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An approximate analytic solution of the nonlinear Riccati differential equation. (English) Zbl 1210.34016

Summary: A hybrid method which combines the Adomian decomposition method (ADM), the Laplace transform algorithm and the Padé approximant is introduced to solve the approximate analytic solutions of the nonlinear Riccati differential equations. This hybrid method demonstrates accurate and reliable results, and has a great improvement in the ADM truncated series solution which diverges rapidly as the applicable domain increases. Three examples herein are given to demonstrate a good accuracy and fast convergence in comparison with the exact solution.

MSC:

34A34 Nonlinear ordinary differential equations and systems
34A45 Theoretical approximation of solutions to ordinary differential equations
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[1] Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0802.65122
[2] Chiu, C. H.; Chen, C. K., A decomposition method for solving the convective longitudinal fins with variable thermal conductivity, Int. J. Heat Mass Transfer, 45, 2067-2075 (2002) · Zbl 1011.80011
[3] Kuang, J. H.; Chen, C. J., Adomian decomposition method used for solving nonlinear pull-in behavior in electrostatic micro-actuators, Math. Comput. Model., 41, 1479-1491 (2005) · Zbl 1180.78015
[4] Hsu, J. C.; Lai, H. Y.; Chen, C. K., Free vibration of non-uniform Euler-Bernoulli beams with general elastically end constraints using Adomian modified decomposition method, J. Sound Vib., 318, 965-981 (2008)
[5] Yang, Y. T.; Chien, S. K.; Chen, C. K., A double decomposition method for solving the periodic base temperature in convective longitudinal fins, Energy Convers. Manage., 49, 2910-2916 (2008)
[6] Tien, W. C.; Chen, C. K., Adomian decomposition method by Legendre polynomials, Chaos Solitons Fractals, 39, 2093-2101 (2009)
[7] Khuri, S. A., A Laplace decomposition algorithm applied to a class of nonlinear differential equations, J. Appl. Math., 1, 141-155 (2001) · Zbl 0996.65068
[8] Yusufoğlu, E., Numerical solution of Duffing equation by the Laplace decomposition algorithm, Appl. Math. Comput., 177, 572-580 (2006) · Zbl 1096.65067
[9] Bittanti, S.; Hernandez, D. B.; Zerbi, G., The simple pendulum and the periodic LQG control problem, J. Franklin Inst., 328, 299-315 (1991) · Zbl 0728.93032
[10] Lee, C. H.; Li, T. H.S.; Kung, F. C., A Riccati equation approach to the robust memoryless stabilization of discrete time-delay systems, J. Franklin Inst., 332, 107-114 (1995) · Zbl 0841.93071
[11] Hendrickson, E., Synthesis of finite-dimensional Riccati-based feedback controls for problems arising in structural acoustics, J. Franklin Inst., 336, 565-588 (1999) · Zbl 0931.49021
[12] Phat, V. N., Switched controller design for stabilization of nonlinear hybrid systems with time-varying delays in state and control, J. Franklin Inst., 347, 195-207 (2010) · Zbl 1298.93290
[13] Ku, Y. H., Solution of the Riccati equation by continued fractions, J. Franklin Inst., 293, 59-65 (1972) · Zbl 0271.34017
[14] El-Tawil, M. A.; Bahnasawi, A. A.; Abdel-Naby, A., Solving Riccati differential equation using Adomian’s decomposition method, Appl. Math. Comput., 157, 503-514 (2004) · Zbl 1054.65071
[15] Tang, B. Q.; Li, X. F., A new method for determining the solution of Riccati differential equations, Appl. Math. Comput., 194, 431-440 (2007) · Zbl 1193.65116
[16] Tan, Y.; Abbasbandy, S., Homotopy analysis method for quadratic Riccati differential equation, Commun. Nonlinear Sci. Numer. Simulat., 13, 539-546 (2008) · Zbl 1132.34305
[17] Geng, F.; Lin, Y.; Cui, M., A piecewise variational iteration method for Riccati differential equations, Comput. Math. Appl., 58, 2518-2522 (2009) · Zbl 1189.65164
[18] Aminikhah, H.; Hemmatnezhad, M., An efficient method for quadratic Riccati differential equation, Commun. Nonlinear Sci. Numer. Simulat., 15, 835-839 (2010) · Zbl 1221.65193
[19] Abbasbandy, S., Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian’s decomposition method, Appl. Math. Comput., 172, 485-490 (2006) · Zbl 1088.65063
[20] Abbasbandy, S., Iterated He’s homotopy perturbation method for quadratic Riccati differential equation, Appl. Math. Comput., 175, 581-589 (2006) · Zbl 1089.65072
[21] Abbasbandy, S., A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials, J. Comput. Appl. Math., 207, 59-63 (2007) · Zbl 1120.65083
[22] Baker, G. A.; Graves-Morris, P., Padé Approximants (1996), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0923.41001
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