New Applications of an Improved (G /G)-expansion Method to Construct the Exact Solutions of Nonlinear PDEs

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1 Freund Publishing House Ltd. International Journal of Nonlinear Sciences & Numerical Simulation (4): New Applications of an Improved (G /G)-epansion Method to Construct the Eact Solutions of Nonlinear PEs Elsayed M. E. Zayed Khaled A. Gepreel Mathematics epartment Faculty of Science Taif University El-Taif El- Hawiyah Kingdom of Saudi Arabia Mathematics epartment Faculty of Science Zagazig University Zagazig Egypt Abstract In the present article we construct traveling wave solutions involving parameters of some nonlinear PEs in mathematical physics namely Konopelchenko-ubrovsky equations Kersten- Krasil Shchik equations Whitham- Broer Kaup equations the fifth order KdV equation using an improved ( G / G ) epansion method where G satisfies a second order linear ordinary differential equation. When these parameters are taken special values the solitary waves are derived from the traveling waves. The eact wave solutions are epressed by hyperbolic trigonometric rational functions. Comparison between this method the ep-function method is presented. Keywords: an improved ( G / G ) - epansion method traveling wave solutions solitary wave solutions nonlinear PEs ep-function method. Introduction In recent years the eact solutions of nonlinear PEs have been investigated by many authors (see for eample [-0] ) who are interested in nonlinear physical phenomena. Many powerful methods have been presented such as the tanh-method [7] the inverse scattering transform [] the Backlund transform [90] the generalized Riccati equation [-6] the Sine-Cosine method[] the F-epansion method [0] the Epfunction method [57] the ( G / G) epansion method [89] so on. In the present article we shall use a new method which is called an improved ( G / G) -epansion method to obtain the traveling wave solutions of Corresponding Author: emezayed@hotmail.com kagepreel@yahoo.com nonlinear PEs. The main idea of this method is that the traveling wave solutions of nonlinear evolution equations can be epressed by a polynomial in ( G / G) where G = G(ξ) satisfies the second order linear ordinary differential equation G ( ξ ) λg ( ξ) μg( ξ ) = 0 ξ = V t while λ μ V are constants. The degree of this polynomial can be determined by considering the homogeneous balance between the highest order derivatives the nonlinear terms appearing in the given nonlinear equations. The coefficients of this polynomial can be obtained by solving a set of algebraic equations resulted from the process of using the proposed method. Recently Bin et al [] Zayed et al [8] have obtained eact solutions of some nonlinear PEs using this method. In this article the improved ( G / G) epansion method will be used to determine the eact wave solutions of Konopelchenko-ubrovsky equations Kersten-

2 74 E.M.E. Zayed & K.A. Gepreel: New applications of improved (G /G)-epansion method to construct eact solutions of nonlinear PEs Krasil Shchik equations Whitham-Broer Kaup equations the fifth order KdV equation in terms of hyperbolic trigonometric rational functions. escription of an improved ( G/G) - epansion method Suppose that we have a nonlinear PE in the following form: F ( u u u u u u u...) = 0. (.) t y tt where uyt ( ) is an unknown unction F is a polynomial in u = u( y t) its partial derivatives in which the highest order derivatives nonlinear terms are involved. Let us now give the main steps for solving Eq. (.) using an improved ( G / G)-epansion method [8] as follows: Step. The traveling wave variable: uyt ( ) = uξ ( ) ξ = y- Vt (.) where V is a constant permits us reducing Eq. (.) to the following OE : Puu ( u u...) = 0. (.) where P is a polynomial in u( ξ ) its total derivatives. Step. Suppose that the solution of (.) can be epressed by a polynomial in ( G / G) as follows: u m i G = i m G (.4) where G = G( ξ ) satisfies the second order linear differential equation where G = G( ξ ) satisfies the second order linear differential equation G ( ξ ) λg ( ξ) μg( ξ) = 0 (.5) where i (i = 0 ± ±... ± m) λ μ are constants to be determined later m 0 or -m 0. The positive integer m can be y determined by considering the homogeneous balance between the highest order derivatives the nonlinear terms appearing in (.). Step. Substituting (.4) into (.) using (.5) collecting all terms with the same order of ( G / G) together then equating each coefficient of the resulted polynomial to zero yield a set of algebraic equations for i λ V μ. Step 4. Since the general solutions of (.5) have been well known for us then substituting i λ V μ the general solutions of (.5) into (.4) we have traveling wave solutions of the nonlinear differential equation (.). Applications In this section we apply the improved ( G / G)- epansion method to construct the traveling wave solutions for some nonlinear partial differential equations in mathematical physics as follows:. Eample. Konopelchenko-ubrovsky equations We start with the following Konopelchenko-ubrovsky equations [80] : ut u 6βuu u u wy u w= 0 w u y= 0 (.) where β are constants. For w y = 0 Eqs. (.) is called the Gardner equation. For = 0 Eqs.(.) is the well- known Kadomtsev- Petviashvili equation for β = 0 it is called modified Kadomtsev- Petviashvili equation. Let us now solve Eqs. (.) by the proposed method. To this end we see that the traveling wave variables uyt ( ) = u( ξ) vyt ( ) = v( ξ) ξ= y Vt (.) permit us to convert Eqs (.) into the following system:

3 ISSN: International Journal of Nonlinear Sciences & Numerical Simulation (4): Vu u βuu u u w u w = w u = (.) Suppose that the solutions of the system (.) can be epressed by two polynomials in terms of ( G / G) as follows: m i G G u= i w= βi G G i= m i= n n (.4) while G = G( ξ ) satisfies the second order linear OE (.5) while i β i are arbitrary constants. Considering the homogeneous balance between the highest order derivative u the nonlinear terms uu uw in (.) we get m= n=. Consequently we have G G 0 u = G G And G G 0 w = β G β G β It is easy to see that G G u = λ μ G G G G λ μ G G λ ( μ λ ) G G ( μ λ ) λμ G μ μλ λ G i (.5) (.6) (.7) G G G u = G G G G G (.8) 4 G G G u = 6 λ (8μ 7 λ ) G G G G G (8 μλ λ ) λ(8 μ λ ) G G G G (8 7 ) μ λ μ λμ G G 4 G 6 μ μλ μ λ μ. G (.9) On substituting (.5)-(.9) into (.) collecting all terms with the same powers of ( G / G) setting them to zero we get a system of algebraic equations which can be solved by the Maple or Mathematica to get the following results: Case 0 = ± [ ± β m λ ] = ± V = [ ( λ 8μ m 6λ ) β 6 β 0 ] μ μ β = ± = ± β = ±. (.0) Case μ 0 = ± [ ± β m λ] = ± V = [ ( λ 4μ m 6λ) β 6 β 0 ] μ β = ± = β = 0 (.) Case 0 = ± [ ± β m λ] = ± V = [ ( λ 4μ m 6λ) β 6 β0 ] β = ± = β = 0 (.) where 0.

4 76 E.M.E. Zayed & K.A. Gepreel: New applications of improved (G /G)-epansion method to construct eact solutions of nonlinear PEs Note that there are other cases which are omitted here. Since the solutions obtained here are so many we just list some eact solutions corresponding to case to illustrate the effectiveness of the improved ( G / G) epansion method. Substituting (.0) into (.5) (.6) yields G μ G u =± ± ± [ ± β m λ ] G G (.) G μ G w=± ± β0 G G where (.4) t ξ = [ ( λ 8μ m6λ) β 6 β0]. (.5) On solving Eq.(.5) substituting the value of the ratio ( G / G) into (.) (.4) we have the following eact solutions: Family. If λ 4μ> 0 then we have ξ ξ h λ w = β 0 m ± ξ ξ B cosh ξ ξ cosh sinh A A μ ± λ ξ ξ sinh cosh A B where = λ μ. Family. 4 If λ 4μ < 0 then we have u =± [ ± β m ] ξ ξ Asin ± ξ ξ Bsin (.7) Asin ξ ξ μ λ ± ξ Bsin ξ (.8) u =± [ ± β m ] h ± h h μ λ ± h (.6) ξ ξ Asin λ w = β0 m ± ξ ξ Bsin ξ ξ Asin μ λ ± ξ ξ Bsin (.9)

5 ISSN: International Journal of Nonlinear Sciences & Numerical Simulation (4): where = μ λ. Family. 4 If λ 4μ= 0 then we have B u =± [ ± β m ] ± ( Bξ A) μ B λ ± Bξ A (.0) λ B μ B λ w= β0 m ± ±. ( Bξ A) Bξ A (.) In particular if B= 0 A 0 λ > 0 μ = 0 then we deduce from (.6) (.7) that: λ λξ u =± [ ± β m ] ± coth (.) λ λ λξ w = β 0 m ± coth (.) while if B 0 B > A λ > 0 μ = 0 then we have: λ λξ u = ± [ ± β m ] ± tanh( ξ ) 0 (.4) λ λ λξ w = β 0 m ± tanh( ξ 0 ). (.5) Note that (.)- (.5) represent the solitary wave solutions of the Konopelchenko- ubrovsky equations (.) where t ξ = [ ( λ m6λ) β 6 β0] tanh A ξ 0 = ( ). (.6) B. Eample. Kersten- Krasil Shchik equations In this section we study the following Kersten- Krasil Shchik equations [6]: u u t w w t 6uu ww w w w w uw wu u w = 0. 6uww = 0 (.7) The traveling wave variables (.) permit us to convert Eqs. (.7) into: Vu u 6uu ww ww w u 6uww = 0 Vw w w w uw wu = 0. (.8) Considering the homogeneous balance between the highest order derivative u the nonlinear terms u u w w in (.8) we get n = m =. Thus the solutions of Eqs. (.8) have the following forms: G G G G 0 u = G G G G (.9) G G 0. w = β G β G β (.0) On substituting (.9) (.0) into (.8) collecting all terms with the same powers of ( G / G) setting them to zero we get a system of algebraic equations which can be solved with the Maple or Mathematica to get the following results: Case = = λ = 0 0 = λ = μ V = λ 8μ β = i β = iμ β 0 = iλ (.)

6 78 E.M.E. Zayed & K.A. Gepreel: New applications of improved (G /G)-epansion method to construct eact solutions of nonlinear PEs Case = = λ 0 = μ = μλ = μ β = i β = iμ β0 = 0 V = λ 8 μ (.) Case = 0 = 0 β = 0 = μ 0 0 = μλ β = iμ β = iλ V = λ 4μ. = μ (.) Note that there are other cases which are omitted here. Since the solutions obtained here are so many we just list some eact solutions corresponding to case. Substituting (.) into (.9) (.0) yield G G u λ λμ G μ G = ± G G G G (.4) G G w = i iμ iλ G G (.5) where ξ = ( λ 8 μ) t i =. Consequently we have the following eact wave solutions: Family. If λ 4μ > 0 then we have cosh sinh A B λ u = h cosh sinh A B λ λμ h cosh sinh A B λ μ h (.6) h Bsinh w=± i h h Bsinh i λ ± μ. h (.7) Family. If λ 4μ > 0 then we have ξ ξ sin cos A B λ u = ξ ξ Bsin ξ ξ Asin λ λμ ξ ξ Bsin ξ ξ Asin λ μ ξ ξ Bsin (.8)

7 ISSN: International Journal of Nonlinear Sciences & Numerical Simulation (4): ξ ξ Asin w=± i ξ ξ Bsin ξ ξ Asin i λ ± μ. ξ ξ Bsin (.9) Family. If λ 4μ = 0 then we have B B λ u = λμ ( Bξ A) Bξ A B λ λ μ Bξ A (.40) ib B λ w = ± iμ. Bξ A Bξ A (.4) In particular if B= 0 A 0 λ > 0 μ = 0 then we get from (.6) (.7) that: λ λξ λξ u = csc h ( ) v= iλ coth( ) (.4). Eample. Whitham- Broer Kaup equations In this section we study the following Whitham- Broer Kaup equations [56]: ut uu v βv = 0 (.44) v ( uv) βv u = 0 t where β are constants. The system (.44) is a complete integrable model which describes the dispersive long wave is shallow water. In this system β are real constants that represent different dispersive powers. We deduce from the homogeneous balance between the highest order derivatives the nonlinear terms in Eqs.(.44) that n = m =. Thus the solutions of Eqs. (.44) have the following forms: G G G G 0 u = G G G G (.45) G G G G 0. v= β G β β G G β G β (.46) On substituting (.45) (.46) into (.44) using the Maple or Mathematica we have the following cases: while if B 0 B > A λ > 0 μ = 0 then we get from (.6) (.7) that: λ λξ u = λξ v = iλ tanh( ξ0 ). sec h ( ξ0 ) (.4) Note that (.4) (.4) represent the solitary wave solutions of the Kersten- Krasil Shchik equations (.7) where ξ = λ t ξ 0 = tanh ( A/ B). Case. μ μ(λ ) = = λ 4μ 0( λ 4 μ) β 6μ 6 μ(5 λ ) = β = 800( λ 4 μ) 4000 ( λ 4 μ) β = = ( λ 4 μ) 600( λ 4 μ) β (0 λ 9λ 80λμ 47 μ) 800( λ 4 μ) 0 = 0λ 0λ 0 λ 80μ0 00μ V = 0( λ 4 μ) C = ( λ 4 μ)

8 80 E.M.E. Zayed & K.A. Gepreel: New applications of improved (G /G)-epansion method to construct eact solutions of nonlinear PEs C [ = 0 μ λ 8000 ( λ 4 μ) 5 0λ 0 μ 4 λ λ μ 4 μ0 λ λμ μ0λ 6 0 λ μ 4 0λ λ 0λ μ 0μ μ0 λ ] = = β = β = 0. (.47) Case. (0 λ ) = = λ 4μ 0( λ 4 μ) 6 β = = 800( λ 4 μ) 600( λ 4 μ) (40 λ λ 60 μ ) β = 800 ( λ 4 μ) 56 C = β = 000 ( λ 4 μ) ( λ 4 μ) (0λ 9λ 80λμ 47 μ) β0 = 800( λ 4 μ) 4 V = [400 λ ( λ 4 μ) 0λ 000λ μ 880λμ 6400μ 0 0μ 99 λ ] C = [ μ 8000 ( λ 4 μ) μλ 44000λ 970 λ 540 μ μλ 9000 μ 0λ μ0λ μ λ λ 4 980λ 0 089λ 0λμ 0 60λμ 6400 μ 0 ] = = β = β = 0 (.48) ± where = λ 4μ 0. 0 (4 μ λ ) We just now list the eact solutions corresponding to the case. On substituting (.47) into (.45) (.46) we get μ(0 λ ) G μ G 0 u = 0( λ 4 μ) G λ 4μ G (0 λ 9λ 80λμ v = 800( λ 4μ) 6μ (5 λ ) 4000 ( λ 4μ) where G G (.49) 47μ) 6μ 800( λ 4μ) G G (.50) t(0λ 0 0λ λ 80μ0 00μ ) ξ =. 0( λ 4μ) Consequently we deduce the following eact wave solutions: Family. If λ 4μ >0 then we have h Bsinh μ(0 λ ) λ u= 0( λ 4 μ) h h Bsinh μ λ 0 λ 4μ h (0λ 9λ 80λμ 47 μ) v= 800( λ 4 μ) h Bsinh 6 μ(5 λ ) λ 4000 ( λ 4 μ) h h Bsinh 6μ λ 800( λ 4 μ) h (.5)

9 ISSN: International Journal of Nonlinear Sciences & Numerical Simulation (4): Family. If λ 4μ <0 then we have ξ ξ Asin μ(0λ ) u λ = 0( λ 4 μ) ξ ξ Bsin ξ ξ Asin μ λ 0 λ 4μ ξ ξ Bsin (.5) (0λ 9λ 80λμ 47 μ) v= 800( λ 4 μ) ξ ξ Asin 6 μ(5 λ ) λ 4000 ( λ 4 μ) ξ ξ Bsin ξ ξ Asin 6μ λ. 800( λ 4 μ) ξ ξ Bsin (.54).4. Eample 4. The fifth order KdV equation In this section we study the following nonlinear fifth order KdV equation [44]: u u u βu u γuu u = 0 (.55) t where β γ are constants. The homogeneous balance of Eq. (.55) gives m =. Consequently the solution of Eq.(.55) has the form (.45). After some reduction we have the following cases: Case 60μ γμ β = 6μ 6 4 V = [44μ 9μ λ = 4 = = 0 where μ 0. Case γμ 0 = μ ] = μ (.56) (56 γ) ( λ 8 μ) 0 β = = = λ 4 4 V = μ μ γμλ γλ 6μλ 6 ( γ 48) γμ ] = = = 0 (.57) where 0. We just now list the eact solutions corresponding to Case. The eact solutions of Eq.(.55) have the following forms: Family If μ < 0 then we have h u = μ μ h μ Family μ ξ Bsinh μ ξ B cosh If μ > 0 then we have μ ξ Bsinh μ ξ B cosh μ ξ μ ξ Asin μ ξ B cos u = μ μ μ ξ B sin sin cos A μ ξ B μ ξ. μ cos sin A μ ξ B μ ξ μ ξ μ ξ. μ ξ μ ξ (.58) (.59)

10 8 E.M.E. Zayed & K.A. Gepreel: New applications of improved (G /G)-epansion method to construct eact solutions of nonlinear PEs get In particular if B = 0 A 0 μ < 0 then we coth tanh u = μ ξ μ ξ μ (.60) while if B 0 B > A μ < 0 then we get u = coth ( μξ ξ0 ) tanh ( μξ ξ0 ) μ (.6) where ξ0 = tanh ( A/ B). Note that (.60) (.6) represent the solitary wave solutions of (.55). 4. Ep-function Method Let us now compare between the ( G / G)- epansion method the other methods such as the Ep- function method introduced by He et al [5] to search for the eact solutions of nonlinear PEs. To this end we apply the Epfunction to the Konopelchenko-ubrovsky equations (.). Now we make the ansatz: a d ep (d ξ )... a-cep (-c ξ ) u= a q ep (q ξ)... a-pep (-p ξ) b k ep (k ξ )... b-mep (-m ξ ) w=. b ep (L ξ )... a ep (-I ξ ) L -I (.6) For the solutions of Eq. (.) we balance the highest order linear derivative u the nonlinear terms u u wu in Eqs (.). After some calculations we have d = q= L= k c= p= m= I. If we choose p = q =. In this case the solutions of Eqs.(.) can be epressed as follows: a ep ( ξ)a 0 aep (- ξ) u= b ep ( ξ)b 0 bep (- ξ) A ep ( ξ )A 0 Aep (- ξ ) w=. B ep ( ξ )B B ep (- ξ ) 0 (.6) Substituting (.6) into (.) collecting the coefficients of ep ( lξ ) l = 0 ±... we get a system of algebraic equations. Solving this system with the aid of the Maple we have the following results: Case i a = β = ( i ) b = b0 ( i 4) b0 ( i ) a = b = ; b0 ( i ) b0( A i ) B = A0 = b0 (i A i A) A = V = 6i A B0 = b0 a0 = 0 (.64) where A b 0 are arbitrary constants i =. The eact solutions take the following form: i b0 ( i 4) ep ( ξ ) ep (-ξ ) u = b0 ( i ) ep ( ξ ) b 0 ep (-ξ ) w= b0 ( i ) ep ( ξ)-b 0 ep (- ξ) b0( A i ) b0( A)( i ) [A ep ( ξ) ep (- ξ)] (.65) where ξ = ( 6i A ) t.

11 ISSN: International Journal of Nonlinear Sciences & Numerical Simulation (4): Case. b0( ab0 5ab0 a0 a0) β = (a b0 a0) ab0 a = B = b = b0 b0 B = b = B0 = b0 Ab 0 b0 A = = (a b a ) 0 0 A0= a0 Ab 0 a b0 V = {(8 ) a b0 aab 0 (a b0 a0) ( ) aba 0 0 a 0Ab 0 a0} (.66) where A b 0 a 0 are arbitrary constants. The eact solutions take the following form: ab0 aep( ξ) a0 ep( ξ) u = b0 ep( ξ) b0 ep( ξ) Ab 0 Aep( ξ ) a0 Ab 0 ab0 ep( ξ ) w = b0 ep( ξ) b0 ep( ξ) (.67) where t ξ = {(8 ) ab 0 aab 0 (a b0 a0) ( ) a ba a Ab a} Note that there are other cases which are omitted here. We think that the Ep- function method is very simple but its results are very cumbersome (see [7] the references cited therein). The results of the ( G / G)-epansion method contain more arbitrary constants compared to the results of the Ep- function method. The performance of the ( G / G) epansion method is reliable simple direct concise gives more eact solutions compared to the Epfunction method. Finally the ( G / G) epansion method allows us to solve more complicated nonlinear PEs compared to the Ep- function method. References [] M.J. Ablowitz P.A. Clarkson Solitons nonlinear Evolution Equations Inverse Scattering Transform Cambridge Univ. Press Cambridge 99. [] Z.Y.Bin L.Chao Commu. Theor. Phys. 5(009) [] E.G. Fan Phys.Lett. A 77 (000) - 8. [4] C.A.Gomez Appl. Math.Comput.89 (007) [5] J.H.He X.H.Wu Chaos Solitons Fractals 0 (006) [6] A.K. Kalkanli S.Y. Sakovich T. Yurdusen J. Math.Phys. 44 (00) [7] N.A. Kudryashov N.B. Loguinova Commu. Nonlinear Sci. Numer. Simul. 4 (009) [8] B.Li Y.Zhang Chaos Solitons Fractals 8 (008) [9] M.R.Miura Backlund Transformation Springer-Verlag Berlin 978. [0] C. Rogers W.F. Shadwick Backlund Transformations Academic Press New York 98. [] M.Wang X.Li Chaos Solitons Fractals 4 (005) [] M.L.WangX.Z.Li J.L.Zhang Phys. Lett. A 7 (008) [] M.Wazwaz Appl. Math. Comput.67 (005) [4] M. Wazwaz Appl. Math.Comput. 84 (007) [5] T.Xu J.Li H.Zhang Y.X.Zhang Z.Z.Yao B. Tian Phys. Lett. A 69 (007) [6] Z.Y.Yan H.Q.Zhang Phys. Lett. A 85 (00) [7] E.M.E. Zayed H.A. Zedan K.A. Gepreel Int. J. Nonlinear Sci. Nume.Simul.5 (004) - 4. [8] E.M.E. Zayed S. Al-Joudi AIP Conference Proceeding Amer. Institute of Phys. Vol 68 (009) [9] E.M.E.Zayed J.Appl. Math. Computing 0 (009) [0] S. Zhang T.C. Xia Appl. Math. Comput. 8 (006)

12 Freund Publishing House Ltd. International Journal of Nonlinear Sciences & Numerical Simulation (4):