) -Expansion Method for Solving (2+1) Dimensional PKP Equation. The New Generalized ( G. 1 Introduction. ) -expansion method

Transcription

1 ISSN (print, (online International Journal of Nonlinear Science Vol.4(0 No.,pp.48-5 The New eneralized ( -Expansion Method for Solving (+ Dimensional PKP Equation Rajeev Budhiraja, R.K. upta, S.S. Bhatia 3 Department of Mathematics Maharishi Marandeshwar University, Mullana, Ambala 3300 (Haryana, India,3 School of Mathematics and Computer Applications Thapar University, Patiala (Punjab, INDIA (Received November 0, accepted July 0 Abstract: A New eneralised ( /-Expansion Method is presented to derive traveling wave solutions for a class of nonlinear partial differential equations.as results, some new exact traveling wave solutions are obtained. Keywords:The (+ Dimensional PKP equation; exact traveling wave solution; the new ( /-expansion method Introduction The non linear phenomena exist in all the fields including either the scientific wor or engineering fields,such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical inematics,chemical physics, and so on. It is well nown that many non linear evolution equations (NLEEs are widely used to describe these complex phenomena. Research on solutions of NLEEs is popular. So, the powerful and efficient methods to find analytic solutions and numerical solutions of non linear equations have drawn a lot of interest by a diverse group of scientists. Many efficient methods have been presented as in [ 7]. Although there are many methods to construct exact solutions of non linear evolution equations such as the F-expansion method [8 0], Tanh-expansion method [, ], The Jacobi elliptic function [3, 4], The homogeneous balance method [5, 6], The Baclund transformation [7, 8], the original ( - expansion method [9] and so on. The objective of this article is to use the New generalized ( /-expansion method to find the exact solutions of non linear evolution equations via the (+ dimensional PKP equation [0]. The main idea of this method is that the traveling wave solutions of the non linear evolution equations can be expressed by polynomials in where = (ξ and based on a first order non linear ordinary differential equation = 5 i= hii with a fifth-degree non linear term, and = ( d dξ. The degree of these polynomial can be determined by considering the homogeneous balance between the highest order derivatives and the non linear terms appearing in the given non linear equations. The coefficients of these polynomials can be obtained by solving a set of algebraic equations resulted from the process of using the proposed method. This method will play an important role in expressing the traveling wave solutions in terms of hyperbolic, trigonometric and the rational functions for the non linear evolution equations in mathematical physics. The New eneralized ( /-expansion method used in this article can be applied to further non linear equations as the difference-differential equations. Description of the new generalized ( -expansion method Consider the non linear partial differential equation in the following form: F (u, u t, u x, u tt, u xt, u xx,... = 0, ( where u = u(x, t is unnown functions, and F is a polynomial in u(x, t and its partial derivatives. In the following,we give the main steps for solving ( using a New eneralized ( /-expansion method. Corresponding author. address: rajeevumarbudhiraja@gmail.com Copyright c World Academic Press, World Academic Union IJNS /637

2 ( R. Budhiraja, R.K. upta, S.S. Bhatia: The New eneralized -Expansion Method for Solving (+ Dimensional 49. Step The traveling wave variable u(x, y, t = u(ξ, ξ = ξ(x, t. ( The traveling wave variable ( permits us reducing ( to an ODE for u = u(ξ. Step F (u, u, u, u,... = 0 (3 Suppose the solution of ODE (3 can be expressed in by a polynomial in as follows: u(ξ = where = (ξ solution of of the first order non linear ODE in the form n i= ( i A i (4 = h 0 + h + h + h h h 5 5 (5 where A i, h 0, h, h, h 3, h 4, h 5 are constants to be detemined and A n 0 where n is called balance number..3 Step 3 The positive integer n can be determined by considering the homogeneous balance between the highest order derivatives and the non linear terms appearing in equation (3..4 Step 4 Substituting (4 into (3 and using the ODE (5, collecting all terms with the same order of together, we get a polynomial in. Equating each coefficient of these polynomial to zero, yeilds a algebraic equations, which can be solved to get A i. Since the general solution of equation (5 is well nown to us, then substituting A i and the general solutions of (5 into (4, we have traveling wave solutions of equation (. Remar It is well nown that (5 admits the follwing solutions. Theorem Suppose that h 0. (i If h 0 = 0, h = 0, h 3 = 0, h 4 = 0 and h 5 0 then equation (5 has the solution ( h 5 + e 4 h ξ B h ( h 5 + e 4 h ξ B h h = ± h 5 + e 4 hξ B h (ii If h 0 0, h = 0, h 3 = 0, h 4 = 0 and h 5 = 0 then equation (5 has the solution = h 0 h + e h ξ B (iii If h 0 = 0, h 0, h 3 = 0, h 4 = 0 and h 5 = 0 then the equation (5 has the solution h = h + e h ξ B 3 h (iv If h 0 = 0, h = 0, h 3 0, h 4 = 0 and h 5 = 0 then the equation (5 has the solution = ± ( h3 + e h ξ B 4 h h h 3 + e hξ B 4 h IJNS homepage:

3 50 International Journal of Nonlinear Science, Vol.4(0, No., pp (v If h 0 = 0, h = 0, h 3 = 0, h 4 0 and h 5 = 0 then the equation (5 has the solution 3 h ( h 4 + e 3 h ξ B 5 h = h 4 + e 3 h. ξ B 5 h 3 Application of the new generalized ( - expansion method for the (+ dimensional PKP equation We start with the following (+ dimensional PKP equation which is written in the following form Suppose that 4 u xxxx + 3 u xu xx u yy + u xt = 0 (6 u(x, y, t = u(ξ, ξ = x + ly + wt (7, l and w are constants that to be determined later. By using (7,(6 is converted into ODE. Integrating (8 with respect to ξ once, we get 4 4 u ( u u u (u + ( 3 l + w u = 0 (8 ( 3 l + w u g = 0 (9 Where g is the integration constant that can be determined later. Considering the homogeneous balance between the highest-order derivatives and non linear terms in (9, we get n =. From(4, we get ( u(ξ = A 0 + A A 0 (0 where A 0, A are constants to be detemined and = (ξ satisfies equation (5. Substituting (0 into (9 we obtained a polynomial in. On equating the coefficients of the polynomial in i (i = 4,..., 6, to zero, we get a system of algebraic equations and solving them by Maple or Mathematica, we obtained five families of solutions for equation (6 as follows: u = a 0 + h e (h ξ B h 0 + h e (h ξ B ( where ξ = x + ly 4 where ξ = x + ly 4 3 l + 4 h t, and h 0 0, h 0 and B, and l is an arbitrary constant. u = a 0 8 B e ( 4 h ξ h h 5 + e ( 4 h ξ B h ( 6 4 h +3 l t, and h 0, h 5 0 and B, and l is an arbitraty constant. u 3 = a 0 h e ( h ξ B 3 h + e ( h ξ B 3 h (3 where ξ = x + ly 4 3 l + 4 h t, and h 0, h 0 and B 3, and l is an arbitrary constant. u 4 = a 0 4 e( hξ h B 4 h 3 + e ( hξ B 4 h (4 IJNS for contribution: editor@nonlinearscience.org.u

4 ( R. Budhiraja, R.K. upta, S.S. Bhatia: The New eneralized -Expansion Method for Solving (+ Dimensional 5 Figure : Wave solution for the PKP equation which is given by eq (3.6. wehere ξ = x + ly 4 3 l +4 4 h t and h 0, h 3 0 and B 4, and l is an arbitrary constant. u 5 = a 0 6 e( 3 h ξ h B 5 h 4 + e ( 3 h ξ B 5 h (5 where ξ = x + ly 3 4 l +3 4 h t, and h 0, h 4 0 and B 5, and l is an arbitrary constant. 4 Conclusion In this paper, The New ernalised ( -expansion method is proposed to obtained more general exact solution of NLEEs. By using the proposed method we have obtained exact solution of (+ dimensional PKP equation. The paper shows that New ernalised ( -Expansion method is direct,effective and can be used for many other NLEEs in mathematical physics. References [] D. Zabala and Aura L. Lopez De Ramos, Effect of the Finite Difference Solution Scheme in a Free Boundary Convective Mass Transfer Model. WSEAS Transactions on Mathematics, 6(007: [] R. Vilums, A. Buiis, Conservative Averaging and Finite Difference Methods for Transient Heat Conduction in 3D Fuse. WSEAS Transactions on Heat and Mass Transfer, 3(008:4. [3] N.E. Mastorais, An Extended Cran-Nicholson Method and its Applications in the Solution of Partial Differential Equations: -D and 3-D Conduction Equations. WSEAS Transactions on Mathematics, 6(007:5 5. [4] N.E. Mastorais, Numerical Solution of Non linear Ordinary Differential Equations via Collocation Method (Finite Elements and enetic Algorithm. WSEAS Transactions on Information Science and Applications, (005: [5] M. Inc, New exact solutions for the ZK-MEW equation by using symbolic computation. Applied Mathematics and Computation, 89(007: [6] Z. Huiqun, Extended Jacobi elliptic function expansion method and its applications. Communications in Non linear Science and Numerical Simulation, (007: [7] A.M. Wazwaz, Multiple-front solutions for the burgers equation and the coupled burgers equations. Applied Mathematics and Computation, 90(007: [8] M.L. Wang and X.Z. Li, Applications of Fexpansion to periodic wave solutions for a new Hamiltonian amplitude equation. Chaos, Solitons and Fractals, 4(005: [9] M.L. Wang and X. Li., Extended F-expansion and periodic wave solutions for the generalized Zaharov equations. Physics Letter A, 343(005: [0] M.L. Wang and Y.B. Zhou, The periodic wave solutions for the Klein-ordon-Schrodinger equations. Physics Letter A, 38(003:84 9. [] E.. Fan, Extended tanh-function method and its applications to non linear equations. Physics Letters. A, 77( IJNS homepage:

5 5 International Journal of Nonlinear Science, Vol.4(0, No., pp [] A.M. Wazwaz, Compactons, solitons and periodic solutions for some forms of non linear Klein-ordon equations. Chaos, Solitons and Fractals, 8(006: [3] E. Fan and Y.C.H. Benny, Double periodic solutions with jacobi elliptic functions for two generalized hirota-satsuma coupled dv systems. Physics Letter A, 9(00: [4] F. Zuntao, L. Shiuo, L. Shida and Z. Qiang, New Jacobi elliptic function expansion and new periodic solutions of non linear wave equations. Physics Letter A, 90(00:7 76. [5] M. Wang, Solitary wave solutions for variantboussinesq equations. Physics Letter A, 99(995:69 7. [6] E.M.E. Zayed, H.A. Zedan and K.A. epreel, On the solitary wave solutions for non linear Hirota-Satsuma coupled KdV equations, Chaos, Solitons and Fractals, (004: [7] M.R. Miura. Baclund Transformation, Springer-Verlag, Berlin, (978. [8] C. Rogers and W. F. Shadwic, Baclund Transformation, Academic, Newor, 98. [9] M.L. Wang, X.Z. Li and J.L. Zhang, The ( -Expansion Method and Travelling Wave Solutions of Nonlinear Evolution Equations in Mathematical Physics. Physics Letter A, 37(008: [0] A. Borhanifar and M.M. Kabir, New periodic and soliton solutions by application of Exp-function method for non linear evolution equations. Journalof Computational and Applied Mathematics, 9(009: [] K. Singh and R.K. upta, Lie symmetries and exact solutions of a new generalized Hirota-Satsuma coupled KdV system with variable coefficients, International Journal of Enginering Science, 44(006:4-55. [] E.M.E. Zayeed and M.A.S. EL-Maly, The eneralised ( / Expansion method for solving non linear Partial Differential Equations in Mathematical Physics. International Journal of Contemporary Mathematical Sciences, 6(0: [3] X. Fan, S. Yang and D. Zhao, Travelling Wave Solutions for the ilson Picering Equation by using the simplified / -Expaqnsion Method. International Journal of Nonlinear Science, 8(009: [4] R.K. upta and Anupma, The Dullin-ottwald-Holm Equation: Classical Lie Approach and Exact Solutions. International Journal of Nonlinear Science, 0 (00:46 5. IJNS for contribution: editor@nonlinearscience.org.u