-Expansion Method For Generalized Fifth Order KdV Equation with Time-Dependent Coefficients

Transcription

1 Math. Sci. Lett. 3 No Mathematical Sciences Letters An International Journal eneralized -Expansion Method For eneralized Fifth Order KdV Equation with Time-Dependent Coefficients Lakhveer Kaur Department of Mathematics Amity Institute of Applied Sciences Amity University Noida India Received: 6 Apr. 04 Revised: 6 May 04 Accepted: 8 May 04 Published online: Sep. 04 Abstract: In this work the generalized -expansion method is presented to seek some new exact s for generalized fifth order KdV equation with time-dependent coefficients. As a result more explicit traveling wave s involving arbitrary parameters are found out which are expressed in terms of hyperbolic functions the trigonometric functions and rational functions. When the parameters are taken special values different types of wave s can be obtained. Keywords: eneralized Fifth Order KdV Equation with Time-Dependent Coefficients eneralized Solutions. -Expansion method Exact Introduction Many phenomena in physics and other fields are often described by nonlinear partial differential equations PDEs. When we want to understand the physical mechanism of natural phenomenon described by nonlinear PDEs exact s for these nonlinear PDEs have to be explored. Thus investigation of exact s of nonlinear PDEs has become one of the most important topics in mathematical physics. Powerful methods that make it possible to generate exact s to nonlinear equations have emerged from the literatures in the past decades. Among them are the tanh-sech method [ ] extended tanh method [3 4] sine-cosine method [5 6] Hirota method [7 8] homogeneous balance method [9 0] Jacobi elliptic function method [ ] F-expansion method [3 4] homotopy perturbation method [5 6] variational iteration method [7 8] Lie symmetry analysis [90] and so on. With the use of the methods mentioned many exact s including the solitary wave s shock wave s and periodic wave s etc. of NLPDEs were obtained. Among the other methods the generalized -expansion method [ ] has been proposed to construct traveling wave s for nonlinear partial differential equations with variable coefficients. The method is based on the homogeneous balance principle and linear ordinary differential equation LODE theory. Being concise and straightforward this method has been applied to various nonlinear partial differential equations with variable coefficients to discover some more general s with some free parameters [ ]. Also it handles nonlinear partial differential equations in a direct manner with no requirement for initial/boundary conditions or initial trial function at the outset. The general fifth order KdV equation reads u t auu xxx bu x u xx cu u x u xxxxx = 0 where ab and c are real constants. This includes the Lax [30] Swada-KoteraSK [3 3] Kaup-Kupershmidt KK [333435] and Ito equations [36]. As the constants ab and c take different values The properties of eq drastically change. For instance the lax equation with a=0 b=0 and c=30 and The SK equation where a=b=c=5 are completely integrable. Corresponding author lakhveerscholar@gmail.com c 04 NSP

2 56 L. Kaur: eneralized -Expansion Method... These two equations have N-soliton s and an infinite set of conservation laws. The KK equation with a=0b= 5 and c=0 is also known to be integrable [34] and to have bilinear representations [37] but the explicit form of its N-soliton is apparently not known. A fourth equation in this class is the Ito equation with a = 3 b = 6 and c = which is not integrable but has a limited number of conservation laws [36]. The variable-coefficient version of nonlinear equations can be considered as generalization of the constant coefficients equations as there are choices for parameters. Due to this much attention has been paid to the study of nonlinear equations with variable coefficients to obtain exact s and some recent contributions can be found in [9 0]. These exact s provide much information about nonlinear phenomena and well-described various aspects of the physical phenomena. In the present paper the general fifth order KdV equation with time-dependent coefficients u t δtuu xxx σtu x u xx βtu u x ρtu xxxxx = 0 which is an important mathematical model in nonlinear physics has been studied for exploring exact s by using generalized -expansion method. Our interest in the present paper is to search for the exact s for equation. In this direction the layout of the paper is as follows: In Section we have summarized the generalized -expansion method. Then this method is applied to variable coefficient version of general fifth order KdV equation in section 3 and a rich variety of exact s are obtained which included the hyperbolic functions the trigonometric functions and rational functions. Finally the conclusions and remarks are given in last section. Description of The eneralized -Expansion Method [3] In this section we have described the generalized -expansion method for finding out some new exact s of nonlinear evolution equations. Consider the nonlinear partial differential equation in the following form: Fuu t u x u y u z...u xt u yt u zt u tt...=0 3 with independent variables X = x y z... t and dependent variable u = ux y z...t is an unknown function F is a polynomial in u = uxyz...t and its various partial derivatives in which the highest order derivatives and nonlinear terms are involved. There are following three main steps of the generalized -expansion method. Step : Suppose that the of Eq. 3 can be expressed by a polynomial in u=α 0 X m i= as follows: i α i X α m X 0 4 where α 0 Xα i Xi=...m and θ = θx are all functions of X to be determined later and = θ satisfies following equation θ θ µθ= 0 5 where θ = ptxqt pt and qt are functions to be determined. Step : In order to determine u explicitly the positive integer m is determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms of u appearing in Eq. 3. Step 3: Substitute 4 into Eq. 3 along with Eq. 5 and collect all terms with the same order of together the left hand side of Eq. 3 is converted into a polynomial in. Then by setting each coefficient of this polynomial to zero we derive a set of over-determined partial differential equations for α 0 Xα i X and ζ. Step 5: The expressions for α 0 Xα i X and ζ can be found by solving the system of partial differential equations obtained in above step and hence the s of Eq. 3 can be derived depending on since the s of Eq. 5 have been well known to us depending on the sign of the discriminant = 4µ. 3 Application of eneralized -Expansion Method to eneralized Fifth Order KdV Equation with Time-Dependent Coefficients In this section some exact s of generalized fifth order KdV equation with time-dependent coefficients have been furnished by generalized -expansion method. As a result hyperbolic function s trigonometric function s and rational s with various parameters are obtained. According to the method described in section the positive integer m is determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms of u in Equation and we found m=. Thus the of Eq according to Eq 4is as follows: u=α 0 tα t α t 6 c 04 NSP

3 Math. Sci. Lett. 3 No / 57 Substituting 6 into and using 5 collecting all terms with the same order of together the left-hand side of Eq. is converted into polynomial in x j j = 0. Setting each coefficient of this polynomial to zero we derived a system of overdetermined differential equations for α 0 tα tα tα 3 tα 4 t pt and qt. Solving this set of equations we have following results: Case : pt=c α 0 t=c α t= α t= σt= 4 δt βt= 6δtc ρt= δt 48c qt= µ 6c 8c µ 5 6 µ c δt dt c 4 7 where c c and c 4 are arbitrary constants. Substituting the general s of 5 into 6 and using 7 we have three types of exact s of as follows: When 4µ > 0 we have obtained hyperbolic function in the form uxt=c 4 µa sinhζa coshζ a sinhζa coshζ 4 µa sinhζa coshζ a sinhζa coshζ 8 where ζ = c x qt 4 µ and qt = µ 6c 8c µ 6 5 µ c δt dt c 4. When 4µ < 0 we have trigonometric function uxt=c 4 µ a sinζa cosζ a sinζa cosζ 4 µ a sinζa cosζ a sinζa cosζ 9 where ζ = c xqt 4 µ and qt = µ 6c 8c µ 5 6 µ c δt dt c 4. Fig. : raphical representation of 9 for = µ = 4c = c = = c 4 = a = a = and δt= When 4µ = 0 we get rational as follows: A uxt=c A A c xqt A 0 c 3 A A c xqt where A A a a are arbitrary constants and qt= c 3c δt dt c 4. Case : Fig. : raphical representation of 8 for = 4 µ = c = c = = c 4 = 4a = a = and δt= t pt=c α 0 t= c 3 c µ α t=c α t=c σt= 3δtc 3δtc c c βt= 44ρtc4 3δtc c c qt= 6 c3 4µ 3ρtc δtc dt where c c and are arbitrary constants. Consequently we have the following three types of exact s of : c 04 NSP

4 58 L. Kaur: eneralized -Expansion Method... where A A a a and are arbitrary constants. Case 3: Fig. 3: raphical representation of 0 for = 5c = c = = c 4 = A = 4A = and δt=t When 4µ > 0 we have obtained hyperbolic function in the form uxt=c 4 µa sinhζa coshζ a sinhζa coshζ 4 µa sinhζa coshζ a sinhζa coshζ 3 c µ c where ζ = c x qt 4 µ and qt= 6 c3 4µ 3ρtc δtc dt. When 4µ < 0 we have trigonometric function uxt=c 4 µ a sinζa cosζ a sinζa cosζ 4 µ a sinζa cosζ a sinζa cosζ 3 c µ c 3 where ζ = c x qt 4 µ and qt= 6 c3 4µ 3ρtc δtc dt. When 4µ = 0 we get rational as follows: uxt= 4 c c A c A A c x 4 A A A c x qt= 6δtc 8δtµc δtc 3 µ dt 6ρtc µ 8ρtc µ dt ρtc 4 δtc 3 µ δtc dt c 4 pt=c α 0 t=c α t= α t= σt= δt 60ρtc βt= 6δtc 5 where c c and are arbitrary constants. Again substituting Equations 5 together with the general Equation 5 into the Equation 6 yields the following exact s of Equation : When 4µ > 0 we have obtained hyperbolic function in the form uxt=c 4 µa sinhζa coshζ a sinhζa coshζ 4 µa sinhζa coshζ a sinhζa coshζ 6 where ζ = c x qt 4 µ and qt= 6δtc 8δtµc δtc 3 µ dt 6ρtc µ 8ρtc µ dt ρtc 4 δtc 3 µ δtc dt c 4. When 4µ < 0 we have trigonometric function uxt=c 4 µ a sinζa cosζ a sinζa cosζ 4 µ a sinζa cosζ a sinζa cosζ 7 where ζ = c x qt 4 µ and qt= 6δtc 8δtµc δtc 3 µ dt 6ρtc µ 8ρtc µ dt ρtc 4 δtc 3 µ δtc dt c 4. When 4µ = 0 we get rational as follows: uxt=c A A A c xqt A A A c xqt 8 where A A a a and are arbitrary constants and qt= 6δtc 3δtc 3 8 δt 4 c 4. c 04 NSP

5 Math. Sci. Lett. 3 No / 59 Case 4: qt= c 44 4µ 44ρtc 4 δtc c βtc dt pt=c α 0 t= c 3 c µ α t=c α t=c σt= 6 δtc c 360ρtc 4 βtc c c 9 where c c and are arbitrary constants. By using the general s of 5 and 9 into 6 we found three types of exact s of as follows: When 4µ > 0 we have obtained hyperbolic function in the form uxt=c c 4 µa sinhζa coshζ a sinhζa coshζ 4 µa sinhζa coshζ a sinhζa coshζ c 3 c µ 0 where ζ = c x qt 4 µ and qt = c 44 4µ 44ρtc 4 δtc c βtc dt. When 4µ < 0 we have trigonometric function uxt=c c 4 µ a sinζa cosζ a sinζa cosζ 4 µ a sinζa cosζ a sinζa cosζ c 3 c µ where ζ = c xqt 4 µ and qt = c 44 4µ 44ρtc 4 δtc c βtc dt. When 4µ = 0 we get rational as follows: uxt=c A A A c xk A A A c xk where A A a a k and are arbitrary constants. Case 5: qt= 7 9 pt=c α t=c δtc 4µ 5δtc 87c δt 4 46 δtc 4c δt 4 46 α 0 t= 4δtc µδtc 6c µδt σt= ρt= 3 48c δt c δt 3δtc 56c δt 4 46 c 4 46 c 4µ c δt 4µ βt= 48c δtc δtc c c c δt α t=c dt 4c δt where c c and are arbitrary constants. The following three types of s of are found by substituting the general s of 5 into 6 and using 9: When 4µ > 0 we have obtained hyperbolic function in the form uxt=c 4µ 4 µa sinhζa coshζ a sinhζa coshζ c 4 µa sinhζa coshζ a sinhζa coshζ α 0 t 4 where ζ = c x qt 4 µ and qt = δtc 4µ 5δtc 87c δt 4 46 dt 7 9 δtc 4c δt 4 46 When 4µ < 0 we have trigonometric function uxt=c 4µ 4 µ a sinζa cosζ a sinζa cosζ 4 µ a sinζa cosζ c a sinζa cosζ α 0 t 5 where ζ = c xqt 4 µ and qt = δtc 4µ 5δtc 87c δt 4 46 dt 7 9 δtc 4c δt When 4µ = 0 we get rational as follows: uxt= 4 c c A c A A c xk A A A c xk 6 where A A a a k and are arbitrary constants. c 04 NSP

6 60 L. Kaur: eneralized -Expansion Method... 4 Conclusions The generalized -expansion method has been successfully used to obtain some new exact s of generalized fifth order KdV equation with time-dependent coefficients. We found a rich variety of exact s which include hyperbolic trigonometric and rational functions involving arbitrary parameters. Also We have presented the graphical representation of obtained s so that they can depict the importance of each obtained and physically interpret their importance. The free parameters c c c 4 k and especially the arbitrary function δt βt ρt in various s can make us discuss the behaviors of s and also provide us enough freedom to construct s that may be related to real physical problem. Note that the nonlinear evolution equation proposed in the present paper is difficult and more general. Therefore the s of generalized fifth order KdV equation with time-dependent coefficients equation in this paper have many potential applications in physics. The authors are grateful to the anonymous referee for a careful checking of the details and for helpful comments that improved this paper. References [] W. Malfliet Solitary wave s of nonlinear wave equations Am. J. Phys [] A. M. Wazwaz The tanh method for travelling wave s of nonlinear equations Appl. Math. Comput [3] S. A. El-Wakil and M. A. Abdou New exact travelling wave s using modified extended tanh-function method Chaos Soliton Fract [4] A. M. Wazwaz The extended tanh method for new soliton s for many forms of the fifth-order KdV equations Appl. Math. Comput [5] A. M. Wazwaz A sine-cosine method for handling nonlinear wave equations Math. Comput. Model [6] A. Bekir New solitons and periodic wave s for some nonlinear physical models by using sine-cosine method Phys. Scr [7] Wazwaz A. M. The Hirota s direct method for multiplesoliton s for three model equations of shallow water waves Appl. Math. Comput [8] A. M. Wazwaz Multiple-soliton s for the KP equation by Hirota s bilinear method and by the tanh-coth method Appl. Math. Comput [9] E. Fan and H. Zhang A note on the homogeneous balance method Phys. Lett. A [0] M. L. Wang Exact s for a compound KdVBurgers equation Phys. Lett. A [] C. Q. Dai and J. F. Zhang Jacobian elliptic function method for nonlinear differential-difference equations Chaos Soliton Fract [] E. Fan and J. Zhang Applications of the Jacobi elliptic function method to special-type nonlinear equations Phys. Lett. A [3] M. A. Abdou The extended F-expansion method and its application for a class of nonlinear evolution equations Chaos Soliton Fract [4] J. L. Zhang M. L. Wang Y. M. Wang and Z. D. Fang The improved F-expansion method and its applications Phys. Lett. A [5] D. D. anji and A. Sadighi Application of He s homotopyperturbation method to nonlinear coupled systems of reaction-diffusion equations Int. J. Nonlinear Sci. Numer. Simul [6] T. Ozis and A. Yildirim Traveling wave of Kortewegde Vries equation using He s homotopy perturbation method Int. J. Nonlinear Sci. Numer. Simul [7] J. H. He Some asymptotic methods for strongly nonlinear equations Int. J. Mod. Phys. B [8] J. H. He and X. H. Wu Construction of solitary and compaction-like by variational iteration method Chaos Soliton Fract [9] S. Kumar K. Singh and R. K. upta Painlevé analysis Lie symmetries and exact s for -dimensional variable coefficients Broer-Kaup equations Commu. Nonlinear Sci. Numer. Simul [0] L. Kaur and R. K. upta Kawahara equation and modified Kawahara equation with time dependent coefficients: symmetry analysis and generalized expansion method Math. Meth. Appl. Sci [] V. Kumar R. K. upta and R. Jiwari Painlevé Analysis Lie Symmetries and exact s for variable coefficients Benjamin-Bona-Mahony-Burger BBMB equation Commun. Theor. Phys [] V. Kumar R. K. upta and R. Jiwari Lie group analysis numerical and nontraveling Wave Solutions for the - Dimensional Diffusion-Advection equation with variable coefficient Chin. Phys. B [3] M. L. Wang X. Li and J. Zhang The -expansion method and travelling wave s of nonlinear evolution equations in mathematical physics Phys. Lett. A [4] H. Naher and F. A. Abdullah The improved - expansion method to the -dimensional breaking soliton equation J. Comput. Anal. Appl [5] H. Naher and F. A. Abdullah The Basic -Expansion Method for the Fourth Order Boussinesq Equation Appl. Math [6] H. Naher and F. A. Abdullah New approach of - expansion method and new approach of generalized expansion method for nonlinear evolution equation AIP Adv [7] S. Zhang L. Dong J. M. Ba and Y. N. Sun The - -expansion method for nonlinear differential difference equations Phys. Lett. A [8] H. Naher and F. A. Abdullah The modified Benjamin- Bona-Mahony equation via the extended generalized Riccati c 04 NSP

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