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Solution of Nonlinear Space-Time Fractional Differential Equations Using the Fractional Riccati Expansion Method

DOI: 10.1155/2013/846283

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The fractional Riccati expansion method is proposed to solve fractional differential equations. To illustrate the effectiveness of the method, space-time fractional Korteweg-de Vries equation, regularized long-wave equation, Boussinesq equation, and Klein-Gordon equation are considered. As a result, abundant types of exact analytical solutions are obtained. These solutions include generalized trigonometric and hyperbolic functions solutions which may be useful for further understanding of the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The periodic and kink solutions are founded as special case. 1. Introduction During recent years, fractional differential equations (FDEs) have attracted much attention due to their numerous applications in areas of physics, biology, and engineering [1–3]. Many important phenomena in non-Brownian motion, signal processing, systems identification, control problem, viscoelastic materials, polymers, and other areas of science are well described by fractional differential equation [4–7]. The most important advantage of using FDEs is their nonlocal property, which means that the next state of a system depends not only upon its current state but also upon all of its historical states [8, 9]. Recently, the fractional functional analysis has been investigated by many researchers [10, 11]. For example, the properties and theorems of Yang-Laplace transforms and Yang-Fourier transforms [12] and their applications to the fractional ordinary differential equations, fractional ordinary differential systems, and fractional partial differential equations have been discussed. Many powerful methods have been established and developed to obtain numerical and analytical solutions of FDEs, such as finite difference method [13], finite element method [14], Adomian decomposition method [15, 16], differential transform method [17], variational iteration method [18–20], homotopy perturbation method [21, 22], the fractional sub-equation method [23], and generalized fractional subequation method [24]. How to extend the existing methods to solve other FDEs is still an interesting and important research problem. Thanks to the efforts of many researchers, several FDEs have been investigated and solved, such as the impulsive fractional differential equations [25], space- and time-fractional advection-dispersion equation [26–28], fractional generalized Burgers’ fluid [29], and fractional heat- and wave-like equations [30], and so forth. The


[1]  A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
[2]  R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000.
[3]  B. J. West, M. Bolognab, and P. Grigolini, Physics of Fractal Operators, Springer, New York, NY, USA, 2003.
[4]  K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
[5]  S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.
[6]  I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
[7]  C. Li, A. Chen, and J. Ye, “Numerical approaches to fractional calculus and fractional ordinary differential equation,” Journal of Computational Physics, vol. 230, pp. 3352–3368, 2011.
[8]  Z. Odibat and S. Momani, “The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics,” Computers and Mathematics with Applications, vol. 58, no. 11-12, pp. 2199–2208, 2009.
[9]  J. H. He, “Some applications of nonlinear fractional differential equations and their approximations,” Bulletin of Science, Technology & Society, vol. 15, no. 2, pp. 86–90, 1999.
[10]  K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, Germany, 2010.
[11]  I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
[12]  X. Yang, “Local fractional integral transforms,” Progress in Nonlinear Science, vol. 4, pp. 1–225, 2011.
[13]  M. Cui, “Compact finite difference method for the fractional diffusion equation,” Journal of Computational Physics, vol. 228, no. 20, pp. 7792–7804, 2009.
[14]  Q. Huang, G. Huang, and H. Zhan, “A finite element solution for the fractional advection–dispersion equation,” Advances in Water Resources, vol. 31, no. 12, pp. 1578–1589, 2008.
[15]  A. M. A. El-Sayed and M. Gaber, “The Adomian decomposition method for solving partial differential equations of fractal order in finite domains,” Physics Letters A, vol. 359, pp. 175–182, 2006.
[16]  A. M. A. El-Sayed, S. H. Behiry, and W. E. Raslan, “Adomian's decomposition method for solving an intermediate fractional advection-dispersion equation,” Computers and Mathematics with Applications, vol. 59, no. 5, pp. 1759–1765, 2010.
[17]  Z. Odibat and S. Momani, “A generalized differential transform method for linear partial differential equations of fractional order,” Applied Mathematics Letters, vol. 21, pp. 194–199, 2008.
[18]  J. H. He, “Variational iteration method for delay differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 2, pp. 235–236, 1997.
[19]  G. Wu and E. W. M. Lee, “Fractional variational iteration method and its application,” Physics Letters A, vol. 374, pp. 2506–2509, 2010.
[20]  S. Guo and L. Mei, “The fractional variational iteration method using He's polynomials,” Physics Letters A, vol. 375, pp. 309–313, 2011.
[21]  J. H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, pp. 257–262, 1999.
[22]  J. H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,” International Journal of Non-Linear Mechanics, vol. 35, pp. 37–43, 2000.
[23]  S. Zhang and H. Q. Zhang, “Fractional sub-equation method and its applications to nonlinear fractional PDEs,” Physics Letters A, vol. 375, pp. 1069–1073, 2011.
[24]  S. Guo , L. Mei, Y. Li, and Y. Sun, “The improved fractional sub-equation method and its applications to the space–time fractional differential equations in fluid mechanics,” Physics Letters A, vol. 376, pp. 407–411, 2012.
[25]  G. M. Mophou, “Existence and uniqueness of mild solutions to impulsive fractional differential equations,” Nonlinear Analysis, Theory, Methods and Applications, vol. 72, no. 3-4, pp. 1604–1615, 2010.
[26]  Q. Huang, G. Huang, and H. Zhan, “A finite element solution for the fractional advection–dispersion equation,” Advances in Water Resources, vol. 31, pp. 1578–1589, 2008.
[27]  W. Jiang and Y. Lin, “Approximate solution of the fractional advection–dispersion equation,” Computer Physics Communications, vol. 181, pp. 557–561, 2010.
[28]  R. K. Pandey, O. P. Singh, and V. K. Baranwal, “An analytic algorithm for the space-time fractional advection-dispersion equation,” Computer Physics Communications, vol. 182, no. 5, pp. 1134–1144, 2011.
[29]  C. Xue, J. Nie, and W. Tan, “An exact solution of start-up flow for the fractional generalized Burgers' fluid in a porous half-space,” Nonlinear Analysis, Theory, Methods and Applications, vol. 69, no. 7, pp. 2086–2094, 2008.
[30]  Y. Molliq R, M. S. M. Noorani, and I. Hashim, “Variational iteration method for fractional heat- and wave-like equations,” Nonlinear Analysis; Real World Applications, vol. 10, no. 3, pp. 1854–1869, 2009.
[31]  K. M. Kolwankar and A. D. Gangal, “Local fractional Fokker-Planck equation,” Physical Review Letters, vol. 80, no. 2, pp. 214–217, 1998.
[32]  Y. Chen, Y. Yan, and K. Zhang, “On the local fractional derivative,” Journal of Mathematical Analysis and Applications, vol. 362, no. 1, pp. 17–33, 2010.
[33]  W. Chen and H. G. Sun, “Multiscale statistical model of fully-developed turbulence particle accelerations,” Modern Physics Letters B, vol. 23, p. 449, 2009.
[34]  J. Cresson, “Scale calculus and the Schr?dinger equation,” Journal of Mathematical Physics, vol. 44, no. 11, pp. 4907–4938, 2003.
[35]  F. Ben Adda and J. Cresson, “About non-differentiable functions,” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 721–737, 2001.
[36]  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.
[37]  G. Jumarie, “New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations,” Mathematical and Computer Modelling, vol. 44, no. 3-4, pp. 231–254, 2006.
[38]  G. Jumarie, “Laplace's transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative,” Applied Mathematics Letters, vol. 22, no. 11, pp. 1659–1664, 2009.
[39]  J.-H. He, “A short remark on fractional variational iteration method,” Physics Letters A, vol. 375, no. 38, pp. 3362–3364, 2011.
[40]  V. I. Karpman, “Radiation by weakly nonlinear shallow-water solitons due to higher-order dispersion,” Physical Review E, vol. 58, no. 4, pp. 5070–5080, 1998.
[41]  H. Leblond and F. Sanchez, “Models for optical solitons in the two-cycle regime,” Physical Review A, vol. 67, no. 1, Article ID 013804, 8 pages, 2003.
[42]  Z. Z. Liu, X. J. Zhou, X. M. Liu, and J. Luo, “Density waves in traffic flow of two kinds of vehicles,” Physical Review E, vol. 67, no. 1, part 2, Article ID 017601, 2003.
[43]  M. A. Manna and V. Merle, “Asymptotic dynamics of short waves in nonlinear dispersive models,” Physical Review E, vol. 57, no. 5, pp. 6206–6209, 1998.
[44]  T. Sakuma and N. Nishiguchi, “Theory of the surface acoustic soliton. V. Approximate soliton solution of the Korteweg de Vries type,” Physical Review B, vol. 41, no. 17, pp. 12117–12121, 1990.
[45]  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 A, vol. 272, no. 1220, pp. 47–78, 1972.
[46]  R. M. Miura, “An inverse scattering method,” Communications on Pure and Applied Mathematics, vol. 21, pp. 467–490, 1974.
[47]  P. A. Clarkson and M. D. Kruskal, “New similarity reductions of the Boussinesq equation,” Journal of Mathematical Physics, vol. 30, no. 10, pp. 2201–2213, 1989.
[48]  M. Toda, “Studies of a non-linear lattice,” Physics Reports, vol. 18, no. 1, pp. 1–123, 1975.
[49]  R. F. Dashen, B. Hasslacher, and A. Neveu, “Nonperturbative methods and extended-hadron models in field theory. II. Two-dimensional models and extended hadrons,” Physical Review D, vol. 10, no. 12, pp. 4130–4138, 1974.


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