The First Integral Method for Solving a System of Nonlinear Partial Differential Equations

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1 ISSN (print), (online) International Journal of Nonlinear Siene Vol.5(008) No.,pp The First Integral Method for Solving a System of Nonlinear Partial Differential Equations Ahmed Hassan Ahmed Ali 1, Kamal Raslan Raslan, 1 Mathematis Department, Faulty of Siene, Menoufia University, Shebein El-Koom, Egypt Community College in Riyadh, King Saud University, Box 8095, Riyadh 117, Saudi Arabia Department of Mathematis, Faulty of Siene, Al-Azhar University, Nasr-City, Cairo, Egypt (Reeived 1 September 007, aepted 5 November 007) Abstrat: We apply the first integral method to study the solutions of the variant Boussinesq and the nonlinear Drinfeld-Sokolov systems. This method is based on the theory of ommutative algebra. The new idea in this paper is to find the solution of a system of nonlinear partial differential equations using the first integral method. Key words: theory of ommutative algebra; the first integral method; the variant Boussinesq system; the Drinfeld-Sokolov system 1 Introdution The study of the solutions of partial differential equations (PDEs) has enjoyed an intense period of ativity over the last forty years from both theoretial and numerial points of view. Improvements in numerial tehniques, together with the rapid advane in omputer tehnology, have meant that many of the PDEs arising from engineering and sientifi appliations, whih were previously intratable, an now be routinely solved. In finite differene methods differential operators are approximated and differene equations are solved [1]. In the finite element method [] the ontinuous domain is represented as a olletion of a finite number N of subdomains known as elements. The olletion of elements is alled the finite element mesh. The differential equations for time dependent problems are approximated by the finite element method to obtain a set of ordinary differential equations (ODEs) in time. These differential equations are solved approximately by finite differene methods. In all finite differene and finite element methods it is neessary to have boundary and initial onditions. However, the Adomian deomposition method, whih has been developed by George Adomian [], depends only on the initial onditions and obtains a solution in series whih onverges to the exat solution of the problem. In reent years, ansatz methods have been developed, suh as the tanh-funtion method [-], extended tanh-funtion method [7, 8], the modified extended tanhfuntion method [9, 10], the generalized hyperboli funtion [11, 1]. Other methods are the variable separation method [1, 1] and the first integral method [15-1]. Wu and He [] solved the variant Boussinesq system using the Exp-funtion. He and Abdou [] derived the solutions of the nonlinear Drinfeld-Sokolov system using the Exp-funtion. Yomba [] proposed the extended Fan s sub-equation method to solve the variant Boussinesq system. Wazwaz [5] used the Sine-Cosine and tanh-funtion methods to obtain the solutions of the nonlinear Drinfeld-Sokolov system. The purpose of this paper is to propose a new approah by applying the theory of ommutative algebra to study the solutions of the variant Boussinesq and the nonlinear Drinfeld-Sokolov systems using the first integral method. Corresponding author. address: ahaali 9@yahoo.om address: kamal raslan@yahoo.om Copyright World Aademi Press, World Aademi Union IJNS /1

2 11 International Journal of Nonlinear Siene,Vol.5(008),No.,pp The first integral method Consider the nonlinear PDE: where u(x, t)is the solution of equation (1). We use the transformations where is onstant. Using the hain rule we obtain (.) = d t F (u, u t, u x, u xx, u xt,...) = 0, (1) u(x, t) = f(ξ), ξ = x t, () d ξ (.), We use () to hange the PDE (1) to ODE: x (.) = Next, we introdue new independent variables This yields a system of ODEs d d ξ (.), x (.) = d (.),... () d ξ G(f, f ξ, f ξξ,...) = 0. () X(ξ) = f(ξ), Y = f ξ (ξ). (5) X ξ (ξ) = Y (ξ), Y ξ (ξ) = F 1 (X(ξ), Y (ξ)). If we an find the integrals to equation () under the same onditions of the qualitative theory of ordinary differential equations [], then the general solutions to () an be solved diretly. However, in general, it is very diffiult for us to realize this even for one first integral, beause for a given plane autonomous system, there is no systemati theory that an tell us how to find its first integrals, nor is there a logial way for telling us what these first integrals are. So, we apply the Division Theorem to obtain one first integral to () whih redues () to a first order integrable ODE. Then, an exat solution to (1) is obtained by solving this equation. Now, let us reall the Division Theorem: Division Theorem: Suppose that P (ω, z) and Q(ω, z) are polynomials in C[ω, z] and P (ω, z) is irreduible in C[ω, z]. If Q(ω, z) vanishes at all zero points of P (ω, z), then there exists a polynomial G(ω, z) in C[ω, z] suh that Q[ω, z] = P [ω, z] G[ω, z]. () Appliations In order to illustrate the solution proedure, we onsider the following two systems, the variant Boussinesq, and the nonlinear Drinfeld-Sokolov systems. All alulations in this artile have been done using the aid of the MATHEMATICA software pakage. Example 1: We start with the variant Boussinesq system [5] Introduing the following transformations u t (x, t) + v x (x, t) + u(x, t) u x (x, t) = 0, v t (x, t) + (u(x, t) v(x, t)) x + u xxx (x, t) = 0. u(x, t) = f(ξ), v(x, t) = g(ξ), (7a) (7b) (8) where, ξ = x t, the system (7) beomes df(ξ) + dg(ξ) + f(ξ) df(ξ) = 0, (9a) IJNS for ontribution: editor@nonlinearsiene.org.uk

3 A. H. A. Ali, K. R. Raslan: The First Integral Method for Solving a System of Nonlinear Partial 11 dg(ξ) Integrating equation (9a), we obtain g(ξ) as + d(f(ξ) g(ξ)) + d f(ξ) = 0. (9b) g(ξ) = f(ξ) 1 (f(ξ)) + α, (10) where α is an arbitrary integration onstant. Integrating equation (9b) and substitute g(ξ) we get d f(ξ) d ξ = (β + α ) + ( α) f(ξ) (f(ξ)) + 1 (f(ξ)). (11) Let X = f(ξ), Y = df, and then equation (11) is equivalent to Ẋ(ξ) = Y (ξ), (1a) Ẏ (ξ) = (β + α ) + ( α) X(ξ) (X(ξ)) + 1 (X(ξ)). (1b) Aording to the first integral method, we suppose that X(ξ) and Y (ξ) are nontrivial solutions of (1), and q(x, Y ) = m a i (X) Y i = 0 is an irreduible polynomial in the omplex domain C[X, Y ] suh that q[x(ξ), Y (ξ)] = m a i (X) Y i = 0, (1) where a i (X)(i = 0, 1,..., m) are polynomials of X and a m (X) 0. Equation (1) is alled the first integral to (1), due to the Division Theorem, there exists a polynomial g(x) + h(x) Y in the omplex domain C[X, Y ] suh that dq d ξ = q X X ξ + q Y Y m ξ = (g(x) + h(x) Y ) a i (X) Y i. (1) In this example, we take two different ases, assuming that m = 1 and m = in equation (1). Case I: Suppose that m = 1, by equating the oeffiients of Y i (i =, 1, 0) on both sides of equation (1), we have ȧ 1 (X) = h(x) a 1 (X), (15a) ȧ 0 (X) = g(x) + h(x) a 0 (X), (15b) a 1 (X)((β + α) + ( α)x(ξ) (X(ξ)) + 1 (X(ξ)) ) = g(x)a 0 (X). (15) Sine a i (X)(i = 0, 1) are polynomials, then from (15a) we dedue that a 1 (X) is onstant and h(x) = 0. For simpliity, take a 1 (X) = 1. Balaning the degrees of g(x) and a 0 (X), we onlude that deg (g(x)) = 1 only. Suppose that g(x) = A 1 X + B 0, and A 1 0, then we find a 0 (X) a 0 (X) = A 0 + B 0 X + 1 A 1X. (1) Substituting a 0 (X), a 1 (X) and g(x) in equation (15) and setting all the oeffiients of powers X to be zero, then we obtain a system of nonlinear algebrai equations and by solving it, we obtain β = 0, α = A 0, A 1 = 1, = B 0, (17a) Using the onditions (17a) in equation (1), we obtain β = 0, α = A 0, A 1 = 1, = B 0. (17b) Y = A 0 B 0 X X. (18) IJNS homepage:

4 11 International Journal of Nonlinear Siene,Vol.5(008),No.,pp Combining (18) with (1), we obtain the exat solution to (11) and then the exat solutions to the variant Boussinesq system (7) an be written as: u(x, t) = B 0 A 0 B0 tan[ A0 B0 (x + B 0 t + ξ 0 )], v(x, t) = A 0 B0 A0 B0 (se[ (x + B 0 t + ξ 0 )]). where ξ 0 is an arbitrary onstant. Similarly, in the ase of (17b), from equation (1) we obtain (19) Y = A 0 B 0 X + X, (0) and the exat solutions to the variant Boussinesq system (7) are given by: u(x, t) = B 0 + A 0 B0 tan[ A0 B0 (x B 0 t + ξ 0 )], v(x, t) = A 0+B0 A0 B0 (se[ (x B 0 t + ξ 0 )]). (1) Case II: Suppose that m =, by equating the oeffiients of Y i (i =,, 1, 0) on both sides of equation (1), we have ȧ (X) = h(x) a (X), ȧ 1 (X) = g(x) a (X) + h(x) a 1 (X), ȧ 0 (X) = a (X) ((β + α ) + ( α) X(ξ) (X(ξ)) + 1 (X(ξ)) ) +g(x) a 1 (X) + h(x) a 0 (X), (a) (b) a 1 (X) ((β + α ) + ( α) X(ξ) (X(ξ)) + 1 (X(ξ)) ) = g(x) a 0 (X). (d) Sine a (X) is a polynomial of X, from (a), we dedue that a (X) is a onstant and h(x) = 0. For simpliity, we take a (X) = 1, and hene () an be rewritten as () a (X) = 1, (a) ȧ 1 (X) = g(x), ȧ 0 (X) = ((β + α ) + ( α) X(ξ) (X(ξ)) + 1 (X(ξ)) ) +g(x) a 1 (X) + h(x), (b) a 1 (X) ((β + α ) + ( α) X(ξ) (X(ξ)) + 1 (X(ξ)) ) = g(x) a 0 (X). (d) Balaning the degrees of g(x), a 1 (X) and a 0 (X), we onlude that deg (g(x)) = 1 only. Now we disuss the ase if deg (g(x)) = 1, suppose that g(x) = A 1 X + B 0, and A 1 0, then we obtain a 1 (X) and a 0 (X) as () a 1 (X) = A 0 + B 0 X + 1 A 1 X, () a 0 (X) = d + 1 (A 1 B 0 + ) X (A 1 ) X + 1 X (A 0 A 1 + B 0 + α) + X (A 0 B 0 α β). where A 0 and d are arbitrary integration onstants. Substituting a 0 (X), a 1 (X), a (X) and g(x) in (d) and setting all the oeffiients of powers X to be zero, then we obtain a system of nonlinear algebrai equations and by solving it, we get (5) d = A 0, β = 0, α = A 0, = B 0, A 1 =, (a) d = A 0, β = 0, α = A 0, = B 0, A 1 =. (b) IJNS for ontribution: editor@nonlinearsiene.org.uk

5 A. H. A. Ali, K. R. Raslan: The First Integral Method for Solving a System of Nonlinear Partial 115 Using the onditions (a) in equation (1), we obtain Y = A 0 B 0 X X. (7) Expression (7) is the first integral of (1). Combining equation (1) with equation (7), we obtain the exat solution to equation (11) as follows: f(ξ) = 1 ( B 0 A 0 B0 tan( A0 B0 (ξ + ξ 0) )). (8) where ξ 0 is an arbitrary integration onstant. Then the exat solutions to the variant Boussinesq system (7) an be written as u(x, t) = 1 ( B 0 A 0 B0 tan( A0 B0 (x+ B 0 t +ξ 0) )), v(x, t) = A 0 B 0 ( B 0 A 0 B0 tan( A0 B0 (x+ B0 t 1 8 ( B 0 A 0 B 0 tan( A0 B 0 (x+ B 0 t +ξ 0) )). Similarly, in the ase of (b), from equation (1) we get +ξ 0) )) (9) Y = A 0 B 0 X + X, (0) and the exat solutions to the variant Boussinesq system (7) are given respetively by u(x, t) = 1 (B 0 + A 0 B0 tan( A0 B0 (x B 0 t +ξ 0) )), v(x, t) = A 0 + B 0 (B 0 + A 0 B0 tan( A0 B0 (x B0 t +ξ 0) )) 1 8 (B 0 + A 0 B0 tan( A0 B0 (x B 0 t +ξ 0) )). All these solutions are new exat solutions. Example : We onsider the nonlinear Drinfeld Sokolov system [5] in the form Introduing the following transformations u t (x, t) + v x(x, t) = 0, v t (x, t) v xxx (x, t) + ( u(x, t) v(x, t)) x = 0. u(x, t) = f(ξ), v(x, t) = g(ξ), (1) (a) (b) () where ξ = x t, the system () redues to df(ξ) + d(g(ξ) ) = 0, (a) dg(ξ) Integrating equation (a), we obtain f(ξ) as d g(ξ) + d( f(ξ) g(ξ)) = 0. (b) f(ξ) = (g(ξ)) α, (5) where α is an arbitrary integration onstant. Substituting f(ξ) into equation (b) yields d g(ξ) d ξ = (g(ξ)) + ( α ) g(ξ) β. () IJNS homepage:

6 11 International Journal of Nonlinear Siene,Vol.5(008),No.,pp Using the first integral method we get the system of ODEs Ẋ(ξ) = Y (ξ), (7a) Ẏ (ξ) = (X(ξ)) + ( α ) X(ξ) β. (7b) Aording to the first integral method, we suppose that X(ξ) and Y (ξ) are the nontrivial solutions of (7), and q(x, Y ) = m a i (X) Y i = 0 is an irreduible polynomial in the omplex domain suh that q[x(ξ), Y (ξ)] = m a i (X) Y i = 0, (8) where a i (X)(i = 0, 1,..., m) are polynomials of X and a m (X) 0. Equation (8) is alled the first integral to (7), due to the Division Theorem, there exists a polynomial g(x) + h(x) Y in the omplex domain C[X, Y ] suh that dq d ξ = q X X ξ + q Y Y m ξ = (g(x) + h(x) Y ) a i (X) Y i. (9) In this example we disuss two different values of m assuming that m = 1 and m = in equation (8). Case I: Suppose that m = 1, by equating the oeffiients of Y i (i =, 1, 0) on both sides of equation (9), we have ȧ 1 (X) = h(x) a 1 (X), (0a) a 1 (X) ( (X(ξ)) + ( α ȧ 0 (X) = g(x) + h(x) a 0 (X), ) X(ξ) β) = g(x) a 0 (X). (0b) Sine a 1 (X) is a polynomial of X, then from equation (0a), we dedue that a 1 (X) is a onstant and h(x) = 0. For simpliity, take a 1 (X) = 1. Balaning the degrees of g(x), a 1 (X) and a 0 (X), we onlude that deg g(x) = 1 only. Suppose that g(x) = A 1 X + B 0, and A 1 0, then we find a 0 (X). (0) a 0 (X) = A 0 + B 0 X + 1 A 1X. (1) Substituting a 0 (X), a 1 (X) and g(x) in equation (0) and setting all the oeffiients of powers X to be zero, then we obtain a system of nonlinear algebrai equations and by solving it, we obtain β = 0, α = A 0, A 1 =, B 0 = 0, A0 β = 0, α =, A 1 =, B 0 = 0. Using the onditions (a) in equation (8), we obtain (a) (b) Y = A 0 + X. () Expression () is the first integral of (7). Combining equation (7) with equation (), we obtain the exat solution to (). Hene the exat solutions to the Drinfeld-Sokolov system () an be expressed as: v(x, t) = A0 tan[ A0 (x t + ξ 0 )], u(x, t) = A0 + + A 0 (tan[ A0 (x t+ξ 0 )]). () IJNS for ontribution: editor@nonlinearsiene.org.uk

7 A. H. A. Ali, K. R. Raslan: The First Integral Method for Solving a System of Nonlinear Partial 117 Similarly, in the ase of (b), from equation (8) we obtain Y = A 0 + X, (5) and the exat solution to the Drinfeld-Sokolov system () an be written as: v(x, t) = A0 tanh[ u(x, t) = A A 0 (tanh[ A0 (x t+ξ 0 )]) A0 (x t + ξ 0 )],. () Case II: Suppose that m =, by equating the oeffiients of Y i (i =,, 1, 0) on both sides of equation (9), we have ȧ (X) = h(x) a (X), (7a) ȧ 0 (X) = a (X) ( (X(ξ)) + ( α ȧ 1 (X) = g(x) a (X) + h(x) a 1 (X), a 1 (X) ( (X(ξ)) + ( α ) X(ξ) β) + g(x) a 1 (X) + h(x) a 0 (X), ) X(ξ) β) = g(x) a 0 (X). (7b) (7) (7d) Sine a (X) is a polynomial of X, from (7a), we dedue that a (X) is a onstant and h(x) = 0. For simpliity, we take a (X) = 1. Balaning the degrees of g(x), a 1 (X) and a 0 (X), we onlude that deg g(x) = 1 only. Now we disuss this ase: if deg g(x) = 1, suppose that g(x) = A 1 X + B 0, then we find a 1 (X), and a 0 (X). a 1 (X) = A 0 + B 0 X + 1 A 1 X, a 0 (X) = d + 1 A 1 B 0 X + (A 1 1) X +X (A 0 B 0 + β), 8 + X (A 0 A 1 +B0 + + α) (8a) (8b) where A 0 and d are arbitrary integration onstants. Substituting a 0 (X), a 1 (X) a (X) and g(x) in (7d) and setting all the oeffiients of powers X to be zero, then we obtain a system of nonlinear algebrai equations and by solving it, we obtain d = A 0, β = 0, α = A0, A 1 =, B 0 = 0, (9a) d = A 0, β = 0, α = A0, A 1 =, B 0 = 0. (9b) Using the onditions (9a) in equation (8), we obtain Y = A 0 X. (50) Expression (50) is the first integral of (7). Combining (7) with (50), we obtain the exat solution to equation () as follows: A0 A0 (ξ + ξ 0 ) g(ξ) = tan( ), (51) 8 where ξ 0 is an arbitrary integration onstant. Then the exat solutions to the nonlinear Drinfeld Sokolov system () an be written as v(x, t) = A0 tan( A 0 (x t+ξ 0 ) ), 8 u(x, t) = A0 + A 0 (tan( A 0 (x t+ξ 0 ) )). (5) IJNS homepage:

8 118 International Journal of Nonlinear Siene,Vol.5(008),No.,pp Similarly, in the ase of (9b), from equation (8) we obtain Y = A 0 + X, (5) and the exat solutions to the nonlinear Drinfeld Sokolov system () are given respetively by v(x, t) = A0 tanh( A 0 (x t+ξ 0 ) ), 8 u(x, t) = A0 + A 0 (tanh( A 0 (x t+ξ 0 ) )). 8 (5) All these solutions are new exat solutions. Conlusion The first integral method is applied suessfully for solving the system of nonlinear partial differential equations whih are the variant Boussinesq and the nonlinear Drinfeld-Sokolov systems exatly. Thus, we dedue that the proposed method an be extended to solve many systems of nonlinear partial differential equations whih are arising in the theory of solitons and other areas. The exat solution of the general system of nonlinear partial differential equations using the first integral method is still an open point of researh. Referenes [1] A. R. Mithell, D. F. Griffiths: The finite differene method in partial differential equations. John Wiley & Sons (1980) [] J. N. Reddy: An introdution to the finite element method. nd Eds., MGraw-Hill (199) [] G. Adomian: Solving frontier problem of physis: The deomposition method. (Boston, MA: Kluwer Aademi) (199) [] E. J. Parkes,B. R. Duffy : An automated tanh-funtion method for finding solitary wave solutions to nonlinear evolution equations. Comput. Phys. Commun. 98: (1998) [5] A. H. Khater et al: The tanh-funtion method, a simple transformation and exat analytial solutions for nonlinear reation-diffusion equations. Chaos, Solitons & Fratals. 1: 51-5 (00) [] D. J. Evans, K. R. Raslan: The tanh funtion method for solving some important non-linear partial differential equation. Int. J. Comput. Math. 8(7): (005) [7] E. Fan : Extended tanh-funtion method and its appliations to nonlinear equations. Phys. Lett. A. 77: 1-8 (000) [8] E. Fan: Traveling wave solutions for generalized Hirota-Satsuma oupled KdV systems. Z Naturforsh. A. 5: 1-18 (001) [9] S. A. Elwakil et al: Modified extended tanh-funtion method for solving nonlinear partial differential equations. Phys. Lett A. 99: (00) [10] A.H.A. Ali : The modified extended tanh-funtion method for solving oupled MKdV and oupled Hirota-Satsuma oupled KdV equations. Phys. Lett. A., : 0-5 (007) [11] Y. T. Gao, B. Tian : Generalized hyperboli funtion method with omputerized symboli omputation to onstrut the solitoni solutions to nonlinear equations of Mathematial Physis. Comput. Phys. Commun. 1: (001) IJNS for ontribution: editor@nonlinearsiene.org.uk

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