Application of Kudryashov method for the Ito equations

Transcription

1 Avilble t Appl. Appl. Mth. ISSN: Vol. 12, Issue 1 June 2017, pp Applictions nd Applied Mthemtics: An Interntionl Journl AAM Appliction of Kudryshov method for the Ito equtions Mozhgn Akbri Deprtment of Pure Mthemtics Fculty of Mthemticl Sciences University of Guiln P.O. Box Rsht, Irn m kbri@guiln.c.ir Received: Jnury 29, 2015; Accepted: Februry 20, 2017 Abstrct In this present work, the Kudryshov method is used to construct exct solutions of the 1+1- dimensionl nd the 1+2-dimensionl form of the generlized Ito integro-differentil eqution. The Kudryshov method is powerful method for obtining exct solutions of nonliner evolution equtions. This method cn be pplied to non-integrble equtions s well s integrble ones. Keywords: Kudryshov Method; The 1+1-dimensionl Form of the Generlized Ito Integro-differentil Eqution; The 1+2-dimensionl Form of the Generlized Ito Integro-differentil Eqution MSC 2010 No.: 34K30, 35R09, 35R10, 47G20 1. Introduction The study of nonliner evolution equtions NLEE hs been going on for the pst few decdes, see Ebdi et l. 2012, Ceser nd Gomez 2010, Li nd Zeng 2007, Li nd Zho 2009, Liu 2000, nd Wzwz During this time, there hs been mesurble progress tht hs been mde. There re lots of equtions tht hve been integrted. There re vrious methods of integrbility tht hve been developed so fr. In ddition to NLEEs, there hs been growing 136

2 AAM: Intern. J., Vol. 12, Issue 1 June interest in the nonliner integro-differentil evolution equtions. Some of these commonly studied integro-differentil evolution equtions re the Ito eqution, the generlized shllow wter wve eqution nd mny others. There re vrious nlyticl methods of solving these NLEEs tht hs lso been developed in the pst couple of decdes. Some of these methods re the expfunction method see He nd Wu 2006, Aminikhh et l. 2009, the F-expnsion method see Abdou 2007, Wng nd Li 2005, Ren nd Zhng 2006, the Jcobi elliptic function expnsion method see Di nd Zhng 2006, Fn nd Zhng 2002, Liu et l. 2001, the modified simplest eqution method see Zyed 2011, Vitnov et l. 2010, Vitnov 2011, Jwd et l. 2010, Akbri 2013, the first integrl method see Rsln 2008, Abbsbndy nd Shirzdi 2010, Feng 2002, Feng nd Wng 2003, the functionl vrible method see Zerrk et l. 2010, Zerrk nd Oumne 2010, Cevikel et l. 2012, nd mny others. In this pper, we propose Kudryshov method to construct exct trvelling wve solutions for nonliner evolution equtions see Kudryshov 2004, Kudryshov 1990, Rybov First, we reduce the nonliner evolution equtions to ODEs by trvelling wve vrible trnsformtion. Secondly, we suppose the solution cn be expressed in polynomil in vrible, where it stisfies the Riccti eqution. At the end, the degree of the polynomil cn be determined by the homogeneous blnce method, nd the coefficients cn be obtined by solving set of lgebric equtions. In this work, by using the Kudryshov method, we im to investigte the 1+1-dimensionl nd the 1+2-dimensionl form of the generlized Ito integro-differentil eqution. This pper is orgnized s follows: In Section 2, we describe briefly the Kudryshov method. In Sections 3 nd 4, we pply the proposed method to solve the 1+1-dimensionl nd the 1+2- dimensionl form of the generlized Ito integro-differentil eqution. In Section 5, the conclusion will be presented. 2. Modifiction of truncted expnsion method We consider generl nonliner prtil differentil eqution PDE in the form P u, u t, u x, u tt, u xt, u xx,... = 0. 1 Using trveling wve ux, t = Uξ, ξ = kx ωt crries eqution 1 into the following ODE: P U, ωu, ku, k 2 U,... = 0. 2 The min steps of the modifiction of the truncted expnsion method re the following: Step 1. Determintion of the dominnt term with highest order of singulrity. To find dominnt terms, we substitute U = ξ p, 3 to ll terms of eqution 2. Then we compre degrees of ll terms of eqution 2 nd choose two or more with the lowest degree. The mximum vlue of p is the pole of eqution 2 nd we denote it s N. This method cn be pplied when N is integer. If the vlue N is non-integer, one cn trnsform the eqution studied.

3 138 Mozhgn Akbri Step 2. We look for exct solution of eqution 2 in the form Uξ = N i Q i ξ, 4 i=0 where i i = 0, 1,..., N re constnts to be determined lter, such tht N 0 while Qξ hs the form 1 Qξ = 1 + d expξ, 5 which is solution to the Riccti eqution where d is rbitrry constnt. Q ξ = Q 2 ξ Qξ, Step 3. We cn clculte the necessry number of derivtives of the function U. It is esy to do using Mple or Mthemtic pckge. Using the cse N = 1 we hve some derivtives of the function Uξ in the form U = Q, U ξ = 1 Q + 1 Q 2, U ξξ = 1 Q 3 1 Q Q 3, 6 U ξξξ = 1 Q Q 2 12Q Q 4. Step 4. We substitute expressions given by equtions 4-6 in eqution 2. Then we collect ll terms with the sme powers of function Qξ nd equte the expressions to zero. As result we obtin lgebric system of equtions. Solving this system we get the vlues of unknown prmeters. 3. New exct trvelling wve solution of the 1+1-dimensionl form of the generlized Ito integro-differentil eqution The 1+1-dimensionl form of the generlized Ito integro-differentil eqution tht is going to be studied in this section is given by q tt + q xxxt + 2q x q t + qq xt + q xx x q t dx = 0, 7 Here, in 7, q is the dependent vrible while x nd t re the independent vribles. The coefficient is constnt. Eqution 7 cn reduced to v ttx + v xxxxt + 2v xx v xt + v x v xxt + v xxx v t = 0, 8 using the potentil q = v x. Eqution 8 is converted to the ODE c 2 eu ce 4 u v + 2ce 3 u u ce 3 u u ce 3 u u = 0. 9

4 AAM: Intern. J., Vol. 12, Issue 1 June Equivlently, cu e 3 u v e 2 u 2 = 0, 10 by the wve vribles v = uξ, ξ = ex ct, where primes denote the derivtives with respect to ξ, nd e, c re rel constnts to be determined lter. Eqution 10 is then integrted twice. This converts it to cu e 3 u e 2 u 2 = The pole order of eqution 11 is N = 1. So we look for the solution of eqution 11 in the uξ = Q. 12 Substituting eqution 12 into eqution 11, we obtin the system of lgebric equtions in the Q 1 : c 1 + e 3 1 = 0, Q 2 : c 1 7e 3 1 e = 0, Q 3 : 12e e = 0, Q 4 : 6e 3 1 e = 0. Solving the lgebric equtions bove, this yields: 1 = 6e, c = e3. 13 From 12 nd 13, we obtin the following trvelling wve solution of eqution 11, uξ = 0 6e where 0 nd d re rbitrry constnts. Then the exct solution to eqution 7 is written s qx, t = 6e2 d d expξ expex e 3 t 1 + d expex e 3 t 2, New exct trvelling wve solution of the 1+2-dimensionl form of the generlized Ito integro-differentil eqution The 1+2-dimensionl form of the generlized Ito integro-differentil eqution to be studied in this section is given by q tt + q xxxt + 2q x q t + qq xt + q xx x q t dx + bq yt + dq xt = 0, 15 Here, in 15, q is the dependent vrible while x, y, nd t re the independent vribles. The coefficient, b, nd d re constnts. Eqution 15 cn reduced to v ttx + v xxxxt + 2v xx v xt + v x v xxt + v xxx v t + bv xyt + dv xxt = 0, 16

5 140 Mozhgn Akbri by using the potentil q = v x. Eqution 16 is converted to the ODE c 2 eu ce 4 u v + 2ce 3 u u ce 3 u u ce 3 u u cbefu cde 2 u = Equivlently, c bf deu e 3 u v e 2 u 2 = 0, 18 by the wve vribles v = uξ, ξ = ex + fy ct, where primes denote the derivtives with respect to ξ, nd e, f, nd c re rel constnts to be determined lter. The eqution 18 is then integrted twice. This converts it to c bf deu e 3 u e 2 u 2 = The pole order of eqution 19 is N = 1. So we look for solution of eqution 19 in the uξ = Q. 20 Substituting eqution 20 into eqution 19, we obtin the system of lgebric equtions in the Q 1 : c bf de 1 + e 3 1 = 0, Q 2 : c bf de 1 7e 3 1 e = 0, Q 3 : 12e e = 0, Q 4 : 6e 3 1 e = 0. Solving the lgebric equtions bove, this yields 1 = 6e, c = e3. 21 From 20 nd 21, we obtin the following trvelling wve solution of eqution 19 uξ = 0 6e where 0 nd d re rbitrry constnts. Then the exct solution to eqution 15 is written s qx, y, t = 6e2 d d expξ expex bf + de + e 3 t 1 + d expex bf + de + e 3 t 2, Conclusion Modifiction of the truncted expnsion method is pplied successfully for solving the Ito eqution, which is nonliner integro-differentil evolution eqution. Compred to the methods used before, one cn see tht this method is direct, concise nd effective. Moreover, the method cn lso be pplied to mny other nonliner evolution equtions.

6 AAM: Intern. J., Vol. 12, Issue 1 June Acknowledgments The uthor is very grteful to the referees for their vluble suggestions nd opinions. The uthor is thnkful to the Editor-in-Chief Professor Alikbr Montzer Hghighi for useful comments nd suggestions towrds the improvement of this pper. REFERENCES Abbsbndy, S. nd Shirzdi, A The first integrl method for modified Benjmin-Bon- Mhony eqution, Commun. Nonliner Sci. Numer. Simul., Vol. 15, pp Abdou, M. A The extended F-expnsion method nd its ppliction for clss of nonliner evolution equtions, Chos Solitons Frctls, Vol. 31, pp Akbri, M Exct solutions of the coupled Higgs eqution nd the Mccri system using the modifed simplest eqution method, Inf. Sci. Lett., Vol. 2, pp Aminikhh, H., Moosei, H. nd Hjipour, M Exct solutions for nonliner prtil differentil equtions vi Exp-function method, Numer. Methods Prtil Differ. Eqution, Vol. 26, No. 6, pp Ceser, A. nd Gomez, S New trveling wves solutions to generlized Kup-Kuperschmidt nd Ito equtions, Applied Mthemtics nd Computtion, Vol. 216, No. 1, pp Cevikel, A. C., Bekir, A., Akr. M. nd Sn, S A procedure to construct exct solutions of nonliner evolution equtions, Prmn-Journl of Physics, Vol. 79, No. 3, pp Di, C. Q. nd Zhng, J. F Jcobin elliptic function method for nonliner differentildifference equtions, Chos Solitons Frctls, Vol. 27, pp Ebdi, G., Kr, A. H., Petkovic, M. D., Yildirim, A. nd Bisws, A Solitons nd conserved quntities of the Ito eqution, Proceedings of the Romnin Acdemy, Vol. 13, No. 3, pp Fn, E. nd Zhng, J Applictions of the Jcobi elliptic function method to specil-type nonliner equtions, Phys. Lett. A, Vol. 305, pp Feng, Z. S The first integrl method to syudy the Burgers-KdV eqution, J. Phys. A. Mth. Gen., Vol. 35, pp Feng, Z. S. nd Wng, X. H The first integrl method to the two-dimensionl Burgers- KdV eqution, Phys. Lett. A, Vol. 308, pp He, J.H. nd Wu, X. H Exp-function method for nonliner wve equtions, Chos Solitons Frctls, Vol. 30, pp Jwd, A. J. M., Petkovic, M. D. nd Bisws, A Modified simple eqution method for nonliner evolution equtions. Appl Mth Comput., Vol. 217, pp Kudryshov, N. A Anlyticl theory of nonliner differentil equtions, Moscow Izhevsk: Institute of Computer Investigtions, pp Kudryshov, N. A Exct solutions of the generlized Kurmoto-Sivshinsky eqution, Phys. Lett. A, Vol. 147, pp Li, C. nd Zeng, Y Soliton solutions to higher order Ito eqution: Pfffin technique, Physics Letters A, Vol. 363, pp. 1-4.

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