International Journal of Nonlinear Analysis and Applications
Solution of Riccati matrix differential equation using new approach of variational iteration method
Document Type : Research Paper
Authors
1 Department of Mathematics and Computer Applications, College of Science, Al-Nahrain University, Iraq.
2 Department of Mathematics, College of Education For Pure Scineces, Ibn Al-Haitham, University of Baghdad, Iraq.
Abstract
To obtain the approximate solution to Riccati matrix differential equations, a new variational iteration approach was proposed, which is suggested to improve the accuracy and increase the convergence rate of the approximate solutons to the exact solution. This technique was found to give very accurate results in a few number of iterations. In this paper, the modified approaches were derived to give modified solutions of proposed and used and the convergence analysis to the exact solution of the derived sequence of approximate solutions is also stated and proved. Two examples were also solved, which shows the reliability and applicability of the proposed approach.
Keywords
Mathematics with Application, 59(2) (2010) 912-918.
[2] R. Abazari, Solution of Riccati types matrix differential equations using matrix differential transform method, J.
Appl. Math. Informatics, 27(5-6) (2009) 1133-1143.
[3] 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, 207(1) (2007) 59-63.
[4] S. Abbasbandy, Homotopy perturbation method for quadratic Riccati differential equationand comparison with
Adomians decomposition method, Applied Mathematics and Computation, 172 (2006) 485-490.
[5] K. H. Al-jizani, N. A. Ahmad and F. S. Fadhel, Variational Iteration Method for Solving Riccati Matrix Differential
Equations, Indonesian Journal of Electrical Engineering and Computer Science, 5(3) (2017) 673-683.
[6] H. Aminikhah, Approximate analytical solution for quadraticRiccati differential equation, Iran J. Numer. Anal.
Optim., 3(2) (2013) 21-31.[7] B. Batiha, A new efficient method for solving quadratic Riccati differential equation, International Journal of
Applied and Mathematical Research, 4(1) (2015 ) 24-29.
[8] R. Bellman and K. Cook, Differential difference equations, Academic Press, Inc., New York, 1963.
[9] E. Davison and M M.aki, The numerical solution of the matrix Ricattidifferential equation, 18(1) (1973) 71-73.
[10] F. Geng, A modified variational iteration method for solving Riccati differential equations, Computers and
Mathematics with Applications, 60(7) (2010) 1868-1872.
[11] A.Ghorbani and S. Momani, An effective variation iteration algorithm for solving Riccati differential equations,
Applied Mathematics Letters, 23(8) (2010) 922-927.
[12] J. H. He, Variational iteration method for delay differential equations, Communications in Nonlinear Science and
Numerical Simulation, 2(4) (1997)235-236.
[13] J. H. He, Variational iteration method - a kind of non-linear analytical technique: some examples, International
Journal of Non-Linear Mechanics, 34(4) 1999,pp.699-708.
[14] J. H.He, Variational iteration method; some recent Results and new interpretations, Journal of Computational
and Applied Mathematics, 207(1) (2007) 3-17.
[15] K. H. Mohammedali, N. A. Ahmad and F. S. Fadhel, He’s Variational Iteration Method for Solving Riccati
Matrix Delay Differential Equations of Variable Coefficients, The 4th International Conference on Mathematical
Sciences. AIP Conference Proceedings, 1830, 020029, (2017) 020029-1 020029-10.
[16] S. Momani and S. Abuasad, Application of He’s variational iteration method to Helmholtz equation, Chaos
Solitons Fractals, 27(5) (2006) 1119-1123.
[17] S. Riaz, M. Rafiq and O. Ahmad, Non standard finite difference method for quadratic Riccati differential equation,
Pakistan, Punjab University J. Math., 47(2), 2015.
[18] M. R. Scott, Invariant imbedding and its applications to ordinary differential equations, Addison-Wesley, 1973.
[19] Sh. Sharma and A. J. Obaid, Mathematical modelling, analysis and design of fuzzy logic controller for the control
of ventilation systems using MATLAB fuzzy logic toolbox, Journal of Interdisciplinary Mathematics, 23(4) (2020)
843-849, DOI: 10.1080/09720502.2020.1727611.
[20] A. M. Wazwaz, The variational iteration method for exact solutions of Laplace equation, Physics letter A, 363(4)
(2007) 260-262.
[21] A. M. Wazwaz, The variational iteration method for analytic treatment of linear and nonlinear ODEs, Applied
Mathematics and Computation, 212(1) (2009) 120-134.
[22] A. M. Wazwaz, The variational iteration method for solving linear and nonlinear Volterra integral and integrodifferential equations, International Journal of Computer Mathematics, 87(5) (2010) 1131-1141.
)
Volume 12, Issue 2
November 2021Pages 1633-1640
- Receive Date: 10 April 2021
- Revise Date: 21 May 2021
- Accept Date: 17 June 2021