Fractional variational iteration method (FVIM) is performed to give an approximate analytical solution of nonlinear fractional Riccati differential equation. Fractional derivatives are described in the Riemann-Liouville derivative. A new application of fractional variational iteration method (FVIM) was extended to derive analytical solutions in the form of a series for these equations. The behavior of the solutions and the effects of different values of fractional order are indicated graphically. The results obtained by the FVIM reveal that the method is very reliable, convenient, and effective method for nonlinear differential equations with modified Riemann-Liouville derivative 1. Introduction In recent years, fractional calculus used in many areas such as electrical networks, control theory of dynamical systems, probability and statistics, electrochemistry of corrosion, chemical physics, optics, engineering, acoustics, viscoelasticity, material science and signal processing can be successfully modelled by linear or nonlinear fractional order differential equations [1–10]. As it is well known, Riccati differential equations concerned with applications in pattern formation in dynamic games, linear systems with Markovian jumps, river flows, econometric models, stochastic control, theory, diffusion problems, and invariant embedding [11–17]. Many studies have been conducted on solutions of the Riccati differential equations. Some of them, the approximate solution of ordinary Riccati differential equation obtained from homotopy perturbation method (HPM) [18–20], homotopy analysis method (HAM) [21], and variational iteration method proposed by He [22]. The He’s homotopy perturbation method proposed by He [23–25] the variational iteration method [26] and Adomian decomposition method (ADM) [27] to solve quadratic Riccati differential equation of fractional order. The variational iteration method (VIM), which proposed by He [28, 29], was successfully applied to autonomous ordinary and partial differential equations and other fields. He [30] was the first to apply the variational iteration method to fractional differential equations. In recent years, a new modified Riemann-Liouville left derivative is suggested by Jumarie [31–35]. Recently, the fractional Riccati differential equation is solved with help of new homotopy perturbation method (HPM) [23]. In this paper, we extend the application of the VIM in order to derive analytical approximate solutions to nonlinear fractional Riccati differential equation: subject to the initial conditions where is fractional
References
[1]
K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.
[2]
I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
[3]
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
[4]
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
[5]
M. Caputo, “Linear models of dissipation whose Q is almost frequency independent, part II,” Geophysical Journal International, vol. 13, no. 5, pp. 529–539, 1967.
[6]
A. A. Kilbas, H. H. Srivastava, and J. J. Trujillo, Theoryand Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
[7]
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993.
[8]
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.
[9]
G. M. Zaslavsky, Hamiltonian Chaosand Fractional Dynamics, Oxford University Press, 2005.
[10]
M. Merdan, A. Y？ld？r？m, and A. G？kdo？an, “Numerical solution of time-fraction Modified Equal Width Wave Equation,” Engineering Computations. In press.
[11]
G. A. Einicke, L. B. White, and R. R. Bitmead, “The use of fake algebraic Riccati equations for co-channel demodulation,” IEEE Transactions on Signal Processing, vol. 51, no. 9, pp. 2288–2293, 2003.
[12]
M. Gerber, B. Hasselblatt, and D. Keesing, “The riccati equation: pinching of forcing and solutions,” Experimental Mathematics, vol. 12, no. 2, pp. 129–134, 2003.
[13]
R. E. Kalman, Y. C. Ho, and K. S. Narendra, “Controllability of linear dynamical systems,” Contributions to Differential Equations, vol. 1, pp. 189–213, 1963.
[14]
S. Bittanti, P. Colaneri, and G. De Nicolao, “The periodic Riccati equation,” in The Riccati Equation, Communications and Control Engineering, pp. 127–162, Springer, Berlin, Germany, 1991.
[15]
S. Bittanti, P. Colaneri, and G. O. Guardabassi, “Periodic solutions of periodic Riccati equations,” IEEE Transactions on Automatic Control, vol. 29, no. 7, pp. 665–667, 1984.
[16]
B. D. O. Anderson and J. B. Moore, Optimal Filtering, Prentice-Hall, Englewood Cliffs, NJ, USA, 1979.
[17]
W. T. Reid, Riccati Differential Equations: Mathematics in Science and Engineering, vol. 86, Academic Press, New York, NY, USA, 1972.
[18]
H. Aminikhah and M. Hemmatnezhad, “An efficient method for quadratic Riccati differential equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 4, pp. 835–839, 2010.
[19]
S. Abbasbandy, “Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian's decomposition method,” Applied Mathematics and Computation, vol. 172, no. 1, pp. 485–490, 2006.
[20]
S. Abbasbandy, “Iterated He's homotopy perturbation method for quadratic Riccati differential equation,” Applied Mathematics and Computation, vol. 175, no. 1, pp. 581–589, 2006.
[21]
Y. Tan and S. Abbasbandy, “Homotopy analysis method for quadratic Riccati differential equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 3, pp. 539–546, 2008.
[22]
S. Abbasbandy, “A new application of He's variational iteration method for quadratic Riccati differential equation by using Adomian's polynomials,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 59–63, 2007.
[23]
N. A. Khan, A. Ara, and M. Jamil, “An efficient approach for solving the Riccati equation with fractional orders,” Computers and Mathematics with Applications, vol. 61, no. 9, pp. 2683–2689, 2011.
[24]
Z. Odibat and S. Momani, “Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order,” Chaos, Solitons and Fractals, vol. 36, no. 1, pp. 167–174, 2008.
[25]
J. Cang, Y. Tan, H. Xu, and S. J. Liao, “Series solutions of non-linear Riccati differential equations with fractional order,” Chaos, Solitons and Fractals, vol. 40, no. 1, pp. 1–9, 2009.
[26]
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 764738, 8 pages, 2010.
[27]
S. Momani and N. Shawagfeh, “Decomposition method for solving fractional Riccati differential equations,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1083–1092, 2006.
[28]
J. H. He, “Variational iteration method—a kind of non-linear analytical technique: some examples,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699–708, 1999.
[29]
J. H. He and X. H. Wu, “Variational iteration method: new development and applications,” Computers and Mathematics with Applications, vol. 54, no. 7-8, pp. 881–894, 2007.
[30]
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.
[31]
G. Jumarie, “Stochastic differential equations with fractional Brownian motion input,” International Journal of Systems Science, vol. 24, no. 6, pp. 1113–1131, 1993.
[32]
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.
[33]
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.
[34]
G. Jumarie, “Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions,” Applied Mathematics Letters, vol. 22, no. 3, pp. 378–385, 2009.
[35]
G. Jumarie, “On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion,” Applied Mathematics Letters, vol. 18, no. 7, pp. 817–826, 2005.
[36]
M.-J. Jang, C.-L. Chen, and Y.-C. Liu, “Two-dimensional differential transform for partial differential equations,” Applied Mathematics and Computation, vol. 121, no. 2-3, pp. 261–270, 2001.
[37]
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, vol. 23, no. 4, pp. 413–417, 2010.
