All Title Author
Keywords Abstract

B?cklund Transformation of Fractional Riccati Equation and Infinite Sequence Solutions of Nonlinear Fractional PDEs

DOI: 10.1155/2014/572052

Full-Text   Cite this paper   Add to My Lib


The B?cklund transformation of fractional Riccati equation with nonlinear superposition principle of solutions is employed to establish the infinite sequence solutions of nonlinear fractional partial differential equations in the sense of modified Riemann-Liouville derivative. To illustrate the reliability of the method, some examples are provided. 1. Introduction Recently, nonlinear fractional differential equations increasingly are used to describe nonlinear phenomena in fluid mechanics, biology, engineering, physics, and other areas of science [1–3]. Much efforts have been spent in recent years to develop various techniques to deal with fractional differential equations. However, for the nonlinear differential equations including fractional calculus, the analytical or numerical results are usually difficult to be obtained. It is therefore needed to find a proper method to solve the problem of nonlinear differential equations containing fractional calculus. In the past, several methods have been formulated, such as Adomian decomposition method [4, 5], variational iteration method [6, 7], homotopy perturbation method [8, 9], differential transform method [10, 11], and fractional subequation method [12–14]. S. Zhang and H.-Q. Zhang [12] first proposed a new direct method called fractional subequation method in solving nonlinear time fractional biological population model and ( )-dimensional space-time fractional Fokas equation, based on the homogeneous balance principle and Jumarie’s modified Riemann-Liouville derivative. In this paper, based on the B?cklund transformation technique and the known seed solutions, we will devise effective way for solving fractional partial differential equations. It will be shown that the use of the B?cklund transformation allows us to obtain new exact solutions from the known seed solutions. 2. B?cklund Transformation of the Fractional Riccati Equation and Nonlinear Superposition Principle Firstly, we give some definitions and properties of the modified Riemann-Liouville derivative [15] which are used in this paper. Assume that , denote a continuous (but not necessarily differentiable) function, and let denote a constant discretization span. Jumarie defined the fractional derivative in the limit form where This definition is close to the standard definition of the derivative (calculus for beginners), and as a direct result, the th derivative of a constant, , is zero. An alternative, which is strictly equivalent to (1) is as follows: Some properties of the fractional modified Riemann-Liouville derivative that were


[1]  K. S. Miller and B. Ross, An Introduction To the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993.
[2]  A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, San Diego, Calif, USA, 2006.
[3]  I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
[4]  J. S. Duan, C. Temuer, and R. Randolph, “The Adomian decomposition method with convergence acceleration techniques for nonlinear fractional differential equations,” Computers & Mathematics with Applications, vol. 66, no. 5, pp. 728–736, 2013.
[5]  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, no. 3, pp. 175–182, 2006.
[6]  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.
[7]  G.-C. Wu and E. W. M. Lee, “Fractional variational iteration method and its application,” Physics Letters A, vol. 374, no. 25, pp. 2506–2509, 2010.
[8]  A. A. Elbeleze, A. Kilicman, and B. M. Taib, “Homotopy perturbation method for fractional black-scholes European option pricing equations using sumudu transform,” Mathematical Problems in Engineering, vol. 2013, Article ID 524852, 7 pages, 2013.
[9]  A. M. A. El-Sayed, A. Elsaid, I. L. El-Kalla, and D. Hammad, “A homotopy perturbation technique for solving partial differential equations of fractional order in finite domains,” Applied Mathematics and Computation, vol. 218, no. 17, pp. 8329–8340, 2012.
[10]  S. Momani, Z. Odibat, and V. S. Erturk, “Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation,” Physics Letters A, vol. 370, no. 5-6, pp. 379–387, 2007.
[11]  Z. Odibat and S. Momani, “A generalized differential transform method for linear partial differential equations of fractional order,” Applied Mathematics Letters, vol. 21, no. 2, pp. 194–199, 2008.
[12]  S. Zhang and H.-Q. Zhang, “Fractional sub-equation method and its applications to nonlinear fractional PDEs,” Physics Letters A, vol. 375, no. 7, pp. 1069–1073, 2011.
[13]  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, no. 4, pp. 407–411, 2012.
[14]  B. Lu, “B?cklund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations,” Physics Letters A, vol. 376, no. 28-29, pp. 2045–2048, 2012.
[15]  G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,” Computers and Mathematics with Applications, vol. 51, no. 9-10, pp. 1367–1376, 2006.
[16]  J. Lee and R. Sakthivel, “New exact travelling wave solutions of bidirectional wave equations,” Journal of Physics, vol. 76, no. 6, pp. 819–829, 2011.
[17]  L. Song, Q. Wang, and H. Zhang, “Rational approximation solution of the fractional Sharma-Tasso-Olever equation,” Journal of Computational and Applied Mathematics, vol. 224, no. 1, pp. 210–218, 2009.


comments powered by Disqus