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On the solutions fractional Riccati differential equation with modified Riemann-Liouville derivative. (English) Zbl 1251.34011

Summary: 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 \(\alpha\) 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.

MSC:

34A08 Fractional ordinary differential equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34A45 Theoretical approximation of solutions to ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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[1] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. · Zbl 0292.26011
[2] I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999. · Zbl 0924.34008
[3] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006. · Zbl 1092.45003
[4] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008
[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. · doi:10.1111/j.1365-246X.1967.tb02303.x
[6] A. A. Kilbas, H. H. Srivastava, and J. J. Trujillo, Theoryand Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006. · Zbl 1092.45003
[7] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993. · Zbl 0789.26002
[8] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993. · Zbl 0818.26003
[9] G. M. Zaslavsky, Hamiltonian Chaosand Fractional Dynamics, Oxford University Press, 2005. · Zbl 1083.37002
[10] M. Merdan, A. Yıldırım, and A. Gökdo\ugan, “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. · Zbl 1369.94390 · doi:10.1109/TSP.2003.815376
[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. · Zbl 1059.34004 · doi:10.1080/10586458.2003.10504488
[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. · Zbl 0151.13303
[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. · Zbl 0541.93037 · doi:10.1109/TAC.1984.1103613
[16] B. D. O. Anderson and J. B. Moore, Optimal Filtering, Prentice-Hall, Englewood Cliffs, NJ, USA, 1979. · Zbl 0688.93058
[17] W. T. Reid, Riccati Differential Equations: Mathematics in Science and Engineering, vol. 86, Academic Press, New York, NY, USA, 1972. · Zbl 0254.34003
[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. · Zbl 1221.65193 · doi:10.1016/j.cnsns.2009.05.009
[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. · Zbl 1088.65063 · doi:10.1016/j.amc.2005.02.014
[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. · Zbl 1089.65072 · doi:10.1016/j.amc.2005.07.035
[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. · Zbl 1132.34305 · doi:10.1016/j.cnsns.2006.06.006
[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. · Zbl 1120.65083 · doi:10.1016/j.cam.2006.07.012
[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. · Zbl 1221.65205 · doi:10.1016/j.camwa.2011.03.017
[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. · Zbl 1152.34311 · doi:10.1016/j.chaos.2006.06.041
[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. · Zbl 1197.34006 · doi:10.1016/j.chaos.2007.04.018
[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. · Zbl 1207.34020 · doi:10.1155/2010/764738
[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. · Zbl 1107.65121 · doi:10.1016/j.amc.2006.05.008
[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. · Zbl 1342.34005
[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. · Zbl 1141.65372 · doi:10.1016/j.camwa.2006.12.083
[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. · Zbl 0771.60043 · doi:10.1080/00207729308949547
[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. · Zbl 1130.92043 · doi:10.1016/j.mcm.2005.10.003
[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. · Zbl 1181.44001 · doi:10.1016/j.aml.2009.05.011
[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. · Zbl 1171.26305 · doi:10.1016/j.aml.2008.06.003
[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. · Zbl 1075.60068 · doi:10.1016/j.aml.2004.09.012
[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. · Zbl 1024.65093 · doi:10.1016/S0096-3003(99)00293-3
[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. · doi:10.1016/j.jksus.2010.07.025
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