0
Research Papers

The First Integral Method for Exact Solutions of Nonlinear Fractional Differential Equations

[+] Author and Article Information
Ahmet Bekir

Art-Science Faculty
Department of Mathematics-Computer,
Eskisehir Osmangazi University,
Eskisehir 26480, Turkey
e-mail: abekir@ogu.edu.tr

Özkan Güner

Department of Management Information Systems,
School of Applied Sciences,
Dumlupinar University,
Kutahya 43100, Turkey
e-mail: ozkan.guner@dpu.edu.tr

Ömer Ünsal

Art-Science Faculty
Department of Mathematics-Computer,
Eskisehir Osmangazi University,
Eskisehir 26480, Turkey
e-mail: ounsal@ogu.edu.tr

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 7, 2014; final manuscript received July 21, 2014; published online January 14, 2015. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 10(2), 021020 (Mar 01, 2015) (5 pages) Paper No: CND-14-1037; doi: 10.1115/1.4028065 History: Received February 07, 2014; Revised July 21, 2014; Online January 14, 2015

In this paper, we establish exact solutions for some nonlinear fractional differential equations (FDEs). The first integral method with help of the fractional complex transform (FCT) is used to obtain exact solutions for the time fractional modified Korteweg–de Vries (fmKdV) equation and the space–time fractional modified Benjamin–Bona–Mahony (fmBBM) equation. This method is efficient and powerful in solving kind of other nonlinear FDEs.

Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.

References

Miller, K. S., and Ross, B., 1993, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York.
Podlubny, I., 1999, Fractional Differential Equations, Academic, San Diego, CA.
Kilbas, A. A., Srivastava, H. M., and Trujillo, J. J., 2006, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, Netherlands.
El-Sayed, A. M. A., and Gaber, M., 2006, “The Adomian Decomposition Method for Solving Partial Differential Equations of Fractal Order in Finite Domains,” Phys. Lett. A, 359(3), pp. 175–182. [CrossRef]
Safari, M., Ganji, D. D., and Moslemi, M., 2009, “Application of He's Variational Iteration Method and Adomian's Decomposition Method to the Fractional KdV–Burgers–Kuramoto Equation,” Comput. Math. Appl., 58(11–12), pp. 2091–2097. [CrossRef]
Sweilam, N. H., Khader, M. M., and Al-Bar, R. F., 2007, “Numerical Studies for a Multi-Order Fractional Differential Equation,” Phys. Lett. A, 371(1–2), pp. 26–33. [CrossRef]
Inc, M., 2008, “The Approximate and Exact Solutions of the Space- and Time-Fractional Burgers Equations With Initial Conditions by Variational Iteration Method,” J. Math. Anal. Appl., 345(1), pp. 476–484. [CrossRef]
Song, L. N., and Zhang, H. Q., 2009, “Solving the Fractional BBM-Burgers Equation Using the Homotopy Analysis Method,” Chaos, Solitons Fractals, 40(4), pp. 1616–1622. [CrossRef]
Arafa, A. A. M., Rida, S. Z., and Mohamed, H., 2011, “Homotopy Analysis Method for Solving Biological Population Model,” Commun. Theor. Phys., 56(5), pp. 797–800. [CrossRef]
Gepreel, K. A., 2011, “The Homotopy Perturbation Method Applied to the Nonlinear Fractional Kolmogorov–Petrovskii–Piskunov Equations,” Appl. Math. Lett., 24(8), pp. 1428–1434. [CrossRef]
Gupta, P. K., and Singh, M., 2011, “Homotopy Perturbation Method for Fractional Fornberg–Whitham Equation,” Comput. Math. Appl., 61(2), pp. 50–254. [CrossRef]
Odibat, Z., and Momani, S., 2008, “Generalized Differential Transform Method for Linear Partial Differential Equations of Fractional Order,” Appl. Math. Lett., 21(2), pp. 194–199. [CrossRef]
Ertürk, V. S., Momani, S., and Odibat, Z., 2008, “Application of Generalized Differential Transform Method to Multi-Order Fractional Differential Equations,” Commun. Nonlinear Sci. Numer. Simul., 13(8), pp. 1642–1654. [CrossRef]
Zhang, S., and Zhang, H.-Q., 2011, “Fractional Sub-Equation Method and Its Applications to Nonlinear Fractional PDEs,” Phys. Lett. A, 375(7), pp. 1069–1073. [CrossRef]
Tong, B., He, Y., Wei, L., and Zhang, X., 2012, “A Generalized Fractional Sub-Equation Method for Fractional Differential Equations With Variable Coefficients,” Phys. Lett. A, 376(38–39), pp. 2588–2590. [CrossRef]
Guo, S., Mei, L., Li, Y., and Sun, Y., 2012, “The Improved Fractional Sub-Equation Method and Its Applications to the Space–Time Fractional Differential Equations in Fluid Mechanics,” Phys. Lett. A, 376(4), pp. 407–411. [CrossRef]
Lu, B., 2012, “The First Integral Method for Some Time Fractional Differential Equations,” J. Math. Anal. Appl., 395(2), pp. 684–693. [CrossRef]
Zhang, S., Zong, Q.-A., Liu, D., and Gao, Q., 2010, “A Generalized Exp-Function Method for Fractional Riccati Differential Equations,” Commun. Fractional Calculus, 1(1), pp. 48–51.
Bekir, A., Güner, Ö., and Cevikel, A. C., 2013, “Fractional Complex Transform and Exp-Function Methods for Fractional Differential Equations,” Abstr. Appl. Anal., 2013, p. 426462. [CrossRef]
Zheng, B., 2012, “(G'/G) “-Expansion Method for Solving Fractional Partial Differential Equations in the Theory of Mathematical Physics,” Commun. Theor. Phys., 58(5), pp. 623–630. [CrossRef]
Gepreel, K. A., and Omran, S., 2012, “Exact Solutions for Nonlinear Partial Fractional Differential Equations,” Chin. Phys. B, 21(11), p. 110204. [CrossRef]
Feng, Z. S., 2002, “The First Integral Method to Study the Burgers-KdV Equation,” J. Phys. A: Math. Gen., 35(2), pp. 343–350. [CrossRef]
Feng, Z. S., and Wang, X. H., 2003, “The First Integral Method to the Two-Dimensional Burgers-KdV Equation,” Phys. Lett. A, 308(2–3), pp. 173–178. [CrossRef]
Raslan, K. R., 2008, “The First Integral Method for Solving Some Important Nonlinear Partial Differential Equations,” Nonlinear Dyn., 53(4), pp. 281–286. [CrossRef]
Abbasbandy, S., and Shirzadi, A., 2010, “The First Integral Method for Modified Benjamin–Bona–Mahony Equation,” Commun. Nonlinear Sci. Numer. Simul., 15(7), pp. 1759–1764. [CrossRef]
Tascan, F., and Bekir, A., 2009, “Travelling Wave Solutions of the Cahn–Allen Equation by Using First Integral Method,” Appl. Math. Comput., 207(1), pp. 279–282. [CrossRef]
Taghizadeh, N., and Mirzazadeh, M., 2011, “The First Integral Method to Some Complex Nonlinear Partial Differential Equations,” J. Comput. Appl. Math., 235(16), pp. 4871–4877. [CrossRef]
Deng, X., 2008, “Exact Peaked Wave Solution of CH-γ Equation by the First-Integral Method,” Appl. Math. Comput., 206(2), pp. 806–809. [CrossRef]
Kurulay, M., and Bayram, M., 2010, “Approximate Analytical Solution for the Fractional Modified KdV by Differential Transform Method,” Commun. Nonlinear Sci. Numer. Simul., 15, pp. 1777–1782. [CrossRef]
Alzaidy, J. F., 2013, “Fractional Sub-Equation Method and Its Applications to the Space–Time Fractional Differential Equations in Mathematical Physics,” Br. J. Math. Comput. Sci., 3(2), pp. 153–163. [CrossRef]
Jumarie, G., 2006, “Modified Riemann–Liouville Derivative and Fractional Taylor Series of Nondifferentiable Functions Further Results,” Comput. Math. Appl., 51(9–10), pp. 1367–1376. [CrossRef]
Li, Z. B., and He, J. H., 2010, “Fractional Complex Transform for Fractional Differential Equations,” Math. Comput. Appl., 15(5), pp. 970–973.
Li, Z. B., and He, J. H., 2011, “Application of the Fractional Complex Transform to Fractional Differential Equations,” Nonlinear Sci. Lett. A: Math. Phys. Mech., 2(3), pp. 121–126, available at: http://works.bepress.com/ji_huan_he/52.
Ding, T. R., and Li, C. Z., 1996, Ordinary Differential Equations, Peking University Press, Peking, China.
Bourbaki, N., 1972, Commutative Algebra, Addison-Wesley, Paris, France.
Feng, Z., and Wang, X., 2001, “Explicit Exact Solitary Wave Solutions for the Kundu Equation and the Derivative Schrödinger Equation,” Phys. Scr., 64(1), pp. 7–14. [CrossRef]
Feng, Z., and Roger, K., 2007, “Traveling Waves to a Burgers–Korteweg–de Vries-Type Equation With Higher-Order Nonlinearities,” J. Math. Anal. Appl., 328(2), pp. 1435–1450. [CrossRef]

Figures

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In