1xx(x, t) =0,u n (x, 0) = 0. If we choose u 0 (x, t) =y 0, and solve the above equations, it s easy to obtain that u 1 (x, t) = t3 3! x2, u 2 (x, t) = t5 5! x2,,u n (x, t) = t2n+1 (2n + 1)! x2. (29)

9 HPM for solving partial differential equations with variable coefficients 1403 Then the exact solution of (28) can be formulated as u(x, t) = lim x +(t + t3 t2n n 3! (2n + 1)! )x2 = x + x 2 sinh t. Example 5. Consider the two-dimensional hyperbolic-like equation with variable coefficients [36] u tt (x, y, t) x2 12 u xx(x, y, t) y2 12 u yy(x, y, t) =0, (30) with the initial condition u(x, y, 0) = x 4,u t (x, y, 0) = y 4. By applying the homotopy perturbation method, we have p 0 : u 0tt (x, y, t) y 0tt =0,y 0 = x 4 + ty 4, p 1 : p 2 : p n : u 1tt (x, y, t) x2 12 u 0xx(x, y, t) y2 12 u 0yy(x, y, t)+y 0tt =0, u 1 (x, y, 0) = 0, u 2tt (x, y, t) x2 12 u 1xx(x, y, t) y2 12 u 1yy(x, y, t) =0, u 2 (x, y, 0) = 0, u ntt (x, y, t) x2 12 u n 1xx(x, y, t) y2 12 u n 1yy(x, y, t) =0, u n (x, y, 0) = 0. Beginning with the initial approximation y 0 = x 4 + ty 4, and solving the above equations, it follows that u 1 (x, y, t) = t2 2! x4 + t3 3! y4, u 2 (x, y, t) = t4 4! x4 + t5 5! y4,, (31) u n (x, y, t) = t2n (2n)! x4 + t2n+1 (2n + 1)! y4. Therefore, the approximate solution of (30) reads u(x, y, t) = lim x 4 (1 + t2 t2n + + n 2! (2n)! )+y4 (t + t3 t2n ! (2n + 1)! ) = x 4 cosh t + y 4 sinh t which gives the exact solution. Example 6. Consider the three-dimensional hyperbolic-like equation with variable coefficients [36] u tt (x 2 + y 2 + z 2 ) 1 2 (x2 u xx + y 2 u yy + z 2 u zz )=0, (32)

10 1404 Lin Jin with the initial condition u(x, y, z, 0) = 0, u t (x, y, z, 0) = x 2 + y 2 z 2. Similarly, by using the homotopy perturbation method, we have p 0 : u 0tt (x, y, z, t) y 0tt =0, y 0 = t(x 2 + y 2 z 2 ), p 1 : u 1tt (x, y, z, t) (x 2 + y 2 + z 2 ) 1 2 (x2 u 0xx (x, y, z, t)+y 2 u 0yy (x, y, z, t) +z 2 u 0zz (x, y, z, t)) + y 0tt =0,u 1 (x, y, z, 0) = 0, p 2 : u 2tt (x, y, z, t) 1 2 (x2 u 1xx (x, y, z, t)+y 2 u 1yy (x, y, z, t) +z 2 u 1zz (x, y, z, t)) = 0, u 1 (x, y, z, 0) = 0, p n : u ntt (x, y, z, t) 1 2 (x2 u n 1xx (x, y, z, t)+y 2 u n 1yy (x, y, z, t) +z 2 u n 1zz (x, y, z, t)) = 0, u n (x, y, z, 0) = 0. Starting with the initial approximation y 0 = t(x 2 + y 2 z 2 ), and solving the above equations, we can obtain the following approximations u 1 (x, y, z, t) = t2 2! (x2 + y 2 + z 2 )+ t3 3! (x2 + y 2 z 2 ), (33) u 2 (x, y, z, t) = t4 4! (x2 + y 2 + z 2 )+ t5 5! (x2 + y 2 z 2 ),, u n (x, y, t) = t2n (2n)! (x2 + y 2 + z 2 )+ t2n+1 (2n + 1)! (x2 + y 2 z 2 ). Therefore, we have u(x, y, z, t) = lim (t + t2 t2n + + n 2! 2n! )(x2 + y 2 ) +( t + t2 t2n+1 + 2! (2n + 1)! )z2 = (x 2 + y 2 )e t + z 2 e t (x 2 + y 2 + z 2 ), which is the exact solution of (32). 6 Conclusions The homotopy perturbation method has been applied for solving the partial differential equations with variable coefficients. Compared with the Adomian decomposition method [36], this method provides the exact solutions of the

11 HPM for solving partial differential equations with variable coefficients 1405 parabolic-like equations and hyperbolic-like equations without the tedious calculation of the Adomian polynomials. Therefore, this method can be seen as a promising and powerful tool for solving the partial differential equations in different dimensions with variable coefficients. References [1] S. Abbasbandy, Application of He s homotopy perturbation method for Laplace transform. Chaos Solitons & Fractals 30(5) (2006), [2] G. Adomian, A review of the decomposition method in applied mathematics. J. Math. Anal. Appl. 135 (1988 ), [3] P.D. Ariel, T. Hayat, and S. Asghar, Homotopy perturbation method and axisymmetric flow over a stretching sheet. Internat. J. Nonlinear. Sci. Numer. Simul. 7(4) (2006 ), [4] A. Belendez, A. Hernandez and T. Belendez, Application of He s homotopy perturbation method to the Duffing-harmonic oscillator. Internat. J. Nonlinear. Sci. Numer. Simul. 8(1) (2007), [5] L. Cveticanin, Homotopy-perturbation method for pure nonlinear differential equation. Chaos Solitons & Fractals 30(5) (2006), [6] D.J. Evans, K.R. Raslan, The Tanh-Function Method for Solving some Important Nonlinear Partial Differential Equation. Internat. J. Comput. Math. 82 (2005), [7] E. Fan, Tanh-Function Method and Its Applications to Nonlinear Equations. Phys. Lett. A 277 (2000), [8] E. Fan, Traveling Wave Solutions for Generalized Hirota-Satsuma Coupled KdV Systems. Z Naturforsch A 56 (2001), [9] D.D. Ganji, A. Sadighi, Application of He s homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations. Internat. J. Nonlinear. Sci. Numer. Simul. 7(4) (2006), [10] A. Ghorbani, J. Saberi-Nadjafi, He s homotopy perturbation method for calculating adomian polynomials. Internat. J. Nonlinear. Sci. Numer. Simul. 8(2) (2007), [11] Q.K. Ghori, M. Ahmed, A.M. Siddiqui, Application of homotopy perturbation method to squeezing flow of a Newtonian fluid. Internat. J. Nonlinear. Sci. Numer. Simul. 8(2) (2007),

12 1406 Lin Jin [12] A. Gorguis, A comparison between Cole-Hopf transformation and the decomposition method for solving Burgers equations. Appl. Math. Comput. 173(1) (2006), [13] J.H. He, Some asymptotic methods for strongly nonlinear equations. Internat. J. Modern Phys. B 20 (2006), [14] J.H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems. Internat. J. Non-linear Mech. 35(1) (2000), [15] J.H. He, Homotopy perturbation technique. Comput. Math. Appl. Mech. Eng. 178(3-4) (1999), [16] J.H. He, New interpretation of homotopy perturbation method. Internat. J. Modern Phys. B 20(18) (2006 ), [17] J.H. He, He JH. Homotopy perturbation method: A new nonlinear technique. Appl. Math. Comput. 135 (2003), [18] J.H. He, Limit cycle and bifurcation of nonlinear problems. Chaos Solitons & Fractals 26 (2005), [19] J.H. He, Application of homotopy perturbation method to nonlinear wave equations. Chaos Solitons & Fractals 26 (2005), [20] J.H. He, Periodic solutions and bifurcations of delay-differential equations. Phys. Lett. A 347 (2005), [21] J.H. He, Homotopy-perturbation method for bifurcation of nonlinear problems. Internat. J. Nonlinear Sci. Numer. Simul. 6 (2005), [22] M.H. Holmes, Introduction to Perturbation Methods. Springer, Berlin, [23] S.J. Liao, An approximate solution technique not depending on small parameters: A special example. Internat. J. Non-Linear Mech. 30(3) (1995), [24] S. Momani, Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method. Appl. Math. Comput. 165 (2005), [25] A.H. Nayfeh, Perturbation Methods. Wiley, New York, [26] T. Ozis, A. Yildirim, Traveling wave solution of Korteweg-de Vries equation using He s homotopy perturbation method. Internat. J. Nonlinear Sci. Numer. Simul. 8(2) (2007),

13 HPM for solving partial differential equations with variable coefficients 1407 [27] T. Ozis, A. Yildirim, A comparative study of He s homotopy perturbation method fordetermining frequency-amplitude relation of a nonlinear oscillator with discontinuities. Internat. J. Nonlinear Sci. Numer. Simul. 8(2) (2007), [28] E.J. Parkes, B.R. Duffy, An Automated Tanh-Function Method for Finding Solitary Wave Solutions to Nonlinear Evolution Equations. Comput. Phys. Commun. 98 (1998), [29] M. Rafei, D.D. Ganji, Explicit solutions of Helmholtz equation and fifthorder KdV equation using homotopy perturbation method. Internat. J. Nonlinear Sci. Numer. Simul. 7(3) (2006), [30] M.A. Rana, A.M. Siddiqui and Q.K. Ghori, Application of He s homotopy perturbation method to Sumudu transform. Internat. J. Nonlinear Sci. Numer. Simul. 8(2) (2007), [31] D.H. Shou, J.H. He, Application of parameter-expanding method to strongly nonlinear oscillators. Internat. J. Nonlinear Sci. Numer. Simul. 8(1) (2007), [32] A.M. Siddiqui, M. Ahmed, Q.K. Ghori, Couette and Poiseuille flows for non-newtonian fluids. Internat. J. Nonlinear Sci. Numer. Simul. 7(1) (2006), [33] A.M. Siddiqui, R. Mahmood, Q.K. Ghori, Thin film flow of a third grade fluid on a moving belt by He s homotopy perturbation method. Internat. J. Nonlinear Sci. Numer. Simul. 7(1) (2006), [34] H. Tari, D.D. Ganji and M. Rostamian, Approximate solutions of K (2,2), KdV and modified KdV equations by variational iteration method, homotopy perturbation method and homotopy analysis method. Internat. J. Nonlinear Sci. Numer. Simul. 8(2) (2007), [35] E. Yusufoglu, Homotopy perturbation method for solving a nonlinear system of second order boundary value problems. Internat. J. Nonlinear Sci. Numer. Simul. 8(3) (2007), [36] A.M. Wazwaz, Exact Solutions for Heat-Like and Wave-Like Equations with Variable Coefficients. Appl. Math. Comput. 149 (2004), Received: February 21, 2008