Fifth Order Predictor-Corrector Method for Solving Quadratic Riccati Differential Equations

Abstract

In this paper, fifth order predictor-corrector method is presented for solving quadratic Riccati differential equations. First, the interval is discretized and then the method is formulated by using the Newton’s backward difference interpolation formula. The stability and convergence of the method have been investigated. To validate the applicability of the proposed method, three model examples with exact solutions have been considered and numerically solved by using MATLAB software. The numerical results are presented in tables and figures for different values of mesh size h. Pointwise absolute errors and maximum absolute errors are also estimated. Concisely, the present method gives better result than some existing numerical methods reported in the literature.  

Keywords

• predictor-corrector method,Riccati differential equations,stability analysis

References

1. Allahviranlooa, T. and Behzadib Sh. S., Application of iterative methods for solving general Riccati equation. Int. J. Industrial Mathematics (ISSN 2008-5621), 4 (4), 389-404, 2012.
2. Baba Seidu, A Matrix system for computing the coefficients of the Adams Bashforth-Moulton Predictor-Corrector formulae. International journal of computational and applied mathematics, 6 (3), 215-220, 2011.
3. Gashu Gadisa and Habtamu Garoma, Comparison of higher order Taylor’s method and Runge-Kutta methods for solving first order ordinary differential equations. Journal of Computer and Mathematical Sciences, 8 (1), 12-23, 2017.
4. Gemechis File and Tesfaye Aga, Numerical solution of quadratic Riccati differential equations. Egyptian journal of basic and applied sciences 3, 392–397, 2016.
5. Vinod Mishra and Dimple Rani, Newton-Raphson based modified Laplace Adomian decomposition method for solving quadratic Riccati differential equations. MATEC Web of Conferences, 57 (05001), 2016.
6. Fateme Ghomanjani and Esmaile Khorram, Approximate solution for quadratic Riccati differential equation. Journal of Taibah University for Science 11, 246–250, 2017.
7. Opanuga Abiodun A., Edeki Sunday O., Okagbue Hilary I. and Akinlabi Grace O., A novel approach for solving quadratic Riccati differential equations. International Journal of Applied Engineering Research, 10 (11), 29121-29126, 2015.
8. Biazar, J. and Eslami, M., Differential Transform method for quadratic Riccati differential equation. International Journal of Nonlinear Science, 9 (4), 444-447, 2010.

9. Changqing Yang, Jianhua Hou and Beibo Qin, Numerical solution of Riccati differential equations by using hybrid functions and tau method. International Scholarly and Scientific Research & Innovation, 6 (8), 871-874, 2012.
10. Geng, F., Lin, Y. and Cui, M., A piecewise variational iteration method for Riccati differential equations. Comput. Math. Appl., 58, 2518-2522, 2009.
11. Gulsu, M., and Sezer, M., On the solution of the Riccati equation by the Taylor matrix method. Applied Mathematics and Computation, 176, 414–421, 2006.
12. Khalid, M., Mariam Sultana, Faheem Zaidi and Uroosa Arshad, An effective perturbation iteration algorithm for solving Riccati differential equations. International Journal of Computer Applications, 111 (10), 1-5, 2015.
13. Tan, Y. and Abbasbandy, S., Homotopy analysis method for quadratic Riccati differential equation. Commun. Nonlinear Sci. Numer. Simul., 13, 539-546, 2008.
14. David Eberly, Stability Analysis for Systems of Differential Equations. Geometric Tools, LLC, 2008.