 Open Journal of Applied Sciences, 2012, 2, 193-197 doi:10.4236/ojapps.2012.23029 Published Online September 2012 (http://www.SciRP.org/journal/ojapps) Approximate Analytical Solutions of Fractional Coupled mKdV Equation by Homotopy Analysis Method Orkun Tasbozan, Alaattin Esen, Nuri Murat Yagmurlu Department of Mathematics, İnonu University, Malatya, Turkey Email: orkun.tasbozan@inonu.edu.tr Received June 1, 2012; revised July 3, 2012; accepted July 15, 2012 ABSTRACT In this paper, the approximate analytical solutions of the fractional coupled mKdV equation are obtained by homotopy analysis method (HAM). The method includes an auxiliary parameter which provides a convenient way of adjusting and controlling the convergence region of the series solution. The suitable value of auxiliary parameter is deter- mined and the obtained results are presented graphically.  Keywords: Homotopy Analysis Method; Approximate Analytical Solution; Fractional Coupled mKdV Equation 1. Introduction Fractional derivatives provide an excellent tool for the description of memory and hereditary characteristics of different materials and processes due to their non-locality characteristics. This is the main advantage of fractional derivatives in comparison with integer order model, in which such effects are in fact neglected . Several defi- nitions of fractional integration and derivation such as Riemann-Liouville’s and Caputo’s have been proposed. The Riemann-Liouville integral operator  having order 0, which is a real number, is defined as     101 d0,xJfxxtt xft (1) and as for 0  0 Jfx fx where the real function ,fx is said to be in the space 0x,CR, if there exists a real number p such that 1 ,pfxxfx nC where and it is said to be in the space (0, ),C1fx  if and only if ,nhC Its fractional derivative of order .Nn0 is gen- erally used   d ,1dnnnDfxJ fxnnx where n is an arbitrary integer. The Riemann-Liouville integral operator has an important role for the develop- ment of the theory of both fractional derivatives and in- tegrals. In spite of this fact, it has certain disadvantages when it comes to modelling real-world phenomena with fractional differential equations. This problem has been solved by M. Caputo first in his article  and then in his book . Caputo definition, which is a modification of Riemann-Liouville definition, can be given as   10 1 =d,Γ 0, 1.nnxnnDfx JDfxxttnfnnt Note that Caputo derivative has the following two im- portant properties  DJf xfx and  10 0! 1.knkkxJDfx fxfknn  In recent years, many important phenomena in various scientific and technological areas have been well de- scribed by fractional differential equations. In general, since these type equations are not exactly solved, their numerical solution techniques have become increasingly important. The HAM, a powerful tool for searching the approximate solutions which was first proposed by Liao [4,5], is one of such numerical solution techniques. Un- like perturbation techniques, the HAM is not limited to any small physical parameters in the considered equation. Therefore, the HAM can overcome the foregoing restrict- tions and limitations of perturbation techniques so it pro- vides us with a powerful tool to analyze strongly nonlin- ear problems . The HAM has been proposed and suc- Copyright © 2012 SciRes. OJAppS O. TASBOZAN ET AL. 194 cessfully applied to solve several fractional differential equations modeling problems arising in science and en- gineering by many authors [6-18] and the references therein. In this paper, we will apply the HAM to frac- tional coupled mKdV equation by using Caputo's defini- tion of fractional differentiation. 2. HAM Solutions of the Time-Fractional Coupled mKdV Equation In this section, we implement the HAM to the fractional coupled mKdV equation defined by 2222333036301,0xxtxtxxx xxxxuuu vvuvDDv vvuuvuuvvt   0 (2) with the initial conditions  ,0tanh 2,,0tanh 2.ux xvx x (3) To investigate the HAM solutions of Equation (2) with the initial conditions given by Equation (3), we can choose the linear operator ,;,; ,1,2itixtq Dxtqi having the property  0ic where are constants. From Equation (2), we can now define nonlinear operators as 1, 2icsi 13211 1132221212 ,; ,;,;1 3,;2,;,; 3,; ,,, ,,; 3,,xtq xtqxtqxtq xtxxtqxtq xxxtq xtxtq xqxtq     21 212322311222212 ,;,,;,;,;,; ,; 3,; 6,;,;,; 3,;,;.xtq xtqxtqxtq xxtq xtqxtxxtqxtq xtqxxtqxtq xtqx Therefore, we construct the zero-order deformation equation as follows 1011 12 1,;,,;, ,;,qxtquxtqxtqxtq (4) 20221 2 1,;,,; ,,;.qxtqvxtqxtqxtq (5) If we choose 0q then we get 1020,;0 ,,0,,;0 ,,0xtuxtuxxtvxtv x and 1q, we obtain  12,;1,, ,;1,.xtuxt xtvxt Thus, as the embedding parameter q increases from 0 to 1, the solutions  ,; 1,2ixtq i also change from the initial values 0,uxt0,vx and to the solutions t,uxt and ,t.ivx If we expand ,;xtq for 1, 2i in Taylor series with respect to the embedding parameter q, we obtain  10 120 1,;,, ,,;,, ,mmmmmmxtquxtuxtqxtqvxtvxtq where   1020,;1,,!,;1,.!mmmqmmmqxtquxt mqxtqvxt mq If the auxiliary linear operator, the initial guess and the auxiliary parameter are properly chosen, as pointed out by Liao [5,8], the above series converges at 1q, and one can have  0 1,,mmuxtu xtuxt,,, (6)   0 1,,mmvxtvxtvxt (7) which should be one of the solutions of the original equ-ation. Let’s define the following vectors    101 101,,,, ,,,,,,, ,,.mnmnuxtuxtuxtvxtvxt vxtuv By differentiating Equations (4) and (5) m times with respect to the embedding parameter q, we obtain the mth- order deformation equations as follows Copyright © 2012 SciRes. OJAppS O. TASBOZAN ET AL. 195 111,1,,mmm mmuxtuxtRuv1,m1m (8)  122,1,, ,mmm mmvxtv xtRuv (9) subject to the initial conditions   ,0tanh 2,,0tanh2mmux xvx x where     1,1 131131–1011001–100 ,,,12, 3,, 3,, 3,,mm mmmmmnnnmn mnknknkmnknk mnnkRuxtuxttxvxtvxtxxuxtuxtuxtxuxtv xtvxtxuv,,     2,1 131131–101–100100–1 ,,,, 3,, 6,, 3,,,, , mm mmmmmnnnmn mnknknkmnknknknkmnRvxtvxttxuxtvxtxxuxtuxtv xtxuxtu vxtxvxvxtxtuvt and 0, 1, 1, 1.mmmBy applying the operator J given by Equation (1), which is the inverse of the operator tD, to the both sides of the mth-order deformation Equations (8) and (9) for we obtain 1,m 111,11,, ,mmm tmmmuxtu xtJRuv,. (10)  122,11,, ,mmm tmmmvxtvxtJ Ruv (11) For the purpose of simplicity, setting 12 and by using Equations (10) and (11) with the initial condi-tions given by Equation (3), we successively obtain     012,tanh2,4,,Γ1cos 2uxt xtuxt hx    2222241,Γ1cos 232tanh 2 ,Γ21cos2tuxt hxtxhx  and     01222222,tanh2,4,,Γ1cos 241,Γ1cos 232tanh 2 ,Γ21cos2vxtxtvxt hxtvxt hxtxhx   etc. Therefore, the series solutions expressed by the HAM given in Equations (6) and (7) can be written in the fol- lowing forms  012012,,,,,,,,uxtuxt uxt uxtvxtvxt vxt vxt,. (12) To demonstrate the efficiency of the method, we com-pare the HAM solutions of fractional coupled mKdV equation given by Equation (12) for 0 with its ex- act solutions    ,tanh24,tanh24uxtx tvxtx t,. The fact that HAM solution series contains the auxil- iary parameter providing us with a simply way to adjust and control the convergence of the solution series should be noted. To obtain an appropriate range for , we consider the so-called -curve to choose a proper value of which ensures that the solution series is convergent, as pointed by Liao , by finding out the valid region of corresponding to the line segments nearly parallel to the horizontal axis. The -curves of 0,0.01u and 0.010v, are given by 3th-order HAM solution given by Equation (12) for various  parame- ters in Figure 1. It can be seen from the figure that the valid range of lies in approximately 1.3 0.7.Figure 2 shows the numerical solutions of ,uxt and ,vxt at x = 2 from t = 0 to t = 0.5 for = –0.7, –1 and –1.3 obtained by 3th-order HAM for 1 and analytical solutions, respectively. Between t = 0 and t = 0.5, it can be seen from the figure that the choice of  –0.7 is an appropriate value. =Figure 3 shows the numerical solutions of ,uxt and ,vxt at x = 2 during for = –0.7 obtained by 3th-order HAM for 00t0.9.5  and 0.8, Copyright © 2012 SciRes. OJAppS O. TASBOZAN ET AL. 196 respectively. Figure 1. The -curves of 3th-order approximate solutions obtained by the HAM.  Figure 2. The results obtained by the HAM for  = 1 and various by 3th-order approximate solution in com- parison with the exact solution at x = 2.  Figure 3. The results obtained by the HAM for  = 0.9,  = 0.8 and = – 0.7 by 3th-order approximate solution at x = 2. 3. Conclusion In this paper, the HAM has been successfully applied to obtain approximate analytical solution of fractional cou-pled mKdV equation. 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