The (G'/G) - Expansion Method for Finding Traveling Wave Solutions of Some Nonlinear Pdes in Mathematical Physics

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1 Vol.3, Issue., Jan-Feb. 3 pp ISSN: The ('/) - Expansion Method for Finding Traveling Wave Solutions of Some Nonlinear Pdes in Mathematical Physics J.F.Alzaidy Mathematics Department, Faculty of Science, Taif University, Kingdom of Saudi Arabia Abstract: In the present paper, we construct the traveling wave solutions involving parameters of some nonlinear PDEs in mathematical physics; namely the variable coefficients KdV (vckdv) equation, the modified dispersive water wave (MDWW) equations and the symmetrically coupled KdV equations by using a simple method which is called the expansion method, where = ξ satisfies the second order linear ordinary differential equation. When the parameters are taken special values, the solitary waves are derived from the traveling waves. The traveling wave solutions are expressed by hyperbolic, trigonometric and rational functions. This method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear PDEs arising in mathematical physics. Keywords: The('/) expansion method; traveling wave solutions; the variable coefficients KdV (vckdv) equation; the modified dispersive water wave (MDWW) equations; symmetrically coupled KdV equations, solitary wave solutions. AMS Subject Classifications: 35K99; 35P5; 35P99. I. Introduction In recent years, the exact solutions of nonlinear PDEs have been investigated by many authors ( see for example [- 3] ) who are interested in nonlinear physical phenomena. Many effective methods have been presented,such as inverse scattering transform method [], a cklund transformation [], Darboux transformation [3], Hirota bilinear method [4], variable separation approach [5], various tanh methods [6 9], homogeneous balance method [], similarity reductions method [,], the reduction mkdv equation method [3], the tri-function method [4,5], the projective Riccati equation method [6], the Weierstrass elliptic function method [7], the Sine- Cosine method [8,9], the Jacobi elliptic function expansion [,], the complex hyperbolic function method [], the truncated Painleve expansion [3], the F-expansion method [4], the rank analysis method [5] and so on. In the present paper, we shall use a simple method which is called the -expansion method [6,7]. This method is firstly proposed by the Chinese Mathematicians Wang et al [8] for which the traveling wave solutions of nonlinear equations are obtained. The main idea of this method is that the traveling wave solutions of nonlinear equations can be expressed by a polynomial in, where = ξ satisfies the second order linear ordinary differential equation " ξ + λ ξ + μ ξ =, where ξ = k x ct,where λ, μ, k and c are constants. The degree of this polynomial can be determined by considering the homogeneous balance between the highest order derivatives and 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. This new method will play an important role in expressing the traveling wave solutions for nonlinear evolution equations via the vckdv equation, the MDWW equations and the symmetrically coupled KdV equations in terms of hyperbolic, trigonometric and rational functions. - II. Description of the Suppose we have the following nonlinear PDE: - expansion method P u, u t, u x, u tt, u xx, u xt, =, () Where u = u(x, t) is an unknown function, P is a polynomial in u = u(x, t) and its various partial derivatives in which the highest order derivatives and nonlinear terms are involved. In the following we give the main steps of a deformation method: Setp. Suppose that u x, t = u ξ, ξ = ξ x, t. () The traveling wave variable () permits us reducing () to an ODE for u = u ξ in the form: where = d dξ. P(u, u, u, ) =, (3) 369 Page

2 Vol.3, Issue., Jan-Feb. 3 pp ISSN: Setp. Suppose that the solution Eq.(3) can be expressed by a polynomial in While = ξ satisfies the second order linear differential equation in the form: Where a i i =,,, n, λ and μ are constants to be determined later. n i= a i as follows: i, (4) " ξ + λ ξ + μ ξ =, (5) Setp3. The positive integer " n " can be determined by considering the homogeneous balance between the highest derivative term and the nonlinear terms appearing in Eq.(3). Therefore, we can get the value of n in Eq.( 4). Setp4. Substituting Eq.( 4) into Eq.( 3) and using Eq.(5), collecting all terms with the same power of equating each coefficient of the resulted polynomial to zero, yield a set of algebraic equations for a i, λ, μ, c and k. Setp5. Solving the algebraic equations by use of Maple or Mathematica, we obtain values for a i, λ, μ, c and k. together and then Setp6. Since the general solutions of Eq. (5) have been well known for us, then substituting the obtained coefficients and the general solution of Eq. (5) into Eq. (4), we have the travelling wave solutions of the nonlinear PDE (). III. Applications of the method In this section, we apply the -expansion method to construct the traveling wave solutions for some nonlinear PDEs, namely the vckdv equation, the MDWW equations and the symmetrically coupled KdV equations which are very important nonlinear evolution equations in mathematical physics and have been paid attention by many researchers. 3.. Example. The cvkdv equation We start with the cvkdv equation [9] in the form: u t + f t uu x + g t u xxx =, (6) Where f t, g t are some given functions.this equation is well-known as a model equation describing the propagation of weakly-nonlinear weakly-dispersive waves in inhomogeneous media.. Obtaining exact solutions for nonlinear differential equations have long been one of the central themes of perpetual interest in mathematics and physics. To study the travelling wave solutions of Eq. (6), we take the following transformation u x, t = u ξ, ξ = x + g t d t, (7) α Where the wave is speed and α is a constant. y using Eq. (7), Eq.(6) is converted into an ODE α u + uu + u =, (8) Where the functions f t and g t in Eq.(6) should satisfy the condition t f t = g t. (9) Integrating Eq.(8) with respect to ξ once and taking the constant of integration to be zero, we obtain α u + u + u =, () Suppose that the solution of ODE () can be expressed by polynomial in terms of as follows: n i= a i i, () Where a i i =,,, n are arbitrary constants, while ξ satisfies the second order linear ODE (5). Considering the homogeneous balance between the highest order derivatives and the nonlinear terms in Eq. (), we get n =. Thus, we have a + a + a, () Where a, a and a are constants to be determined later. Substituting Eq.() with Eq.(5) into Eq.() and collecting all terms with the same power of system of algebraic equations:. Setting each coefficients of this polynomial to be zero, we have the following 37 Page

3 : International Journal of Modern Engineering Research (IJMER) Vol.3, Issue., Jan-Feb. 3 pp ISSN: a α + a + λμa + μ a =, : λ a + μa + a α + a a + 6λμa =, : 3λa + a + 4λ a + 8μa + a α + a a =, 3 : a + λa + a a =, 4 : 6a + a =, (3) On solving the above algebraic Eqs. (3) y using the Maple or Mathematica, we have Where M = λ 4μ. Substituting Eq. (4) into Eq.() yields Where ξ = x M a = 6μ, a = 6λ, a = 6, = αm, (4) 6μ 6λ 6, (5) t g t d t. (6) On solving Eq.( 5), we deduce that = M M A sin M ξ + sinh M ξ A sinh M ξ + cosh M ξ λ M ξ + cos M ξ A cos M ξ + sin M ξ λ ξ + A λ if M >, if M <, (7) if M =, Where A and are arbitrary constants and M = λ 4μ. On substituting Eq.(7) into Eq.(5), we deduce the following three types of traveling wave solutions: Case. If M >, Then we have the hyperbolic solution 3M M ξ + sinh M ξ A sinh M ξ + cosh M ξ, (8) Case. If M <, Then we have the trigonometric solution 3M + A sin M ξ + cos M ξ A cos M ξ + sin M ξ, (9) Case3. If M =, Then we have the the rational solution 3 λ 6 μ +, () ξ + A In particular, if we set =, A, λ >, μ =, in Eq.(8), then we get 3λ csch λ ξ, () While, if, λ >, A > and μ =, then we deduce that: 37 Page

4 Vol.3, Issue., Jan-Feb. 3 pp ISSN: Where ξ = tanh A 3λ λ sech ξ + +ξ, ().The solutions () and () represent the solitary wave solutions of Eq. (6). 3.. Example. The MDWW equation In this subsection, we study the MDWW equations [3] in the forms: u t = 4 v xx + uv x, (3) v t = u xx uu x + 3 vv x. (4) The traveling wave variables below u x, t = u ξ, v x, t = v ξ, ξ = k x + t, (5) permit us converting the equations (3) and (4) into ODEs for u x, t = u ξ and v x, t = v ξ as follows: 4 kv + uv u =, (6) ku uu + 3 vv v =, (7) Where k and are the wave number and the wave speed, respectively. On integrating Eqs.(6) and (7) with respect to ξ once, yields k 4 kv + uv u =, (8) Where k and k is an integration constants. k ku u v v =, (9) Suppose that the solutions of the ODEs (8) and (9) can be expressed by polynomials in terms of v ξ = n i= m i= a i b i as follows: i, (3) i. (3) Where a i i =,,, n and b i i =,,, m are arbitrary constants, while ξ satisfies the second order linear ODE (5).Considering the homogeneous balance between the highest order derivatives and the nonlinear terms in Eqs.(8) and (9), we get n = m =. Thus, we have a + a, a, (3) v ξ = b + b, b, (33) Where a, a, b and b are arbitrary constants to be determined later. Substituting Eqs.(3),(33) with Eq.(5) into Eqs.(8) and (9), collecting all terms with the same power of algebraic equations: : a + a b + 4 kμb + k =, : a + a b + 4 kλb + a b =, : kb 4 + a b =, and setting them to zero, we have the following system of 37 Page

5 Vol.3, Issue., Jan-Feb. 3 pp ISSN: : a + kμa + b + 3b 4 + k =, : kλa a a + b + 3b b =, : ka a + 3b Solving the above algebraic Eqs.(34) by using the Maple or Mathematica, yields 4 =. (34) a = λa, a = a, b o = λa, b = a, k = 4 Ma, k = Ma, k = a. Substituting Eq.(35) into Eqs.(3) and (33) we obtain (35) λa + a, (36) v ξ = λa + a,, (37) where ξ = a (x + t). (38) From Eqs.(7),(36) and (37), we deduce the following three types of traveling wave solutions: Case. If M >, then we have the hyperbolic solution a M M ξ + sinh M ξ A sinh M ξ + cosh M ξ, (39) v ξ = a M M ξ + sinh M ξ A sinh M ξ + cosh M ξ. (4) Case. If M <, then we have the trigonometric solution a M A sin M ξ + cos M ξ A cos M ξ + sin M ξ, (4) v ξ = a M A sin M ξ + cos M ξ A cos M ξ + sin M ξ. (4) Case3. If M =, then we have the rational solution a v ξ = a ξ + A, (43) ξ + A. (44) In particular if =, A, λ > and μ =, then we deduce from Eqs.(39) and (4) that: a λ coth λ ξ, (45) v ξ = a λ coth λ ξ, (46) While, if, A >, λ > and μ =, then we deduce that: a λ tanh λ ξ + ξ, Page

6 Vol.3, Issue., Jan-Feb. 3 pp ISSN: v ξ = a λ tanh λ ξ + ξ, 48 Whereξ = tanh A. The solutions (45)- (48) represent the solitary wave solutions of Eqs.(3) and (4) Example 3. The symmetrically coupled KdV equations In this subsection, we consider the symmetrically coupled KdV equations [3] in the forms: u t = u xxx + v xxx + 6uu x + 4uv x + u x v =, (49) v t = u xxx + v xxx + 6vv x + 4vu x + v x u =. (5) The traveling wave variable (5) permits us converting Eqs.(49) and (5) into the following ODEs: u + k u + v + 6uu + 4uv + u v =, (5) v + k u + v + 6vv + 4vu + v u =. (5) Considering the homogeneous balance between highest order derivatives and nonlinear terms in Eqs.(5) and (5), we have, a, (53) a + a v ξ = b + b + a + b, b, (54) Where a, a, a, b, b and b are arbitrary constants to be determined later. Substituting Eqs.(53) and (54) with and setting them to zero, we have the follow- Eq.(5) into Eqs.(5) and (5), collecting all terms with the same power of ing system of algebraic equations: : k λ μa k μ a + μa 6μa a 6k λμ a k λ μb k μ b 4μa b 6k λμ b =, : k λ 3 a 8k λμa + λa 6λa a 6μa 4k λ μa 6k μ a + μa μa a k λ 3 b 8k λμb 4λa b 6μa b 4k λ μb 6k μ b 8μa b =, : 7k λ a 8k μa + a 6a a 6λa 8k λ 3 a 5k λμa + λa λa a 8μa a 7k λ b 8k μb 4a b 6λa b 8μa b 8k λ 3 b 5k λμb 8λa b μa b =, 3 : k λa 6a 38k λ a 4k μa + a a a 8λa a μa k λb 6a b 8λa b 38k λ b 4k μb 8a b λa b μa b =, 4 : 6k a 54k λa 8a a λa 6k b 8a b 54k λb a b λa b =, 5 : 4k a a 4k b a b =, : k λ μa k μ a 6k λμ a k λ μb k μ b + μb μa b 6k λμ b =, : k λ 3 a 8k λμa 4k λ μa 6k μ a k λ 3 b 8k λμb + λb λa b 6μa b 6μb 4k λ μb 6k μ b + μb 4μa b =, : 7k λ a 8k μa 8k λ 3 a 5k λμa 7k λ b 8k μb + b a b 6λa b μa b 6λb 8k λ 3 b 5k λμb + λb 4λa b 8μa b 8μb b =, 3 : k λa 38k λ a 4k μa k λb 6a b λa b 6b 38k λ b 4k μb + b 4a b 8λa b μa b 8λb b μb =, 4 : 6k a 54k λa 6k b a b 54k λb 8a b λa b 8b b λb =, 5 : 4k a 4k b a b b = Page

7 Vol.3, Issue., Jan-Feb. 3 pp ISSN: Solving the above algebraic Eqs.(55) by using the Maple or Mathematica, we have a = b =, a = λ λ + 8μ, a = λ + 8μ, b = λ λ + 8μ, (56) Substituting Eq.(56) into Eqs.(53) and (54) yields λ λ + 8μ v ξ = λ λ + 8μ Where ξ = ± b = λ + 8μ, k = ± λ + 6μ. λ +6μ λ + 8μ λ + 8μ, (57), (58) (x + t). (59) From Eqs. (7), (57) and (58), we deduce the following three types of traveling wave solutions: Case. If M >, Then we have the hyperbolic solution 4 λ + 8μ λ M M ξ + sinh M ξ A sinh M ξ + cosh M ξ, (6) v ξ = 4 λ + 8μ λ M M ξ + sinh M ξ A sinh M ξ + cosh M ξ. (6) Case. If M <, Then we have the trigonometric solution 4 λ + 8μ λ + M M ξ + sinh M ξ A sinh M ξ + cosh M ξ, (6) v ξ = 4 λ + 8μ λ + M M ξ + sinh M ξ A sinh M ξ + cosh M ξ. (63) Case3. If M =, Then we have the rational solution v ξ = 4 λ + 8μ λ 4 4 λ + 8μ λ 4 ξ + A ξ + A, (64). (65) In particular if =, A, λ > and μ =, then we deduce from Eq.(6) and Eq.(6) that: 4 csch λ ξ, (66) v ξ = 4 csch λ ξ, (67) while, if, A >, λ > and μ =, then we deduce that: 4 sech λ ξ + +ξ, (68) v ξ = 4 sech λ ξ + +ξ, (69) Where ξ = tanh A. The solutions (66) - (69) represent the solitary wave solutions of Eqs. (49) and (5). 375 Page

8 Vol.3, Issue., Jan-Feb. 3 pp ISSN: IV. Conclusion In this work, we have seen that three types of traveling wave solutions in terms of hyperbolic, trigonometric and rational functions for the vckdv equation, the MDWW equations and the symmetrically coupled KdV equations are successfully found out by using the -expansion method. From our results obtained in this paper, we conclude that the - expansion method is powerful, effective and convenient. The performance of this method is reliable, simple and gives many new solutions. The -expansion method has more advantages: It is direct and concise. Also, the solutions of the proposed nonlinear evolution equations in this paper have many potential applications in physics and engineering. Finally, this method provides a powerful mathematical tool to obtain more general exact solutions of a great many nonlinear PDEs in mathematical physics. References [] M. J. Ablowitz and P.A. 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