Exact Solutions of the Time Fractional BBM-Burger Equation by Novel <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.1173pt" id="M1" height="18.8468pt" version="1.1" viewBox="-0.0657574 -15.7295 49.6991 18.8468" width="49.6991pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.018,0,0,-0.018,6.188,0)"><path id="g113-72" d="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z"/></g><g transform="matrix(.013,0,0,-0.013,18.421,-7.899)"><path id="g50-31" d="M310 541L304 571C290 586 211 619 185 610L80 76L131 52L310 541Z"/></g><g transform="matrix(.018,0,0,-0.018,23.613,0)"><path id="g113-48" d="M368 703H309L44 -163H104L368 703Z"/></g><g transform="matrix(.018,0,0,-0.018,30.99,0)"><use xlink:href="#g113-72"/></g><g transform="matrix(.018,0,0,-0.018,43.222,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg>-Expansion Method

Shakeel, Muhammad; Ul-Hassan, Qazi Mahmood; Ahmad, Jamshad; Naqvi, Tauseef

doi:https://doi.org/10.1155/2014/181594

Advances in Mathematical Physics

On this page

Abstract Introduction Discussion Conclusions References Copyright Related Articles

Research Article | Open Access

Volume 2014 | Article ID 181594 | https://doi.org/10.1155/2014/181594

Exact Solutions of the Time Fractional BBM-Burger Equation by Novel -Expansion Method

Muhammad Shakeel,¹Qazi Mahmood Ul-Hassan,¹Jamshad Ahmad,¹and Tauseef Naqvi¹

Academic Editor: Hossein Jafari

Received15 Feb 2014

Accepted27 May 2014

Published11 Sept 2014

Abstract

The fractional derivatives are used in the sense modified Riemann-Liouville to obtain exact solutions for BBM-Burger equation of fractional order. This equation can be converted into an ordinary differential equation by using a persistent fractional complex transform and, as a result, hyperbolic function solutions, trigonometric function solutions, and rational solutions are attained. The performance of the method is reliable, useful, and gives newer general exact solutions with more free parameters than the existing methods. Numerical results coupled with the graphical representation completely reveal the trustworthiness of the method.

1. Introduction

Differential equations of noninteger order are generalizations of conventional differential equations of integer order [1]. Exploration and applications of integrals and derivatives of random order are efficiently dealt with the field of mathematical analysis, called as fractional calculus, which has engrossed in considerable interest in many disciplines, nowadays. The behavior of many physical systems can be perfectly defined by the fractional theory. In recent years, we cannot rebuff the importance of fractional differential equations because of their numerous applications in the areas of physics and engineering. For example, the nonlinear fluctuation of earthquakes can be modeled with the help of fractional derivatives and the fluid-dynamic traffic model with fractional derivatives can eradicate the problems arising from the huge traffic flow [2, 3]. We apply a generalized fractional complex transform [4–7] to convert the given fractional order differential equation to ordinary differential equation. On account of development of the computer and its exact description of countless real-life problems, fractional calculus has touched the height of fame and success nowadays, although it was invented three centuries ago by Newton and Leibniz. Many important phenomena in electromagnetic, acoustics, viscoelasticity, electrochemistry, and material science are better described by differential equations of noninteger order [8–11]. A physical interpretation of the fractional calculus was given in [12–14]. With the development of symbolic computation software, likeMaple, many researchers developed and established many numerical and analytical methods to search for exact solutions of nonlinear evolution equations (NLEEs), for example, Cole-Hopf transformation [15], Tanh-function method [16–19], inverse scattering transform method [20], variational iteration method [21, 22], Exp-function method [23–26], homogeneous balance method [27, 28], and F-expansion method [29, 30], which are used for searching the exact solutions.

Lately, a straight and crisp method, called -expansion method, was introduced by Wang et al. [31] and confirmed that it is a powerful method for seeking analytic solutions of nonlinear evolution equations (NLEEs). For additional references see the articles [32–37]. In order to establish the efficiency and diligence of -expansion method and to extend the range of applicability, further research has been carried out by several researchers. For illustration, Zhang et al. [38] made an overview of -expansion method for the evolution equations with variable coefficients. Zhang et al. [39] also presented an improved -expansion method to look for more broad traveling wave solutions. Zayed [40] offered a new technique of -expansion method where gratifies the Jacobi elliptic equation, , where , , and are random constants, and obtained new exact solutions. Zayed [41] for a second time offered a different approach of this method in which satisfies the Riccati equation , where and are casual constants.

The -expansion method and the transformed rational function method used by Ma and Lee [42] have a common idea. That is, we initially put the given nonlinear evolution equation (NLEE) into the equivalent ordinary differential equation (ODE), and then ODE can be transformed into a system of arithmetical polynomials with the influential constants. By the solutions of the ordinary differential equation, we can obtain the exact traveling solutions and rational solution of the nonlinear evolution equations.

In this article, we will apply novel -expansion method introduced by Alam et al. [43] to solve the time fractional BBM-Burger equation, whereas, the modified Riemann-Liouville derivative given by Jumarie [44] is used and abundant new families of exact solutions are found. The Jumarie’s modified Riemann-Liouville derivative of order is defined by the following expression: Some important properties of Jumarie’s derivative are

2. Description of the Method

Consider the fractional partial differential equation in the form, where is Jumarie’s modified Riemann-Liouville derivatives of is an unknown function, and is a polynomial in and its various partial derivatives including fractional derivatives in which the highest order derivatives and nonlinear terms are involved.

The main steps of the method are as follows.

Step 1. Li and He [5] projected a fractional complex transformation to convert fractional partial differential equation into ordinary differential equation (ODE), so all analytical methods devoted to the advanced calculus can be easily applied to the fractional calculus. The following complex transformation: where are arbitrary constants with , permits us to convert (5) into an ordinary differential equation of integer order in the form where the primes stand for the ordinary derivatives with respect to .

Step 2. Integrate (7) term by term one or more times, if possible; capitulate constant(s) of integration which can be calculated later on.

Step 3. Suppose that the solution of (7) can be expressed as where Herein or may be zero, but both of them cannot be zero simultaneously. and are constants to be determined later and satisfies the second order nonlinear ordinary differential equation where primes denote the derivative with respect ; , , and are genuine constants.
The Cole-Hopf transformation reduces (10) into the Riccati equation Equation (11) has individual twenty-five solutions (see Zhu, [45] for details).

Step 4. The positive integer can be calculated by balancing the highest order linear term with the highest order nonlinear term in (7).

Step 5. Inserting (8) together with (9) and (10) into (7), we obtain polynomials in and , . Collecting each coefficient of the resulted polynomials to zero acquiesces an overdetermined set of algebraic equations for , , , and .

Step 6. After solving the system of algebraic equations obtained in Step 5, we can obtain the values of the constants. The solutions of (10) together with the obtained values of the constants yield abundant exact traveling wave solutions of the nonlinear evolution equation (5).

3. Application of the Method to the Time Fractional BBM-Burger Equation

Consider the following BBM-Burger equation in time fractional operator form as: S. Kumar and D. Kumar [46] used new fractional homotopy analysis transform method to time fractional BBM-Burger equation and found the series solution of this equation. Song and Zhang [47] and Fakhari et al. [48] find different type of solutions of the fractional BBM-Burger equation by using homotopy analysis method.

By the use of (4), (12) is converted into an ordinary differential equation of integer order and after integrating once, we obtain where is an integration constant.

Considering the homogeneous balance between and in (13), we obtain . Therefore, the trial solution (8) becomes Using (14) into (13), left-hand side is converted into polynomials in and . Equating the coefficients of same power of the resulted polynomials to zero, we attain a set of algebraic equations (which are omitted for the sake of simplicity) for , , , , , , , , and . Solving the overdetermined set of algebraic equations by using the symbolic computation software, such as Maple 13, we obtain the following solution sets:

The Set 1. We consider that where , , , , , and are arbitrary constants.

The Set 2. We consider that where , , , , , and are arbitrary constants.

The Set 3. We consider that where , , , , and are arbitrary constants.

Substituting (15)–(17) into (14), we obtain where Substituting the solutions of (10) into (18) and simplifying, we obtain the following solutions.

When and (or ), and its graph is shown in Figure 1. Consider where and are real constants. Consider the following: When and (or ), where and are real constants such that . Consider the following: When and , and its graph is shown in Figure 2. Consider where is an arbitrary constant.

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

When and , the solution of (12) is where is an arbitrary constant.

Substituting the solutions of (10) into (19) and simplifying, we obtain the following solutions.

When and (or ), the other families of exact solutions of (12) are omitted for convenience.

When and (or ), and its graph is given in Figure 3. Consider When and , the solution of (12) is where is an arbitrary constant.

(a)

(b)

(c)

(d)

We can write down the other families of exact solutions of (12) which are omitted for practicality.

Finally, substituting the solutions of (10) into (20) and simplifying, we obtain the following solutions.

When and (or ), Others families of exact solutions are omitted for the sake of simplicity.

When and (or ), and its graph is given in Figure 4. Consider When and , the solution of (12) is where is an arbitrary constant.

(a)

(b)

(c)

(d)

Other exact solutions of (12) are omitted here for convenience.

4. Discussion

If we replace by and by and put in (10), then the novel -expansion method coincides with Akbar et al.’s generalized and improved -expansion method [36]. On the other hand if we put in (8) and in (10) then the method is identical to the improved -expansion method presented by Zhang et al. [39]. Again if we set , and the negative exponents of are zero in (8), then the method turn out into the basic -expansion method introduced by Wang et al. [31]. At the end, if we put in (10) and are functions of and instead of constants then the method is transformed into the generalized -expansion method developed by Zhang et al. [38]. Thus the methods presented in [31, 36, 38, 39] are only special cases of the novel -expansion method. So our method is more general having more free parameters than the existing methods.

5. Conclusions

A novel -expansion method is applied to fractional partial differential equation successfully. As applications, abundant new exact solutions for time fractional BBM-Burger equation have been successfully obtained. The nonlinear fractional complex transformation for is very important, which ensures that a certain fractional partial differential equation can be converted into another ordinary differential equation of integer order. The obtained solutions are more general with more free parameters. Thus novel -expansion method would be a powerful mathematical tool for solving nonlinear evolution equations. To the best of our knowledge, the solutions obtained in this article are not reported in literature previously.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

A. G. Nikitin and T. A. Barannyk, “Solitary wave and other solutions for nonlinear heat equations,” Central European Journal of Mathematics, vol. 2, no. 5, pp. 840–858, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
View at: MathSciNet
J. H. He, “Some applications of nonlinear fractional differential equations and their applications,” Bulletin of Science, Technology & Society, vol. 15, no. 2, pp. 86–90, 1999.
View at: Google Scholar
Z. B. Li and J. H. He, “Application of the fractional complex transform to fractional differential equations,” Nonlinear Science Letter A, vol. 2, no. 3, pp. 121–126, 2011.
View at: Google Scholar
Z. Li and J. He, “Fractional complex transform for fractional differential equations,” Mathematical & Computational Applications, vol. 15, no. 2, pp. 970–973, 2010.
View at: Google Scholar | MathSciNet
J. He, S. K. Elagan, and Z. B. Li, “Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus,” Physics Letters A, vol. 376, no. 4, pp. 257–259, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
R. W. Ibrahim, “Fractional complex transforms for fractional differential equations,” Advances in Difference Equations, vol. 2012, article 192, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
H. Jafari, A. Kadem, D. Baleanu, and T. Yilmaz, “Solutions of the fractional davey-stewartson equations with variational iteration method,” Romanian Reports in Physics, vol. 64, no. 2, pp. 337–346, 2012.
View at: Google Scholar
H. Jafari and S. Seifi, “Solving a system of nonlinear fractional partial differential equations using homotopy analysis method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 1962–1969, 2009.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Das, R. Kumar, P. K. Gupta, and H. Jafari, “Approximate analytical solutions for fractional space- and time-partial differential equations using homotopy analysis method,” Applications and Applied Mathematics, vol. 5, no. 10, pp. 1641–1659, 2010.
View at: Google Scholar | MathSciNet
H. Jafari and C. M. Khalique, “Analytical solutions of nonlinear fractional differential equations using variational iteration method,” Journal of Nonlinear Systems and Applications, vol. 2, no. 3-4, pp. 148–151, 2011.
View at: Google Scholar
Y. Hao, H. M. Srivastava, and H. Jafari, “Helmholtz and diffusion equations associated with local fractional derivative operators involving the CANtorian and CANtor-type cylindrical coordinates,” Advances in Mathematical Physics, vol. 2013, Article ID 754248, 5 pages, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
X. J. Yang, “Local fractional integral transforms,” Progress in Nonlinear Science, vol. 4, pp. 1–225, 2011.
View at: Google Scholar
X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science, New York, NY, USA, 2012.
A. H. Salas and C. A. Gómez S., “Application of the Cole-Hopf transformation for finding exact solutions to several forms of the seventh-order KdV equation,” Mathematical Problems in Engineering, vol. 2010, Article ID 194329, 14 pages, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
W. Malfliet, “The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations,” Journal of Computational and Applied Mathematics, vol. 164-165, pp. 529–541, 2004.
View at: Publisher Site | Google Scholar
M. A. Abdou, “The extended tanh method and its applications for solving nonlinear physical models,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 988–996, 2007.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. M. Wazwaz, “The extended tanh method for new compact and noncompact solutions for the KP-BBM and the ZK-BBM equations,” Chaos, Solitons and Fractals, vol. 38, no. 5, pp. 1505–1516, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
E. G. Fan, “Extended tanh-function method and its applications to nonlinear equations,” Physics Letters A, vol. 277, no. 4-5, pp. 212–218, 2000.
View at: Publisher Site | Google Scholar | MathSciNet
M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, vol. 149 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, UK, 1991.
View at: Publisher Site | MathSciNet
H. Jafari and H. Tajadodi, “He's variational iteration method for solving fractional Riccati differential equation,” International Journal of Differential Equations, vol. 2010, Article ID ID 764738, 8 pages, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
N. Faraz, Y. Khan, H. Jafari, A. Yildirim, and M. Madani, “Fractional variational iteration method via modified Riemann-Liouville derivative,” Journal of King Saud University-Science, vol. 23, no. 4, pp. 413–417, 2011.
View at: Publisher Site | Google Scholar
J. H. He and X. H. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons & Fractals, vol. 30, no. 3, pp. 700–708, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
H. Naher, F. A. Abdullah, and M. Ali Akbar, “New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the exp-function method,” Journal of Applied Mathematics, vol. 2012, Article ID 575387, 14 pages, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
S. T. Mohyud-Din, “Solutions of nonlinear differential equations by Exp-function method,” World Applied Sciences Journal, vol. 7, pp. 116–147, 2009.
View at: Google Scholar
S. T. Mohyud-Din, M. A. Noor, and A. Waheed, “Exp-function method for generalized travelling solutions of calogero-degasperis-fokas equation,” Zeitschrift fur Naturforschung A: Journal of Physical Sciences, vol. 65, no. 1, pp. 78–84, 2010.
View at: Google Scholar
X. Zhao, L. Wang, and W. Sun, “The repeated homogeneous balance method and its application to nonlinear partial differential equations,” Chaos, Solitons and Fractals, vol. 28, no. 2, pp. 448–453, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
F. Zhaosheng, “Comment on ‘On the extended applications of homogeneous balance method’,” Applied Mathematics and Computation, vol. 158, no. 2, pp. 593–596, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
M. Wang and X. Li, “Applications of $F$ -expansion to periodic wave solutions for a new Hamiltonian amplitude equation,” Chaos, Solitons and Fractals, vol. 24, no. 5, pp. 1257–1268, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
M. A. Abdou, “The extended $F$ -expansion method and its application for a class of nonlinear evolution equations,” Chaos, Solitons & Fractals, vol. 31, no. 1, pp. 95–104, 2007.
View at: Publisher Site | Google Scholar | MathSciNet
M. Wang, X. Li, and J. Zhang, “The $(G^{ʹ} / G)$ -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp. 417–423, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
M. A. Akbar, N. H. M. Ali, and S. T. Mohyud-Din, “The alternative $G^{'}$ /G-expansion method with generalized Riccati equation: application to fifth order (1 + 1)-dimensional Caudrey-Dodd-Gibbon equation,” International Journal of Physical Sciences, vol. 7, no. 5, pp. 743–752, 2012.
View at: Google Scholar
H. Jafari, N. Kadkhoda, and E. Salehpour, “Application of (G′/G)-expansion method to nonlinear Lienard equation,” Indian Journal of Science and Technology, vol. 5, no. 4, pp. 2554–2556, 2012.
View at: Google Scholar
E. J. Parkes, “Observations on the basic $(G' / G)$ -expansion method for finding solutions to nonlinear evolution equations,” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1759–1763, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
M. Ali Akbar and N. H. M. Ali, “The modified alternative (G′/G)-expansion method for finding the exact solutions of nonlinear PDEs in mathematical physics,” International Journal of Physical Sciences, vol. 6, no. 35, pp. 7910–7920, 2011.
View at: Publisher Site | Google Scholar
M. A. Akbar, N. H. M. Ali, and E. M. E. Zayed, “A generalized and improved (G'/G)-expansion method for nonlinear evolution equations,” Mathematical Problems in Engineering, vol. 2012, Article ID 459879, 22 pages, 2012.
View at: Publisher Site | Google Scholar
E. Salehpour, H. Jafari, and N. Kadkhoda, “Application of (G'/G)-expansion method to nonlinear Lienard equation,” Indian Journal of Science and Technology, vol. 5, no. 4, pp. 2554–2556, 2012.
View at: Google Scholar
J. Zhang, X. Wei, and Y. Lu, “A generalized $G^{'} / G$ -expansion method and its applications,” Physics Letters A, vol. 372, no. 20, pp. 3653–3658, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
J. Zhang, F. Jiang, and X. Zhao, “An improved $(G' / G)$ -expansion method for solving nonlinear evolution equations,” International Journal of Computer Mathematics, vol. 87, no. 8, pp. 1716–1725, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
E. M. E. Zayed, “New traveling wave solutions for higher dimensional nonlinear evolution equations using a generalized (G'/G)-expansion method,” Journal of Physics A: Mathematical and Theoretical, vol. 42, no. 19, Article ID 195202, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
E. M. E. Zayed, “The G′/G-expansion method combined with the Riccati equation for finding exact solutions of nonlinear PDEs,” Journal of Applied Mathematics Informatics, vol. 29, no. 1-2, pp. 351–367, 2011.
View at: Google Scholar
W. Ma and J. Lee, “A transformed rational function method and exact solutions to the $3 + 1$ dimensional Jimbo-Miwa equation,” Chaos, Solitons & Fractals, vol. 42, no. 3, pp. 1356–1363, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
M. N Alam, M. A. Akbar, and S. T. Mohyud-Din, “A novel ( $G^{'}$ /G)-expansion method and its application to the Boussinesq equation,” Chinese Physics B, vol. 23, no. 2, Article ID 020203, 2013.
View at: Publisher Site | Google Scholar
G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1367–1376, 2006.
View at: Publisher Site | Google Scholar
S. Zhu, “The generalizing Riccati equation mapping method in non-linear evolution equation: application to $(2 + 1)$ -dimensional Boiti-Leon-Pempinelle equation,” Chaos, Solitons and Fractals, vol. 37, no. 5, pp. 1335–1342, 2008.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Kumar and D. Kumar, “Fractional modelling for BBM-Burger equation by using new homotopy analysis transform method,” Journal of the Association of Arab Universities for Basic and Applied Sciences, 2013.
View at: Publisher Site | Google Scholar
L. Song and H. Zhang, “Solving the fractional BBM-Burgers equation using the homotopy analysis method,” Chaos, Solitons and Fractals, vol. 40, no. 4, pp. 1616–1622, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
A. Fakhari, G. Domairry, and E. Brahimpour, “Approximate explicit solutions of nonlinear BBMB equations by homotopy analysis method and comparison with the exact solution,” Physics Letters A, vol. 368, no. 1-2, pp. 64–68, 2007.
View at: Publisher Site | Google Scholar | MathSciNet

Copyright

Copyright © 2014 Muhammad Shakeel et al. 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.

PDF Download Citation

Download other formats

Order printed copies

Views

1615

Downloads

1550

Citations