THE SOLUTION OF COUPLED MODIFIED KDV SYSTEM BY THE HOMOTOPY ANALYSIS METHOD

Transcription

1 TWMS Jour. Pure Appl. Mah., V.3, N.1, 1, pp THE SOLUTION OF COUPLED MODIFIED KDV SYSTEM BY THE HOMOTOPY ANALYSIS METHOD M. GHOREISHI 1, A.I.B.MD. ISMAIL 1, A. RASHID Absrac. In his paper, he Homoopy Analysis Mehod HAM is presened for obaining he approximae soluion of new coupled modified Koreweg-de Vries MKdV sysem. The approximae analyical soluion is obained by using his mehod in he form of a convergen power series wih componens ha are easily compuable. For he problems considered, only a few erms are required o obain an approximae soluion ha is accurae and efficien. Keywords: homoopy analysis mehod, new coupled MKdV sysem, auxiliary parameer. AMS Subjec Classificaion: 35Q53, 55Pxx. 1. Inroducion By considering 4 4 marix specral problem wih hree poenials Wu e al. [14] derived a new hierarchy of nonlinear evoluion equaions. Two new coupled modified Koreweg-de Vries MKdV equaions are included in his hierarchy. These equaions are governed by he following parial differenial equaions: and u = 1 u xxx 3u u x + 3 vz xx + 3uvz x, v = v xxx 3vu x x 3v z x + 6uvu x + 3u v x, z = z xxx 3zu x x 3z v x + 6uzu x + 3u z x, { u = 1 u xxx 3u u x + 3 v xx + 3uv x 3λu x, v = v xxx 3vv x 3u x v x + 3u v x + 3λv x. In general, analyical soluions of 1 and canno be obained because he sysems are very complicaed. Therefore he HAM is proposed for solving 1 and, which is an approximae analyical mehod. Fan [7] used an exended anh mehod wih symbolic compuaion and found he solion soluions, riangular periodic soluions and raional soluions for new coupled MKdV sysem 1. The Riccai equaion and symbolic compuaion have been also used by Fan [8] o find wo kinds of he solion soluion for he coupled MKdV sysem. Fan s [8] approach was o ake advanage of a Riccai equaion involving a parameer and use is soluions o replace he anh-funcion in he anh-funcion mehod. The Adomian Decomposiion Mehod ADM and Variaional Ieraion Mehod VIM have been used by Inc and Cavlak [9] o obain approximae analyical soluions for he new coupled MKdV sysem 1. Raslan [13] developed he ADM o approximaely solve he coupled MKdV sysem. 1 1 School of Mahemaical Science, Universii Sains Malaysia, Penang, Malaysia, Deparmen of Mahemaics, Gomal Universiy, Dera Ismail Khan, Pakisan, prof.rashid@yahoo.com Manuscrip received December 11. 1

2 M. GHOREISHI, A.I.B.MD. ISMAIL, A. RASHID: COUPLED MODIFIED KDV SYSTEM The basic aim of his sudy is o obain he approximae analyical soluion for he coupled sysem 1 and using HAM. Approximae analyical schemes such as ADM, VIM, Homoopy Perurbaion Mehod HPM and HAM have been widely used o solve operaor equaions. These schemes generae an infinie series soluion and do no have he problem of rounding error. The soluion obained by using hese mehods show heir applicabiliy, accuracy and efficiency in solving many nonlinear problems in physics, engineering and various branches of mahemaics [1]. The HAM was developed by Liao [1] who uilized he idea of homoopy in opology. Since hen numerous auhors have used HAM o solve various differenial equaions bu as far as we are aware sysems 1 and have no been solved by HAM. HAM has an advanage over perurbaion mehods ha i does no depend on small or large parameers. Compared wih oher analyical mehods such as ADM, VIM, HAM allows for fine-uning of convergence region and rae of convergence by allowing an auxiliary parameer o vary [1, 6]. This paper is organized as follows: In secion, we presen a descripion of he HAM, which has been described by ohers, in paricular [, 3, 5]. In secion 3, we employ he HAM for solving wo examples of he new coupled MKdV sysems 1 and and compare he approximae soluion obained wih he exac soluion. Finally, in secion 4, we give he conclusions of he sudy.. A descripion of he HAM To apply he HAM, he new coupled MKdV sysem 1 is considered. The following deformaion equaion was consruced by Liao [1] 1 pl u [φx, ; p u x, ] = p H 1 x, N 1 [φx, ; p], 3 1 pl v [ϕx, ; p v x, ] = p H x, N [ϕx, ; p], 4 1 pl z [ψx, ; p z x, ] = p H 3 x, N 3 [ψx, ; p], 5 where N 1, N and N 3 are hree nonlinear operaors, x and denoe he independen variables, H 1 x,, H x, and H 3 x, are auxiliary funcion, p [, 1] is he embedding parameer, is an auxiliary parameer, u x,, v x, and z x, are iniial guesses of ux,, vx, and zx,, respecively. The funcions φx, ; p, ϕx, ; p and ψx, ; p are known funcions and L u, L v and L z are auxiliary operaors ha are defined as follows which saisfies L u [ux, ] = u, L v[vx, ] = v, L z[zx, ] = z, L u [c 1 x] =, L v [c x] =, L z [c 3 x] =, where c 1 x, c x and c 3 x are inegral consans. When p = and p = 1, we ge φx, ; = u x,, ϕx, ; = v x,, ψx, ; = z x,, φx, ; 1 = ux,, ϕx, ; 1 = vx,, ψx, ; 1 = zx,. Hence as p increase from o 1, he soluion of coupled MKdV sysem 1 will vary from he iniial guesses u x,, v x, and z x, o he exac soluion ux,, vx, and zx,. Expanding φx, ; p, ϕx, ; p and ψx, ; p as a Taylor series wih respec o p, gives φx, ; p = φx, ; + p m m φx, ; p m! p m p=,

3 14 TWMS JOUR. PURE APPL. MATH., V.3, N.1, 1 Thus where ϕx, ; p = ϕx, ; + ψx, ; p = ψx, ; + φx, ; p = u x, + ϕx, ; p = v x, + ψx, ; p = z x, + p m m ϕx, ; p m! p m p=, p m m ψx, ; p m! p m p=. u m x, p m, 6 v m x, p m, 7 z m x, p m, 8 u m x, = 1 m φx, ; p m! p m p=, 9 v m x, = 1 m ϕx, ; p m! p m p=, 1 z m x, = 1 m ψx, ; p m! p m p=. 11 According o [4], he suiable choice of he iniial condiion u x,, v x, and z x,, he auxiliary linear parameer L u, L v and L z, he non-zero auxiliary parameer and he auxiliary funcion H 1 x,, H x, and H 3 x, will guaranee ha 1 The soluion φx, ; p, ϕx, ; p ψx, ; p of he zero-order deformaion equaion 3-5 exiss for all p [, 1]. The deformaion derivaive m φx, ; p p m p=, m ϕx, ; p p m p= and m ψx, ; p p m p= exiss for m = 1,, The power series 6-8 converges a p = 1. If p = 1 φx, ; 1 = ux, = u x, + ϕx, ; 1 = vx, = v x, + ψx, ; 1 = zx, = z x, + u m x,, 1 v m x,, 13 z m x,, 14 which mus be one of he soluions of new coupled MKdV sysem 1. According o he definiion 9-11, he governing equaions can be deduced from he zero-order deformaion equaions 3-5. For furher analysis, he vecors u n x, = {u x,, u 1 x,,..., u n x, }, v n x, = {v x,, v 1 x,,..., v n x, }, z n x, = {z x,, z 1 x,,..., z n x, },

4 M. GHOREISHI, A.I.B.MD. ISMAIL, A. RASHID: COUPLED MODIFIED KDV SYSTEM are defined. Differeniaing equaions 3-5 m-imes wih respec o he parameer p and hen dividing by m! and seing p =, gives he linear equaions [1] wih he iniial condiions where and we have L u [u m x, χ m u x, ] = R 1m u, 15 L v [v m x, χ m v x, ] = R m v, 16 L z [z m x, χ m z x, ] = R 3m z, 17 R 1m u = R m v = R 3m z = u m x, =, v m x, =, z m x, =, 1 m 1! N 1 φx, ; p p, 18 1 m 1! N ϕx, ; p p, 19 1 m 1! N 3 ψx, ; p p, χ m = {, m 1, 1, m > 1. Now, he soluion of he m-order deformaion equaions for m 1 becomes u m x, = χ m u x, + v m x, = χ m v x, + z m x, = χ m z x, + H 1 x, R 1m u d, 1 H x, R m v d, H 3 x, R 3m z d. 3 The deailed analysis of he convergence of he HAM is discussed by Liao in [11]. We noe ha he HAM only uiliies he iniial and makes no use of he boundary condiions. 3. Numerical soluions In his secion, he HAM will be demonsraed on examples of new coupled MKdV. For our numerical compuaion, le he expression ψ m x, = k= u k x,, 4 denoe he m-erm HAM approximaion o ux,. We compare he approximae analyical soluion is obained using HAM for our new coupled MKdV wih he exac soluion. We define E m x, o be he absolue error beween he exac soluion and m-erm approximae HAM soluion ψ m x, as follows E m x, = ux, ψ m x,. 5

5 16 TWMS JOUR. PURE APPL. MATH., V.3, N.1, 1 Example 3.1. For he firs example, we presen he new coupled MKdV equaions wih analyical soluion o show he efficiency of HAM, which has been described in he secion. We consider he sysem 1 wih iniial condiions as follows [7, 9] ux, = anh x, vx, = 1 1 anh x, 4 zx, = anh x. I can be verified ha he exac soluions of his example are ux, = anh x 11, vx, = anh x 11, zx, = anh x 11. We sar wih he following zeroh-componens u x, = anh x, 6 v x, = 1 1 anh x, 4 7 z x, = anh x. 8 According o secion, we can define he operaors N 1, N and N 3 as N 1 [φ] = φ 1 φ xxx + 3φ φ x 3 ϕψ xx 3φϕψ x, 9 N [ϕ] = ϕ + ϕ xxx + 3ϕφ x x + 3ϕ ψ x 6φϕφ x 3φ ϕ x, 3 N 3 [ψ] = ψ + ψ xxx + 3ψφ x x + 3ψ ϕ x 6φψφ x 3φ ψ x, 31 Now, we can obain he zeroh- where φ = φx, ; p, ϕ = ϕx, ; p and ψ = ψx, ; p. deformaion equaion as R 1m u = u 1 u xxx i= v i z i R m v = v + v xxx i= v i i k= xx v k z i k x 6 3 i= i= i= i= i= i k= i k= v i u i x i k= i k= u k u i k x v k z i k x + v k u i k x u k v i k x, x, 3 33

6 and M. GHOREISHI, A.I.B.MD. ISMAIL, A. RASHID: COUPLED MODIFIED KDV SYSTEM R 3m z = z + z xxx i= z i i k= z k v i k x 6 3 i= i= i= z i u i x i k= i k= x + z k u i k x u k z i k x We sar wih he iniial condiions 6-8. By means of he 1-3, we can obain direcly he oher componens of he HAM soluion in he series form of Thus, we have and u 1 x, = 3 1 sec h x anh x 1 anh x 4 sech x 1 anh x 4 v 1 x, = 3 sec h x 1 anh x anh x 1 + anh x 1 4 sec h x anh x1 + anh x + 3 sec h x1 + anh x 3 sec h4 x sec h x 1 anh x anh x sec h x sec h4 x sec h x 1 anh x anh x sec h4 x 4 sec h x... z 1 x, = sec h 4 x 3 4 sec h x anh x 3 sec h x anh x1 + anh x + 3 sec h x 1 + anh x 4 sec h x anh x sec h4 x sec h x anh x anh x. and so on. Thus ux, and vx, can be wrien as follows ux, = u n x, = u x, + u 1 x, + u x, +..., vx, = n= v n x, = v x, + v 1 x, + v x, n= Table 1, and 3 show he absolue error beween soluion obained using HAM wih five erms and he exac soluion for ux,, vx, and zx,, respecively, a =.7. The errors are very small in hese ables. The resuls show ha he homoopy analysis mehod is very useful o obain explici approximae analyical soluions. A very good approximaion wih he acual soluion of he equaions were achieved by using five erms only. We have ascerained ha he overall errors can be made smaller by adding new erms of he HAM series Table 1: Absolue error E 5 for variables x and a =.7 for ux,. /x

7 18 TWMS JOUR. PURE APPL. MATH., V.3, N.1, 1 Table : Absolue error E 5 for variables x and a =.7 for vx,. /x Table 3: Absolue error E 5 for variables x and a =.7 for zx,. /x Absolue error u forħ.7 a x 15 Absolue error v forħ.7 a x E E Figure 1. The absolue error beween HAM soluion and he exac soluion for =.7 a x = Absolue error u forħ.7 a x E Figure. The absolue error beween HAM soluion and he exac soluion for =.7 a x = 15. In figure 1 and, he absolue error beween soluion obained using HAM wih five erms and he exac soluion have been ploed for ux,, vx, and zx, for case of =.7 a x = 15. Similarly, we show his error in figure 3 and 4 a x = 15. In figure 5 and 6, we have shown he -curve for u,, v, and z,. As we know, he auxiliary parameer can be employed o adjus he convergence region of he homoopy analysis soluion [11]. According o hese -curve, i is easy o discover he valid region of which corresponds o he line segmen

8 M. GHOREISHI, A.I.B.MD. ISMAIL, A. RASHID: COUPLED MODIFIED KDV SYSTEM nearly parallel o he horizonal axis. From figures 5 and 6, i is clear ha he HAM series is convergen when 1.4 < <.6. Absolue error u forħ.7 a x 15 Absolue error u forħ.7 a x E E Figure 3. The absolue error beween HAM soluion and he exac soluion for =.7 a x = 15. Absolue error u forħ.7 a x E Figure 4. The absolue error beween HAM soluion and he exac soluion for =.7 a x = 15. ħ curve for u, 15 u, ħ Figure 5. The -curve for u, given by he 5-order HAM approximaion soluion when H 1x, = 1.

9 13 TWMS JOUR. PURE APPL. MATH., V.3, N.1, 1 ħ curve for v, ħ curve for z, 4 4 v, 6 8 z, ħ ħ Figure 6. The -curve for v, and z, given by he 5-order HAM approximaion soluion when H x, = H 3 x, = 1. Example 3.. Consider he coupled MKdV equaion wih he iniial condiions [8, 13] ux, = k anhkx, vx, = 1 4k + λ k anh kx. I can be verified ha he exac soluions of his sysem are ux, = k anh kx k k + 3λ, vx, = 1 4k + λ k anh kx k k + 3λ. We sar wih he iniial condiions u x, = k anhkx, 35 v x, = 1 4k + λ k anh kx. 36 According o secion, we can define he operaors N 1 and N as N 1 [φ] = φ 1 φ xxx + 3φ φ x 3 ϕ xx 3φϕ x + 3λφ x 37 N [ϕ] = ϕ + ϕ xxx + 3ϕϕ x + 3φ x ϕ x 3φ ϕ x 3λϕ x, 38 where φ = φx, ; p and ϕ = ϕx, ; p. equaion 15 and 16 ha are Now, we can obain he zeroh-order deformaion R 1m u = u 1 u xxx v xx 3 i= i k= v i i= u k u i k x x + 3λu x, 39

10 M. GHOREISHI, A.I.B.MD. ISMAIL, A. RASHID: COUPLED MODIFIED KDV SYSTEM and R m v = v + v xxx +3 i= 3 i= v i v i x + 3 i k= x v i x i= u k v i k x 3λv x. 4 We sar wih he iniial condiions 35 and 36. By means of he 1 and, we can easily obain direcly he oher componen of he HAM soluion in he series form of 1 and 13. Thus, we have u 1 x, = [3λk sec h kx + 3k 4 sec h kx anh kx + 3k k sec h 4 kx 4k sec h kx anh kx k k3 sec h 4 kx + 4k 3 sec h kx anh kx 3 4k sec h kx anh kx + k sec h kx 1 λ + 4k k anh kx]. v 1 x, = [1λk 3 sec h kx anhkx 1k 5 sec h 4 kx anhkx + 1k 5 sec h kx anh 3 kx 1 sec h kx anhkx 1 4k + λ k anh kx k 16k 3 sec h 4 kx anhkx + 8k 3 sec h kx anh 3 kx] and so on. Thus ux, and vx, can be wrien as follows ux, = vx, = u n x, = u x, + u 1 x, + u x, +..., n= v n x, = v x, + v 1 x, + v x, n= Table 4 and 5 show he absolue error beween soluion obained using HAM wih four erms and he exac soluion for ux, and vx,, respecively, a = 1, k =.1 and λ =.1. The errors are very small in hese ables. The resuls show ha he homoopy analysis mehod is very useful o obain explici approximae analyical soluions. A very good approximaion wih he acual soluion of he equaions was achieved by using four erms only. We have ascerained ha he overall errors can be made smaller by adding new erms of he HAM series 1 and 13. Table 4: Absolue error E 4 for variables x and a = 1, for ux,. x/

11 13 TWMS JOUR. PURE APPL. MATH., V.3, N.1, Absolue error u forħ 1, k.1,λ.1 a x Absolue error u forħ 1, k.1,λ.1 a x E E Figure 7. The absolue error E 4 for = 1, k =.1, λ =.1 a x = 1. Absolue error u forħ 1, k.1,λ.1 a x Absolue error u forħ 1, k.1,λ.1 a x E E Figure 8. The absolue error E 4 for = 1, k =.1, λ =.1 a x =..5 ħ curve for u,.1 ħ curve for v.5, u,..1 v.5, ħ ħ Figure 9. The -curve for u, and v.5, given by he 4h-order HAM approximaion soluion when H 1x, = H x, = 1.

12 M. GHOREISHI, A.I.B.MD. ISMAIL, A. RASHID: COUPLED MODIFIED KDV SYSTEM Table 5: Absolue error E 4 for variables x and a = 1, for vx,. x/ In figures 7 and 8, we have displayed he absolue error beween soluion obained using HAM wih four erms and he exac soluion for = 1, k =.1, λ =.5 a x = 1 and x =, respecively. The -curves have been ploed in figure 9 for u, and v.5,. As we have expressed, i is easy o check ha he HAM soluion is convergen o he exac soluions when 1.4 < < Conclusion In his paper, we have illusraed how HAM can be used for he new coupled modificaion of KdV sysem. The mehod was esed on wo examples. The HAM soluion conains he auxiliary parameer provides a mehod o adjus and conrol he convergence region of he infinie series for large value of. The obained resuls show ha he HAM is a very accurae and effecive echnique for he soluion of he new MKdV sysem. References [1] Alomari, A.K., Noorani, M.S.M., Nazar, R., Li, C.P., 1, Homoopy analysis mehod for solving fracional Lorenz sysem, Communicaions in Nonlinear Science and Numerical Simulaion, 15, pp [] Alomari, A. K., Noorani, M. S. M., Nazar, R.,9, Comparison beween he homoopy analysis mehod and homoopy paerurbaion mehod o solve coupled Schrodinger-KdV equaion, Journal of Applied Mahemaics and Compuaion, 31, pp.1 1. [3] Alomari, A.K., Noorani, M.S.M., Nazar, R., 8, The homoopy analysis mehod for he exac soluions of he K,, Burgers and coupled Burgers equaions, Applied Mahemaical Sciences, 4, pp [4] Awawdeh, F., Adawi, A., Musafa, Z., 9, Soluions of he SIR models of epidemics using HAM, Chaos, Solions and Fracals, 4, pp [5] Baaineh, A.S., Noorani, M.S.M., Hashim, I., 9, On a new reliable modificaion of homoopy analysis mehod, Communicaions in Nonlinear Science and Numerical Simulaion, 14, pp [6] Baaineh, A.S., Noorani, M.S.M., Hashim, I., 9, Modified homoopy analysis mehod for solving sysems of second-order BVPs, Communicaions in Nonlinear Science and Numerical Simulaion, 14, pp [7] Fan, E.,, Using symbolic compuaion o exacly solve a new coupled MKdV sysem, Physics Leer A, 991, pp [8] Fan, E., 1, Solion soluions for a generalized HiroaSasuma coupled KdV equaion and a coupled MKdV equaion, Physics Leers A, 8, pp.18. [9] Inc, M., Cavlak, E., 8, On numerical soluions of a new coupled MKdV sysem by using he Adomian decomposiion mehod and He s variaional ieraion mehod, Physica Scripa, 78, pp.1 7. [1] Liao, S.-J., 199, The Proposed Homoopy Analysis Techniques for he Soluion of Nonlinear Problems, Ph.D.disseraion, Shanghai Jiao Tong Universiy, Shanghai, China. [11] Liao, S.-J., 3, Beyond Perurbaion: Inroducion o he Homoopy Analysis Mehod, CRC Press, Boca Raon, Chapman and Hall. [1] Rashid, A., Izani, A.M.I., 1, A Chebyshev specral collocaion mehod for he coupled nonlinear Schrdinger equaions, Inernaional Journal of Applied and Compuaional Mahemaics, 91, pp [13] Raslan, K.R., 4, The decomposiion mehod for a Hiroa-Sasuma coupled KdV equaion and a coupled MKdV equaion, Inernaional Journal of Compuer Mahemaics, 811, pp

13 134 TWMS JOUR. PURE APPL. MATH., V.3, N.1, 1 [14] Wu, Y.T., Geng, X.G., Hu, X.B., Zhu, S., 1999, A generalized HiroaSasuma coupled Korewegde Vries equaion and Miura ransformaions, Physics Leers A, 554-6, pp Seyed Mohammad Ghoreishi was born in Tehran-Iran. He obained his B.Sc. from Universiy of Sisan and Baluchesan in He received his M. Sc. from Universiy of Science and Technology, Iran in He worked as a lecurer a several Universiies of Iran in 1997 o 7. He received his Ph.D. in Applied Mahemaics from School of Mahemaical Sciences, Universii Sains Malaysia, Penang, Malaysia in 11. He sudied he numerical and approximae analyical soluion of parabolic equaions wih nonlocal boundary condiions. Ahmad Izani Md. Ismail joined he School of Mahemaical Sciences, Universii Sains Malaysia, in 1984 and was appoined a Professor in 11. He has augh undergraduae and graduae courses on numerical analysis, mahemaical modeling, fluid mechanics and compuer programming. His primary research ineres is in mahemaical modeling and he is presenly supervising and co-supervising 8 Ph.D. sudens. Abdur Rashid joined he Deparmen of Mahemaics, Gomal Universiy, Dera Ismail Khan, Pakisan in He compleed Ph.D. from Shanghai Universiy of Science Technology, Shanghai, P. R. China, in He was appoined a Professor a Deparmen of Mahemaics, Gomal Universiy, Dera Ismail Khan, Pakisan, in 8. His research ineres is Numerical Soluion of Parial Differenial Equaions by Specral mehods.