HOMOTOPY ANALYSIS METHOD FOR SOLVING COUPLED RAMANI EQUATIONS

Transcription

1 HOMOTOPY ANALYSIS METHOD FOR SOLVING COUPLED RAMANI EQUATIONS A. JAFARIAN 1, P. GHADERI 2, ALIREZA K. GOLMANKHANEH 3, D. BALEANU 4,5,6 1 Department of Mathematics, Uremia Branch, Islamic Azan University, Uremia, Iran jafarian5594@yahoo.com 2 Department of Mathematics, Uremia Branch, Islamic Azan University, Uremia, Iran ghaderi.pariya@gmail.com 3 Department of Physics, Uremia Branch, Islamic Azan University, P.O.BOX 969, Uremia, Iran alireza@physics.unipune.ac.in 4 Çankara University, Faculty of Art and Sciences, Department of Mathematics and Computer Sciences, Balgat 0630, Ankara, Turkey 5 Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box: 80204, Jeddah, 21589, Saudi Arabia 6 Institute of Space Sciences, P.O. BOX MG-23, RO , Magurele-Bucharest, Romania dumitru@cankaya.edu.tr Received May 7, 2013 In this manuscript, coupled Ramani equations are solved by means of an analytic technique, namely the homotopy analysis method (HAM). The HAM is a capable and a straightforward analytic tool for solving nonlinear problems and does not need small parameters in the governing equations and boundary/initial conditions. The result of this study presents the utility and sufficiency of HAM method. Comparisons demonstrate that there is a good agreement between the HAM solutions and the exact solutions. Key words: Homotopy analysis method, Coupled Ramani equations, Approximate solution. 1. INTRODUCTION It is well known that most of the phenomena that appear in mathematical physics and different branches of science and engineering can be described by partial differential equations (PDEs) and nonlinear evolution equations (NLEEs). The nonlinear evolution equations are prominent in modeling several physical phenomena. The coupled evolution equations attracted a significant amount of research work in the literature, mainly for the specification of soliton solutions or periodic solutions. In [1], a coupled Ramani equation of the following form has been studied: u xxxxxx + 15u xx u xxx + 15u x u xxxx + 45u 2 xu xx 5(u xxxt + 3u xx u t + 3u x u xt ) 5u tt + 18v x = 0 (1) RJP Rom. 59(Nos. Journ. Phys., 1-2), Vol , Nos. (2014) 1-2, P. (c) 26 35, Bucharest, 2014 v t v xxx 3v x u x 3vu xx = 0. (2)

2 2 Homotopy analysis method for solving coupled Ramani equations 27 Multi-soliton solutions of the coupled Ramani equations were derived and represented using Pfaffians in a compact form in [2] and the three-soliton solution of this coupled system is derived by Hu et al. [1]. Exact solutions of the coupled Ramani equations were determined using the tanh method in [3 5]. In [6], the Hirota s bilinear method was used to determine multiple soliton solutions and multiple singular soliton solutions for Ramani equation. In the past decades, many powerful and direct methods have been developed to find special solutions of NLEEs [7 17]. Among them are the tanh method [4], the exp-function method [18], the tanh-coth method [19], (G /G)-expansion method [20], the sine-cosine method [21], variational iteration method [3], Adomian decomposition method [22,23], and many other methods [23], which have been used in a reasonable way to obtain exact solutions to NLEEs. However, a novel approach called homotopy analysis method (HAM) [24] has been introduced in order to solve nonlinear partial differential equations. Recently, Jafarian et al. [25] used homotopy analysis method to study the coupled harmonic waves nonlinear magneto-thermoelasticity equations under influence of rotation and a similar method was used for the Cauchy problem arising in one dimensional nonlinear thermoelasticity; see Ref. [26]. In another work, Golmankhaneh et al. [27] used this method to find solutions to the second-order random differential equations. Also, the HAM was used by Jafarian et al. [28] in order to obtain the approximate solution of Kadomtsev-Petviashvili-II equation. In this manuscript, we use the homotopy analysis method to obtain exact solutions for the coupled Ramani equations. Different from all other analytic methods, it provides us with a simple way to adjust and control the convergence region of solution series by choosing proper values for auxiliary parameter h. This paper is arranged in the following manner. In Section 2, the basic ideas of the present approach are described. Then in Section 3, by choosing special forms of initial conditions, the proposed method is applied to study coupled Ramani equations. Further, in Section 4, numerical results are given to illustrate the capability and accuracy of the proposed method. Finally, conclusions are given. 2. BASIC IDEA OF HAM Let us consider the following differential equation [24] N[u(x,t)] = 0, (3) where N is a nonlinear operator, t is the independent variable, and u(x,t) is an unknown function. For simplicity, we ignore all boundary or initial conditions, which can treated in a similar way. By means of generalizing the traditional homotopy method, Liao [24] constructs the so-called zero-order deformation equation (1 q)l[φ(x,t;q) u 0 (x,t)] = qhh(x,t)n[φ(x,t;q)], (4)

3 28 A. Jafarian et al. 3 where q [0,1] is the embedding parameter, h 0 is a non-zero auxiliary parameter, H(x,t) 0 is a nonzero auxiliary function, L is an auxiliary linear operator, u 0 (x,t) is an initial guess of u(x,t), and φ(x,t;q) is a unknown function. It is important that ones has great freedom to choose auxiliary parameters in HAM. Obviously, when q = 0 and q = 1, it holds φ(x,t;0) = u 0 (x,t), φ(x,t;1) = u(x,t), (5) Thus, as q increases from 0 to 1, the solution φ(x,t;q) varies from the initial guess u 0 (x,t) to the solution u(x,t). Expanding by Taylor series with respect to q, we have φ(x,t;q) = u 0 (x,t) + u m (x,t) q m, (6) where u m (x,t) = 1 m φ(x,t;q) m! q m q=0. (7) If the auxiliary linear operator, the initial guess, the auxiliary parameter h, and the auxiliary function are so properly chosen, the series (6) converges at q = 1, then we have u(x,t) = u 0 (x,t) + u m (x,t), (8) which must be one of solutions for the original nonlinear equation, as proved by Liao [24]. As h = 1 and H(x,t) = 1, equation (4) becomes m=1 m=1 (1 q)l[φ(x,t;q) u 0 (x,t)] + qn[φ(x,t;q)] = 0, (9) which is used mostly in the homotopy perturbation method (HPM). According to the definition (7), the governing equation can be deduced from the zeroth-order deformation equation (4). Let us define the vector u n = {u 0 (x,t),u 1 (x,t),...,u n (x,t)}. Differentiating equation (4) m times with respect to the embedding parameter q, then setting q = 0 and finally dividing them by m!, we have the so-called m th -order deformation equation L[u m (x,t) χ m u (x,t)] = hh(x,t)r m ( u ), (10) where 1 N[φ(x,t;q)] R m ( u ) = (m 1)! q q=0, (11) and 0 m 1, χ m = (12) 1 m > 1.

4 4 Homotopy analysis method for solving coupled Ramani equations 29 It should be emphasized that u m (x,t) for m 1 is governed by the linear equation (10) under the linear boundary condition that come from original problem, which can be easily symbolically solved by the MATLAB computer software. 3. APPLICATION OF HAM TO A COUPLED RAMANI EQUATION To solve the coupled Ramani equations (1) and (2) by means of HAM, we start with the following initial approximations [5] u 0 (x,t) = a 0 + 2αcoth(αx) + 2tβα 2 csch 2 (αx), v 0 (x,t) = (4/9)βα 4 (16/27)α 6 + (5/9)β 2 α 2 (5/54)β 3 + ((20/9)βα 4 + (16/9)α 6 (5/9)β 2 α 2 )coth(x) 2, and the auxiliary linear operators L 1 [φ 1 (x,t;q)] = 2 φ 1 (x,t;q) t 2, L 2 [φ 2 (x,t;q)] = φ 2(x,t;q), t with property L 1 [c 1 + tc 2 ] = 0, L 2 [c 3 ] = 0, where c 1, c 2 and c 3 are constant coefficients, and φ 1 (x,t;q) and φ 2 (x,t;q) are real functions. Furthermore, the nonlinear operators N 1 and N 2 are defined as N 1 [φ 1,φ 2 ] = 2 φ 1 t φ 1 5 x 6 φ 1 3 φ x 2 x 3 3 φ 1 4 φ 1 x x 4 9( φ 1 x )2 2 φ 1 x φ 1 x 3 t + φ 1 φ x 2 t + 3 φ 1 2 φ 1 x x t 18 5 φ 2 x. N 2 [φ 1,φ 2 ] = φ 2 t 3 φ 2 x 3 3 φ 1 φ 2 x x 3φ 2 φ 1 2 x 2. where q [0,1], φ 1 (x,t;q) and φ 2 (x,t;q) are real functions of x, t and q. Using above definitions, with assumption H 1 (x,t) = 1, H 2 (x,t) = 1, we develop the zeroth-order deformation equations as follows (1 q)l 1 [φ 1 (x,t;q) u 0 (x,t)] = qh 1 N 1 [φ 1 (x,t;q),φ 2 (x,t;q)], (13) (1 q)l 2 [φ 2 (x,t;q) v 0 (x,t)] = qh 2 N 2 [φ 1 (x,t;q),φ 2 (x,t;q)]. (14) Obviously, when q = 0 and q = 1, it is clear that φ 1 (x,t;0) = u 0 (x,t), φ 1 (x,t;1) = u(x,t),

5 30 A. Jafarian et al. 5 φ 2 (x,t;0) = v 0 (x,t), φ 2 (x,t;1) = v(x,t). Both of h 1 and h 2 are properly chosen so that the terms u m (x,t) = 1 m φ 1 (x,t;q) m! q m q=0, v m (x,t) = 1 m φ 2 (x,t;q) m! q m q=0. exist for m 1 and the power series of q in the following forms u(x,t) = u 0 (x,t) + u m (x,t), (15) v(x,t) = v 0 (x,t) + m=1 v m (x,t). (16) Differentiating equations (13)-(14) m times with respect to the embedding parameter q, then setting q = 0 and finally dividing them by m!, we have the so-called m th -order deformation equations where m=1 L 1 [u m (x,t) χ m u (x,t)] = h 1 R 1m ( u, v ), (17) L 2 [v m (x,t) χ m v (x,t)] = h 2 R 2m ( u, v ), (18) R 1m ( u, v ) = 2 u t u 5 x u j x 3 4 u j x 4 9 n=0 2 u j u j x 2 t 2 u n x 2 [ + 3 u j x n u j x 2 u t x u j x 2 3 u j x 3, u n j x ] 4 u t x 3 + v x, R 2m ( u, v ) = v t 3 v x 3 3 v j u j x x 3 v j u 2 j x 2. and χ m is defined by (12). Now, the solution of the m th -order deformation equations (17)-(18) becomes u 0 (x,t) = a 0 + 2αcoth(αx) + 2tβα 2 (csch 2 (αx)),

6 6 Homotopy analysis method for solving coupled Ramani equations 31 Table 1 Absolute error of u(x,t) with h = 1 and t = 20. x u HAM u Exact Absolute error e e e e e e e e e e-009 Table 2 Absolute error of v(x,t) with h = 1 and t = 20. x v HAM v Exact Absolute error e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-013 u 1 (x,t) = u 0 (x,t) + h t τ 0 0 R 11( u 0, v 0 ) dµ dτ = (α 3 βht 2 (210β sinh(2αx) 210β sinh(4αx) + 90β sinh(6αx) 15β sinh(8αx) + 800α 5 t 1088α 5 tcosh(2αx)+ 256α 5 tcosh(4αx) + 64α 5 tcosh(6αx)

7 32 A. Jafarian et al. 7 32α 5 tcosh(8αx) 34560α 7 β 2 t α 3 βt α 6 βt 2 sinh(2αx) α 6 βt 2 sinh(4αx) α 6 βt 2 sinh(6αx) 41472α 7 β 2 t 3 cosh(2αx) 6912α 7 β 2 t 3 cosh(4αx) 2400α 3 βtcosh(2αx) 480α 3 βtcosh(4αx)+ 480α 3 βtcosh(6αx)))/(2(15cosh 5 (2αx) 75cosh 4 (2αx)+150cosh 3 (2αx) 150cosh 2 (2αx) + 75cosh(2αx) 15)),. v 0 (x,t) = (4/9)βα 4 (16/27)α 6 (5/9)β 2 α 2 (5/54)β 3 + ((20/9)βα 4 + (16/9)α 6 (5/9)β 2 α 2 )coth(x) 2, v 1 (x,t) = v 0 (x,t) + h t τ 0 0 R 21( u 0, v 0 ) dµ dτ =. (8α 3 βht(16α α 2 β 5β 2 )(sinh(4αx) 2sinh(2αx)+ 60α 3 t + 56α 3 tcosh(2αx) + 4α 3 tcosh(4αx) 3αβt + 2αβtcosh(2αx)+ αβt cosh(4αx)))/(9(15 cosh(2αx) 6 cosh(4αx) + cosh(6αx) 10)). Finally, we have u(x,t) = u 0 (x,t) + u 1 (x,t) + u 2 (x,t) +, v(x,t) = v 0 (x,t) + v 1 (x,t) + v 2 (x,t) +.

8 8 Homotopy analysis method for solving coupled Ramani equations u HAM 1.1 u Exact 1.05 u x Fig. 1 Comparison between exact results for u and the 2th-order HAM approximate results for u (using h = 1) when t = x v v HAM v Exact Fig. 2 Comparison between exact results for v and the 2th-order HAM approximate results for v (using h = 1) when t = 20. x

9 34 A. Jafarian et al RESULTS In this section, in order to verify numerically whether the proposed methodology leads to higher accuracy, we appraise the numerical solution of the equations (1) and (2), with the following arbitrary constants: a 0 = 1, α = 0.01, β = Furthermore, to show the efficiency of the present method for our problem in comparison with the exact solution, we report absolute error which is defined by error = u Exact u HAM. Tables 1 and 2 show the absolute errors for differences between the exact solutions [4] and the 2th-order approximate solutions obtained by HAM at some points. Besides, the behavior of the exact and approximate solutions are illustrated in Figures 1 and 2. Comparison of the result obtained by HAM with exact solution displays the accuracy of the new method. It is obvious that the overall errors can be made smaller by adding new terms from the iteration formulas. The obtained results show that our method is an efficient one for solving nonlinear evolution problems. 5. CONCLUSIONS In this paper, the homotopy analysis method has been successfully used to find out the approximate solutions of coupled Ramani equations. The proposed method prepares one with a expedient way to control the convergence of approximate solution series. The basic ideas of this approach can be widely utilized to solve other strongly nonlinear evolution problems. Our numerical results indicated that the obtained approximate solutions were in suitable agreement with the exact solutions, thus demonstrating the remarkable efficiency of the homotopy analysis method. REFERENCES 1. X.B. Hu, D.L. Wang, and H.W. Tam, Appl. Math. Lett. 13, (2000). 2. J.X. Zhao and H.W. Tam, Appl. Math. Lett. 16, (2006). 3. M. Akbarzade and J. Langari, Int. J. Math. Anal. 5, (2011). 4. E. Yusufoglu and A. Bakir, Chaos, Solitons Fract. 37, (2008). 5. E.M.E. Zayed and H.M. Abdel Rahman, WSEAS Trans. Math. 11, (2012). 6. A.M. Wazwaz and H. Triki, Appl. Math. Comput. 216, (2010). 7. D. Baleanu, Alireza K. Golmankhaneh, and Ali K. Golmankhaneh, Rom. J. Phys. 54, (2009). 8. Alireza K. Golmankhaneh, Ali K. Golmankhaneh, and D. Baleanu, Rom. Rep. Phys. 63, (2011); D. Baleanu et al., Rom. Rep. Phys. 64, (2012); H. Jafari et al., Rom. Rep. Phys. 64, (2012); A. Razminia and D. Baleanu, Proc. Romanian Acad. A 13, (2012).

10 10 Homotopy analysis method for solving coupled Ramani equations Alireza K. Golmankhaneh, Ali K. Golmankhaneh, and D. Baleanu, Signal Processing. 91, (2011). 10. Alireza K. Golmankhaneh, Ali K. Golmankhaneh, and D. Baleanu, Rom. Rep. Phys. 63, (2011). 11. A.K. Golmankhaneh, T. Khatuni, N.A. Porghoveh, and D. Baleanu, Cent. Eur. J. Phys. 10, (2012). 12. A. Kadem and D. Baleanu, Rom. J. Phys. 56, (2011). 13. A. Kadem and D. Baleanu, Rom. J. Phys. 56, (2011). 14. G. Ebadi et al., Rom. J. Phys. 58, 3-14 (2013); G. Ebadi et al., Rom. Rep. Phys. 65, (2013); A.G. Johnpillai, A. Yildirim, and A. Biswas, Rom. J. Phys. 57, (2012); G. Ebadi et al., Rom. Rep. Phys. 64, (2012); H. Triki et al., Rom. Rep. Phys. 64, (2012); H. Triki et al., Rom. Rep. Phys. 64, (2012); G. Ebadi et al., Rom. Rep. Phys. 64, (2012). 15. A. Salas et al., Proc. Romanian Acad. A 14, (2013); A. Biswas et al., Proc. Romanian Acad. A 13, (2012); H. Triki et al., Proc. Romanian Acad. A 13, (2012); G. Ebadi et al., Proc. Romanian Acad. A 13, (2012). 16. H. Leblond and D. Mihalache, Phys. Reports 523, (2013); D. Mihalache, Rom. J. Phys. 57, (2012); H. Leblond, H. Triki, and D. Mihalache, Rom. Rep. Phys. 65, (2013). 17. A.M. Wazwaz, Rom. J. Phys. 58, (2013); A.M. Wazwaz, Rom. Rep. Phys. 65, (2013); A.M. Wazwaz, Proc. Romanian Acad. A 14, (2013). 18. A.G. Davodi, D.D. Ganji, A.G. Davodi, and A. Asgari, Appl. Math. Comput. 217, (2010). 19. A.H. Salas and C.A. Gomez, AAM: Intern. J. 5, (2010). 20. I.E. Inan, Cankaya Univ. J. Sci. Eng. 7, (2010). 21. M.T. Alquran, Appl. Math. Inf. Sci. 6, (2012). 22. M.A. Al-Mazmumy, Appl. Math. 2, (2011). 23. X.G. Luo, Q.B. Wu, and B.Q. Zhang, J. Math. Anal. Appl. 321, (2006). 24. S.J. Liao, Beyond perturbation: Introduction to homotopy analysis method (Chapman &Hall/CRC Press, Boca Raton, 2003). 25. A. Jafarian and P. Ghaderi, Rep. Math. Phys. 2013, (in press). 26. A. Jafarian, P. Ghaderi, Alireza K. Golmankhaneh, and D. Baleanu, Rom. J. Phys. 58, (2013). 27. Alireza K. Golmankhaneh, Neda A. Porghoveh, and D. Baleanu, Rom. Rep. Phys. 65, (2013). 28. A. Jafarian, P. Ghaderi, and Alireza K. Golmankhaneh, Rom. Rep. Phys. 65, (2013).