## Abstract and Applied Analysis

### A New Method for Riccati Differential Equations Based on Reproducing Kernel and Quasilinearization Methods

#### Abstract

We introduce a new method for solving Riccati differential equations, which is based on reproducing kernel method and quasilinearization technique. The quasilinearization technique is used to reduce the Riccati differential equation to a sequence of linear problems. The resulting sets of differential equations are treated by using reproducing kernel method. The solutions of Riccati differential equations obtained using many existing methods give good approximations only in the neighborhood of the initial position. However, the solutions obtained using the present method give good approximations in a larger interval, rather than a local vicinity of the initial position. Numerical results compared with other methods show that the method is simple and effective.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 603748, 8 pages.

Dates
First available in Project Euclid: 1 April 2013

https://projecteuclid.org/euclid.aaa/1364845168

Digital Object Identifier
doi:10.1155/2012/603748

Mathematical Reviews number (MathSciNet)
MR2898051

Zentralblatt MATH identifier
1237.65090

#### Citation

Geng, F. Z.; Li, X. M. A New Method for Riccati Differential Equations Based on Reproducing Kernel and Quasilinearization Methods. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 603748, 8 pages. doi:10.1155/2012/603748. https://projecteuclid.org/euclid.aaa/1364845168

#### References

• W. T. Reid, Riccati Differential Equations, Academic Press, New York, NY, USA, 1972.
• J. F. Carinena, G. Marmo, A. M. Perelomov, and M. F. Z. Rañada, “Related operators and exact solu-tions of Schrödinger equations,” International Journal of Modern Physics A, vol. 13, no. 28, pp. 4913–4929, 1998.
• M. R. Scott, Invariant Imbedding and Its Applications to Ordinary Differential Equations: an Introduction, Addison-Wesley, London, UK, 1973.
• M. A. El-Tawil, A. A. Bahnasawi, and A. Abdel-Naby, “Solving Riccati differential equation using Adomian's decomposition method,” Applied Mathematics and Computation, vol. 157, no. 2, pp. 503–514, 2004.
• S. Abbasbandy, “Homotopy perturbation method for quadratic Riccati differential equation and com-parison with Adomian's decomposition method,” Applied Mathematics and Computation, vol. 172, no. 1, pp. 485–490, 2006.
• S. Abbasbandy, “A new application of He's variational iteration method for quadratic Riccati differen-tial equation by using Adomian's polynomials,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 59–63, 2007.
• S. Abbasbandy, “Iterated He's homotopy perturbation method for quadratic Riccati differential equa-tion,” Applied Mathematics and Computation, vol. 175, no. 1, pp. 581–589, 2006.
• M. Lakestani and M. Dehghan, “Numerical solution of Riccati equation using the cubic B-spline scaling functions and Chebyshev cardinal functions,” Computer Physics Communications, vol. 181, no. 5, pp. 957–966, 2010.
• F. Z. Geng, Y. Z. Lin, and M. G. Cui, “A piecewise variational iteration method for Riccati differential equations,” Computers & Mathematics with Applications, vol. 58, no. 11-12, pp. 2518–2522, 2009.
• B. Q. Tang and X. F. Li, “A new method for determining the solution of Riccati differential equations,” Applied Mathematics and Computation, vol. 194, no. 2, pp. 431–440, 2007.
• A. Ghorbani and S. Momani, “An effective variational iteration algorithm for solving Riccati diff-erential equations,” Applied Mathematics Letters, vol. 23, no. 8, pp. 922–927, 2010.
• 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.
• Z. Odibat and S. Momani, “Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order,” Chaos, Solitons & Fractals, vol. 36, no. 1, pp. 167–174, 2008.
• S. H. Hosseinnia, A. Ranjbar, and S. Momani, “Using an enhanced homotopy perturbation method in fractional differential equations via deforming the linear part,” Computers & Mathematics with Ap-plications, vol. 56, no. 12, pp. 3138–3149, 2008.
• F. Mohammadi and M. M. Hosseini, “A comparative study of numerical methods for solving quad-ratic Riccati differential equations,” Journal of the Franklin Institute, vol. 348, no. 2, pp. 156–164, 2011.
• M. Cui and Y. Lin, Nonlinear Numerical Analysis in the Reproducing Kernel Space, Nova Science Publish-ers Inc., New York, NY, USA, 2009.
• A. Berlinet and C. Thomas-Agnan, Reproducing Kernel Hilbert Spaces in Probability and Statistics, Kluwer Academic Publishers, Boston, Mass, USA, 2004.
• F. Z. Geng, “New method based on the HPM and RKHSM for solving forced Duffing equations with integral boundary conditions,” Journal of Computational and Applied Mathematics, vol. 233, no. 2, pp. 165–172, 2009.
• F. Z. Geng, “Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method,” Applied Mathematics and Computation, vol. 215, no. 6, pp. 2095–2102, 2009.
• F. Z. Geng and M. Cui, “Solving a nonlinear system of second order boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 1167–1181, 2007.
• H. M. Yao and Y. Z. Lin, “Solving singular boundary-value problems of higher even-order,” Journal of Computational and Applied Mathematics, vol. 223, no. 2, pp. 703–713, 2009.
• C. L. Li and M. G. Cui, “How to solve the equation $\text{AuBu}+\text{Cu}=\text{f}$,” Applied Mathematics and Com-putation, vol. 133, no. 2-3, pp. 643–653, 2002.