Numerical Solution of Riccati Differential Equations by Using Hybrid Functions and Tau Method

Authors: Changqing Yang, Jianhua Hou, Beibo Qin

Abstract:

A numerical method for Riccati equation is presented in this work. The method is based on the replacement of unknown functions through a truncated series of hybrid of block-pulse functions and Chebyshev polynomials. The operational matrices of derivative and product of hybrid functions are presented. These matrices together with the tau method are then utilized to transform the differential equation into a system of algebraic equations. Corresponding numerical examples are presented to demonstrate the accuracy of the proposed method.

Keywords: Hybrid functions, Riccati differential equation, Blockpulse, Chebyshev polynomials, Tau method, operational matrix.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1333847

References:

[1] W.T.Reid, Riccati Differential Equations, New York, USA: Academic Press,1972.
[2] J. F. Carinena, G. Marmo , A. M. Perelomov, and M. F. Ranada, Related operators and exact solutions of Schr¨odinger equations, International Journal of Modern Physics A 13(1998) 4913-4929.
[3] S. Bittanti, P. Colaneri and G.O. Guardabassi, Periodic solutions of periodic Riccati equations. IEEE Trans. Autom. Control, 29 (7) (1984) 665-667.
[4] B.D.O. Anderson,J.B. Moore, Optimal Filtering, Prentice-Hall, Englewood Cliffs, NJ (1979).
[5] M.A.El-Tawil, A.A.Bahnasawi, A.Abdel-Naby, Solving Riccati differential equation using Adomians decomposition method, Appl. math. comput. 157(2004)503-514.
[6] P. Y. Tsai, C.K. Chen, An approximate analytic solutionof the nonlinear Riccati differential equation, J.Franklin Inst. 347(2011)1850-1862.
[7] S.Abbasbandy, Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomians decomposition method, Appl. math. comput. 172(2006)485-490.
[8] S.Abbasbandy, Iterated He-s homotopy perturbation method for quadratic Riccati equation, Appl. math. comput. 175(2006)581-589.
[9] S.Abbasbandy, A new application of He-s variational iteration method for quardratic Riccati differential equation by using Adomian-s polynomials, J. Comput. Appl. Math. 207(2007)59-63.
[10] M. G¨ulsu, M.Sezer, On the solution of the Riccati equation by the Taylor matrix method, Appl. math. comput. 176(2006)414-421.
[11] F. Mohammadi, M.M. Hosseini, A comparative study of numerical methods for quadratic Riccati differential equations, J.Franklin Inst. 348(2011)156-164.
[12] C. Yang, Numerical Solution of Nonlinear Fredholm Integrodifferential Equations of Fractional Order by Using Hybrid of Block-Pulse Functions and Chebyshev Polynomials, Mathematical Problems in Engineering, vol. 2011, Article ID 341989, 2011. doi:10.1155/2011/341989
[13] M. Sezer, M. Kaynak, Chebyshev polynomials solutions of linear differential equations, Int. J. Math. Educ. Sci. Technol. 27(4) (1996) 607-618.
[14] C. Canuto, M.Y. Hussaini, A.Quarteroni, T.A. Zhang, Spectral Methods on Fluid Dynamics, Springer-Verlag, New York, (1988).
[15] K.Maleknejad, M.Tavassoli Kajani, Solving linear integro-differential equations system by Galerkin methods with hybrid functions, Appl. Math. Comput. 159 , 603-612(2004).