logo

Mathematical Problems in Engineering

PDF
Mathematical Problems in Engineering / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 498249 | https://doi.org/10.1155/2010/498249

Cesar A. Gómez, Alvaro H. Salas, "Exact Solutions for the Generalized BBM Equation with Variable Coefficients", Mathematical Problems in Engineering, vol. 2010, Article ID 498249, 10 pages, 2010. https://doi.org/10.1155/2010/498249

Exact Solutions for the Generalized BBM Equation with Variable Coefficients

Cesar A. Gómez1 and Alvaro H. Salas2,3

1Department of Mathematics, Universidad Nacional de Colombia, Bogotá, Colombia

2Department of Mathematics and Statistics, Universidad Nacional de Colombia, Manizales, Cll 45, Cra 30, Colombia

3Department of Mathematics, Universidad de Caldas, Manizales, Colombia

Academic Editor: Jihuan He
Received23 Nov 2009
Accepted21 Jan 2010
Published15 Mar 2010

Abstract

The variational iteration algorithm combined with the exp-function method is suggested to solve the generalized Benjamin-Bona-Mahony equation (BBM) with variable coefficients. Periodic and soliton solutions are formally derived in a general form. Some particular cases are considered.

1. Introduction

The BBM equation

𝑢𝑡+𝑢𝑢𝑥+𝑢𝑥−𝜇𝑢𝑥𝑥𝑡=0,(1.1) which describes approximately the unidirectional propagation of long waves in certain nonlinear dispersive systems, has been proposed by Benjamin et al. in 1972 [1] as a more satisfactory model than the KdV equation [2]

𝑢𝑡+𝑢𝑢𝑥+𝑢𝑥𝑥𝑥=0.(1.2) It is easy to see that (1.1) can be derived from the equal width EW-equation [3]:

𝑢𝑡+𝑢𝑢𝑥−𝜇𝑢𝑥𝑥𝑡=0,(1.3) by means of the change of variable 𝑢=𝑢+1, that is, by replacing 𝑢 with 𝑢+1. This last equation is considered as an equally valid and accurate model for the same wave phenomena simulated by (1.1) and (1.2). On the other hand, some researches analyzed the generalized KdV equation with variable coefficients

𝑢𝑡+𝜎(𝑡)𝑢𝑝𝑢𝑥+𝜇(𝑡)𝑢𝑥𝑥𝑥=0,(1.4) because this model has important applications in several fields of science [4–7].

Motivated by these facts, we will consider here the generalized EW-equation with variable coefficients

𝑢𝑡+𝜎(𝑡)𝑢𝑝𝑢𝑥−𝜇(𝑡)𝑢𝑥𝑥𝑡=0.(1.5) Using the solutions of (1.5) we obtain exact solutions to the generalized BBM equation

𝑢𝑡+𝜎(𝑡)(𝑢+1)𝑝𝑢𝑥−𝜇(𝑡)𝑢𝑥𝑥𝑡=0,(1.6) of order 𝑝>0.

2. Exact Solutions to Generalized BBM Equation

2.1. The Variational Iteration Method

Consider the following nonlinear equation:

𝐿𝑢(𝑥,𝑡)+𝑁𝑢(𝑥,𝑡)=𝑔(𝑥,𝑡),(2.1) where 𝐿 and 𝑁 are linear and nonlinear operators, respectively, and 𝑔(𝑥,𝑡) is an inhomogeneous term. According to the variational iteration method (VIM) [8–14], a functional correction to (2.1) is given by

𝑢𝑛+1(𝑥,𝑡)=𝑢𝑛(𝑥,𝑡)+𝑡0𝜃(𝜏)𝐿𝑢𝑛(𝑥,𝜏)+𝑁̃𝑢𝑛(𝑥,𝜏)−𝑔(𝑥,𝜏)𝑑𝜏,(2.2) where 𝜃(𝜏) is a general Lagrange's multiplier, which can be identified via the variational theory; the subscript 𝑛≥0 denotes the 𝑛th order approximation and ̃𝑢 is a restricted variation which means 𝛿̃𝑢=0. In this method, we first determine the Lagrange multiplier 𝜃(𝜏) that will be identified optimally via integration by parts. The successive approximation 𝑢𝑛+1 of the solution 𝑢 will be readily obtained upon using the determined Lagrangian multiplier and any selective function 𝑢0. One of the advantages of the VIM, is the free choice of the initial solution 𝑢0(𝑥,𝑡). If we consider a special form to 𝑢0 with arbitrary parameters, using the relations

𝑢𝑛(𝑥,𝑡)=𝑢𝑛+1𝜕(𝑥,𝑡),𝑘𝜕𝑡𝑘𝑢𝑛𝜕(𝑥,𝑡)=𝑘𝜕𝑡𝑘𝑢𝑛+1(𝑥,𝑡),(2.3) we can obtain a set of algebraic equations in the unknowns given by the parameters that appear in 𝑢0. Solving this system, we have exact solutions to (2.1). To solve (1.5), we construct the following functional equation

𝑢𝑛+1(𝑥,𝑡)=𝑢𝑛(𝑥,𝑡)+𝑡0𝜃(𝜏)𝐿𝑢𝑛(𝑥,𝜏)+𝑁̃𝑢𝑛(𝑥,𝜏)𝑑𝜏,(2.4) where

𝐿𝑢𝑛𝑢(𝑥,𝜏)=𝑛𝜏(𝑥,𝜏),𝑁̃𝑢𝑛(𝑥,𝜏)=𝜎(𝜏)(̃𝑢+1)𝑝̃𝑢𝑥(𝑥,𝜏)−𝜇(𝜏)̃𝑢𝑥𝑥𝜏(𝑥,𝜏).(2.5) Taking in (2.4) variation with respect to the independent variable 𝑢𝑛, and noticing that 𝛿𝑁̃𝑢𝑛=0 we have

𝛿𝑢𝑛+1(𝑥,𝑡)=𝛿𝑢𝑛(𝑥,𝑡)+𝛿𝑡0𝜃(𝜏)𝐿𝑢𝑛(𝑥,𝜏)+𝑁̃𝑢𝑛(𝑥,𝜏)𝑑𝜏=𝛿𝑢𝑛(𝑥,𝑡)+𝜃(𝑡)𝛿𝑢𝑛(𝑥,𝑡)−𝑡0𝜃(𝜏)𝛿𝑢𝑛(𝑥,𝜏)𝑑𝜏=0.(2.6) This yields the stationary conditions

𝜃1+𝜃(𝑡)=0,(𝑡)=0.(2.7) Therefore,

𝜃(𝑡)=−1.(2.8) Substituting this value into (2.4) we obtain the formula

𝑢𝑛+1(𝑥,𝑡)=𝑢𝑛(𝑥,𝑡)−𝑡0𝐿𝑢𝑛(𝑥,𝜏)+𝑁̃𝑢𝑛(𝑥,𝜏)𝑑𝜏.(2.9)

Using the wave transformation

𝜉=𝑥+𝜆𝑡+𝜉0,(2.10) setting

𝜕𝑢𝜕𝑡1𝜕(𝜉)=𝑢𝜕𝑡0(𝜉),(2.11) and performing one integration, (2.9) reduces to

𝜆𝑢0(𝜉)+𝜎(𝑡)𝑢𝑝+10𝑝+1(𝜉)−𝜆𝜇(𝑡)𝑢0(𝜉)=0,(2.12) where for sake of simplicity we set the constant of integration equal to zero. With the change of variable

𝑢0(𝜉)=𝑣2/𝑝(𝜉),(2.13) equation (2.12) converts to

𝜆𝑣2(𝜉)−2𝜇(𝑡)(2−𝑝)𝑝2𝜆𝑣2−2𝜇(𝑡)𝑝𝜆𝑣(𝜉)𝑣(𝜉)+𝜎(𝑡)𝑝+1𝑣(𝜉)4=0.(2.14) Observe that if 𝑣(𝜉) is a solution to (2.14), then −𝑣(𝜉) is also a solution to this equation.

2.2. The Exp-Function Method

Recently, He and Wu [15] have introduced the Exp-function method to solve nonlinear differential equations. In particular, the Exp-function method is an effective method for solving nonlinear equations with high nonlinearity. The method has been used in a satisfactory way by other authors to solve a great variety of nonlinear wave equations [15–21]. The Exp-function method is very simple and straightforward, and can be briefly revised as follows: Given the nonlinear partial differential equation

𝐹𝑢,𝑢𝑥,𝑢𝑡,𝑢𝑥𝑥,𝑢𝑥𝑡,𝑢𝑡𝑡,…=0,(2.15) it is transformed to ordinary differential equation

𝐹𝑢,𝑢,𝑢,𝑢,𝑢𝑥𝑡,…=0,(2.16) by mean of wave transformation 𝜉=𝑥+𝜆𝑡+𝜉0. Solutions to (2.16) can then be found using the expression

∑𝑢(𝜉)=𝑑𝑛=−𝑐𝑎𝑛exp(𝑛𝜉)∑𝑞𝑛=−𝑝𝑏𝑛,exp(𝑛𝜉)(2.17) where 𝑐,𝑑,𝑝, and 𝑞 are positive integers which are unknown to be determined later, 𝑎𝑛 and 𝑏𝑛 are unknown constants.

After balancing, we substitute (2.17) into (2.16) to obtain an algebraic systems in the variable 𝜁=exp(𝑛𝜉). Solving the algebraic system we can obtain exact solutions to (2.16) and reversing, solutions to (2.15) in the original variables.

3. Solutions to (2.14) by the Exp-Function Method

Using the Exp-function method, we suppose that solutions to (2.14) can be expressed in the form

∑𝑣(𝜉)=1𝑛=−1𝑎𝑛exp(𝑛𝑟𝜉)∑1𝑚=−1𝑏𝑚=𝑎exp(𝑚𝑟𝜉)−1exp(−𝑟𝜉)+𝑎0+𝑎1exp(𝑟𝜉)𝑏−1exp(−𝑟𝜉)+𝑏0+𝑏1exp(𝑟𝜉).(3.1) We obtain following solutions to (2.14): 𝑣1=±2𝜆𝑘(𝑝+1)(𝑝+2)√2(𝑝+1)(𝑝+2)𝜆exp𝑝/2𝜉𝜇(𝑡)−𝑘2−√𝜎(𝑡)exp𝑝/2𝜉𝑣𝜇(𝑡),𝜆=𝜆(𝑡),2=±2𝜆𝑘(𝑝+1)(𝑝+2)𝜎(𝑡)𝑘2√exp𝑝/2𝜉−√𝜇(𝑡)−2𝜆(𝑝+1)(𝑝+2)exp𝑝/2𝜉𝜇(𝑡),𝜆=𝜆(𝑡).(3.2) Some special solutions are obtained if

𝑘𝜆=𝜆(𝑡)=±22𝑝2+3𝑝+2𝜎(𝑡).(3.3) This choice gives solutions

𝑣3𝑘=±2𝑝csch2√𝜉𝑘𝜇(𝑡),𝜆=22𝑝2𝑣+3𝑝+2𝜎(𝑡),(3.4)4=𝑘2𝑝sech2√𝜉𝑘𝜇(𝑡),𝜆=−22𝑝2𝑣+3𝑝+2𝜎(𝑡),(3.5)5𝑘=±2𝑝csc2√𝜇𝜉𝑘(𝑡),𝜆=−22𝑝2𝜎+3𝑝+2(𝑡).(3.6) Solution (3.6) follows from (3.4) with the identifications 𝜇(𝑡)→−𝜇(𝑡) and √𝑘→−𝑘−1.

𝑣6𝑘=−2𝑝sec2√𝜉𝑘𝜇(𝑡),𝜆=−22𝑝2+3𝑝+2𝜎(𝑡).(3.7) Solution (3.7) follows from (3.4) with the identifications 𝜇(𝑡)→−𝜇(𝑡) and 𝑘→−𝑘.

4. Particular Cases

4.1. Case 1: Solutions to (2.14) When 𝑝=2

Equation (2.14) takes the form

𝜆𝑣2(𝜉)−𝜆𝜇(𝑡)𝑣(𝜉)𝑣1(𝜉)+3𝜎(𝑡)𝑣(𝜉)4=0.(4.1) From (3.2) with 𝑝=2:

𝑣7=±24𝜆𝑘√24𝜆exp1/𝜇𝜉(𝑡)−𝑘2𝜎−√(𝑡)exp1/𝜇𝜉,𝑣(𝑡)8=±24𝜆𝑘𝑘2√𝜎(𝑡)exp1/𝜉−√𝜇(𝑡)−24𝜆exp1/𝜉.𝜇(𝑡)(4.2) From (3.3)–(3.7) with 𝑝=2:

𝑣9𝑘=±21csch√𝜉𝑘𝜇(𝑡),𝜆=2𝑣24𝜎(𝑡),10𝑘=±21sech√𝜉𝑘𝜇(𝑡),𝜆=−2𝑣24𝜎(𝑡),11𝑘=±21csc√𝜉𝑘−𝜇(𝑡),𝜆=−2𝜎𝑣24(𝑡),12𝑘=±21sec√𝜉𝑘𝜇(𝑡),𝜆=−224𝜎(𝑡).(4.3) Other exact solutions are:

𝑣13=±3𝑎22√exp√−2/𝜇(𝑡)𝜉+2√55𝑎exp𝑘−2/𝜇(𝑡)𝜉−223𝑎22√exp√−2/𝜇(𝑡)𝜉+22𝑎exp1−2/𝜇(𝑡)𝜉+22,𝜆=−3𝑘2𝑣𝜎(𝑡),14=±3𝑎2√±2√55𝑎exp2√−2/𝜇(𝑡)𝜉−22exp𝑘−2/𝜇(𝑡)𝜉3𝑎2√+22𝑎exp2√−2/𝜇(𝑡)𝜉+22exp1−2/𝜇(𝑡)𝜉,𝜆=−3𝑘2𝑣𝜎(𝑡),15⎛⎜⎜⎝√=±𝑘1−448+55√3𝑎11+√55exp√−2/𝜇(𝑡)𝜉+228+⎞⎟⎟⎠155,𝜆=−3𝑘2𝑣𝜎(𝑡),16𝑘√=±𝑎±sinh−2/𝜇(𝑡)𝜉√𝑎2√+1±cosh1−2/𝜇(𝑡)𝜉,𝜆=−3𝑘2𝑣𝜎(𝑡),17𝑘√=±𝑎±cosh−2/𝜇(𝑡)𝜉√𝑎2√+1±sinh1−2/𝜇(𝑡)𝜉,𝜆=−3𝑘2𝑣𝜎(𝑡),18√=±𝑘cos2/𝜇(𝑡)𝜉√1±sin𝑘2/𝜇(𝑡)𝜉,𝜆=23𝜎(𝑡).(4.4)

4.2. Case 2: Solutions to (2.14) When 𝑝=4

Equation (2.14) takes the form

𝜆𝑣2𝑣(𝜉)+𝜇(𝑡)𝜆2−𝜆2𝜇(𝑡)𝑣(𝜉)𝑣1(𝜉)+5𝜎(𝑡)𝑣(𝜉)4=0.(4.5) From (3.2) with 𝑝=4:

𝑣19=±60𝜆𝑘√60𝜆exp2/𝜇𝜉(𝑡)−𝑘2𝜎−√(𝑡)exp2/𝜇𝜉𝑣(𝑡),𝜆=𝜆(𝑡),20=±60𝜆𝑘𝜎(𝑡)𝑘2√exp2/𝜉−√𝜇(𝑡)−60𝜆exp2/𝜉𝜇(𝑡),𝜆=𝜆(𝑡).(4.6) From (3.3)–(3.7) with 𝑝=4:

𝑣21𝑘=±22csch√𝜉𝑘𝜇(𝑡),𝜆=2𝑣60𝜎(𝑡),22𝑘=±22sech√𝜉𝑘𝜇(𝑡),𝜆=−2𝑣60𝜎(𝑡),23𝑘=±22csc√𝜉𝑘−𝜇(𝑡),𝜆=−2𝜎𝑣60(𝑡),24𝑘=±22sec√𝜉𝑘−𝜇(𝑡),𝜆=−260𝜎(𝑡).(4.7) Other exact solutions are:

𝑣25𝑘√=±𝑎exp2/𝜉−𝜇(𝑡)−42𝑎2√exp4/𝜉√−𝜇(𝑡)+16𝑎exp2/𝜉1−𝜇(𝑡)+16,𝜆=−5𝑘2𝑣𝜎(𝑡),26𝑘√=±4exp2/−𝜇(𝑡)𝜉−𝑎2𝑎2√+16𝑎exp2/𝜉√−𝜇(𝑡)+16exp4/𝜉1−𝜇(𝑡),𝜆=−5𝑘2𝑣𝜎(𝑡),27⎛⎜⎜⎝3=±2𝑘1−√2±cos2/𝜉⎞⎟⎟⎠4𝜇(𝑡),𝜆=−5𝑘2𝑣𝜎(𝑡),28⎛⎜⎜⎝3=±2𝑘1−√2±sin2/𝜉⎞⎟⎟⎠4𝜇(𝑡),𝜆=−5𝑘2𝜎(𝑡).(4.8) It is clear that using (2.13) we obtain solutions to (1.5). Finally, observe that if 𝑢0(𝑥,𝑡) is a solution of (1.5), then the solutions 𝑢(𝑥,𝑡) to the generalized BBM equation (1.6) are obtained as follows:

𝑢(𝑥,𝑡)=𝑢0(𝑥,𝑡)−1.(4.9)

5. Conclusions

We have considered the generalized EW-equation with variable coefficients and the generalized BBM-equation with variable coefficients. We obtained analytic solutions by using the variational iteration method combined with the exp-function method. With the aid of Mathematica we have derived a lot of different types of solutions for these two models. Combined formal soliton-like solutions as well as kink solutions have been formally derived. The results obtained show that the technique used here can be considered as a powerful method to analyze other types of nonlinear wave equations.

According to [22], there are alternative iteration alorithms, which might be useful for future work. Furthermore, various modifications of the exp-function method have been appeared in open literature, for example, the double exp-function method [23, 24].

Other methods for solving nonlinear differential equations may be found in [25–35].

We think that the results presented in this paper are new in the literature.

References

  1. T. B. Benjamin, J. L. Bona, and J. J. Mahony, “Model equations for long waves in nonlinear dispersive systems,” Philosophical Transactions of the Royal Society of London. Series A, vol. 272, no. 1220, pp. 47–78, 1972. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  2. D. J. Korteweg and G. de Vries, “On the change of long waves advancing in a rectangular canal and a new type of long stationary wave,” Philosophical Magazine, vol. 39, pp. 422–443, 1835. View at: Google Scholar
  3. P. J. Morrison, J. D. Meiss, and J. R. Cary, “Scattering of regularized-long-wave solitary waves,” Physica D, vol. 11, no. 3, pp. 324–336, 1984. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  4. R. M. Miura, “Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation,” Journal of Mathematical Physics, vol. 9, pp. 1202–1204, 1968. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  5. N. Nirmala, M. J. Vedan, and B. V. Baby, “Auto-Bäcklund transformation, Lax pairs, and Painlevé property of a variable coefficient Korteweg-de Vries equation. I,” Journal of Mathematical Physics, vol. 27, no. 11, pp. 2640–2643, 1986. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  6. Z. Liu and C. Yang, “The application of bifurcation method to a higher-order KdV equation,” Journal of Mathematical Analysis and Applications, vol. 275, no. 1, pp. 1–12, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. Y. Zhang, S. Lai, J. Yin, and Y. Wu, “The application of the auxiliary equation technique to a generalized mKdV equation with variable coefficients,” Journal of Computational and Applied Mathematics, vol. 223, no. 1, pp. 75–85, 2009. View at: Publisher Site | Google Scholar
  8. J.-H. He, “Variational iteration method for autonomous ordinary differential systems,” Applied Mathematics and Computation, vol. 114, no. 2-3, pp. 115–123, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  9. J.-H. He, “Homotopy perturbation method: a new nonlinear analytical technique,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 73–79, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  10. J.-H. He, “The homotopy perturbation method nonlinear oscillators with discontinuities,” Applied Mathematics and Computation, vol. 151, no. 1, pp. 287–292, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  11. A.-M. Wazwaz, “The variational iteration method for rational solutions for KdV, K(2,2), Burgers, and cubic Boussinesq equations,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 18–23, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  12. E. Yusufoglu and A. Bekir, “The variational iteration method for solitary patterns solutions of gBBM equation,” Physics Letters. A, vol. 367, no. 6, pp. 461–464, 2007. View at: Publisher Site | Google Scholar | MathSciNet
  13. J.-M. Zhu, Z.-M. Lu, and Y.-L. Liu, “Doubly periodic wave solutions of Jaulent-Miodek equations using variational iteration method combined with Jacobian-function method,” Communications in Theoretical Physics, vol. 49, no. 6, pp. 1403–1406, 2008. View at: Publisher Site | Google Scholar | MathSciNet
  14. C. A. Gómez and A. H. Salas, “The variational iteration method combined with improved generalized tanh-coth method applied to Sawada-Kotera equation,” Applied Mathematics and Computation, 2009. In press. View at: Publisher Site | Google Scholar
  15. J.-H. He and X.-H. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 30, no. 3, pp. 700–708, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  16. S. Zhang, “Exp-function method exactly solving the KdV equation with forcing term,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 128–134, 2008. View at: Publisher Site | Google Scholar | MathSciNet
  17. J.-H. He and L.-N. Zhang, “Generalized solitary solution and compacton-like solution of the Jaulent-Miodek equations using the Exp-function method,” Physics Letters. A, vol. 372, no. 7, pp. 1044–1047, 2008. View at: Publisher Site | Google Scholar | MathSciNet
  18. S. Zhang, “Application of Exp-function method to a KdV equation with variable coefficients,” Physics Letters. A, vol. 365, no. 5-6, pp. 448–453, 2007. View at: Publisher Site | Google Scholar | MathSciNet
  19. A. H. Salas, “Exact solutions for the general fifth KdV equation by the exp function method,” Applied Mathematics and Computation, vol. 205, no. 1, pp. 291–297, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  20. A. H. Salas, C. A. Gómez, and J. E. Castillo Hernańdez, “New abundant solutions for the Burgers equation,” Computers & Mathematics with Applications, vol. 58, no. 3, pp. 514–520, 2009. View at: Publisher Site | Google Scholar | MathSciNet
  21. S. Zhang, “Exp-function method: solitary, periodic and rational wave solutions of nonlinear evolution equations,” Nonlinear Science Letters A, vol. 1, pp. 143–146, 2010. View at: Google Scholar
  22. J.-H. He, G.-C. Wu, and F. Austin, “The variational iteration method which should be followed,” Nonlinear Science Letters A, vol. 1.1, pp. 1–30, 2010. View at: Google Scholar
  23. Z.-D. Dai, C.-J. Wang, S.-Q. Lin, D.-L. Li, and G. Mu, “The three-wave method for nonlinear evolution equations,” Nonlinear Science Letters A, vol. 1, no. 1, pp. 77–82, 2010. View at: Google Scholar
  24. H.-M. Fu and Z.-D. Dai, “Double Exp-function method and application,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 7, pp. 927–933, 2009. View at: Google Scholar
  25. A. H. Salas, “Some solutions for a type of generalized Sawada-Kotera equation,” Applied Mathematics and Computation, vol. 196, no. 2, pp. 812–817, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  26. A. H. Salas and C. A. Gómez, “Computing exact solutions for some fifth KdV equations with forcing term,” Applied Mathematics and Computation, vol. 204, no. 1, pp. 257–260, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  27. A. H. Salas, C. A. Gómez, and J. G. Escobar, “Exact solutions for the general fifth order KdV equation by the extended tanh method,” Journal of Mathematical Sciences: Advances and Applications, vol. 1, no. 2, pp. 305–310, 2008. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  28. A. H. Salas and C. A. Gómez, “A practical approach to solve coupled systems of nonlinear PDE's,” Journal of Mathematical Sciences: Advances and Applications, vol. 3, no. 1, pp. 101–107, 2009. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  29. A. H. Salas and C. A Gómez, “El software Mathematica en la búsqueda de soluciones exactas de ecuaciones diferenciales no lineales en derivadas parciales mediante el uso de la ecuación de Riccati,” in Memorias del Primer Seminario Internacional de Tecnologías en Educación Matemática, vol. 1, pp. 379–387, Universidad Pedagógica Nacional, Santafé de Bogotá, Colombia, 2005. View at: Google Scholar
  30. A. H. Salas, H. J. Castillo, and J. G. Escobar, “About the seventh-order Kaup-Kupershmidt equation and its solutions,” September 2008, http://arxiv.org, arXiv:0809.2865. View at: Google Scholar
  31. A. H. Salas and J. G. Escobar, “New solutions for the modified generalized Degasperis-Procesi equation,” September 2008, http://arxiv.org/abs/0809.2864. View at: Google Scholar
  32. A. H. Salas, “Two standard methods for solving the Ito equation,” May 2008, http://arxiv.org/abs/0805.3362. View at: Google Scholar
  33. A. H. Salas, “Some exact solutions for the Caudrey-Dodd-Gibbon equation,” May 2008, http://arxiv.org/abs/0805.3362. View at: Google Scholar
  34. A. H. Salas S and C. A. Gómez S, “Exact solutions for a third-order KdV equation with variable coefficients and forcing term,” Mathematical Problems in Engineering, vol. 2009, Article ID 737928, 13 pages, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  35. C. A. Gómez S and A. H. Salas, “The Cole-Hopf transformation and improved tanh-coth method applied to new integrable system (KdV6),” Applied Mathematics and Computation, vol. 204, no. 2, pp. 957–962, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright © 2010 Cesar A. Gómez and Alvaro H. Salas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

More related articles

PDF Download Citation Citation
Download other formatsMore
XML
Order printed copiesOrder
Views1073
Downloads635
Citations

Related articles