- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 603748, 8 pages
A New Method for Riccati Differential Equations Based on Reproducing Kernel and Quasilinearization Methods
Department of Mathematics, Changshu Institute of Technology, Changshu, Jiangsu 215500, China
Received 1 December 2011; Accepted 17 December 2011
Academic Editor: Shaher Momani
Copyright © 2012 F. Z. Geng and X. M. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- 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 solutions 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 comparison 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 differential 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 equation,” 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 differential 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 Applications, vol. 56, no. 12, pp. 3138–3149, 2008.
- F. Mohammadi and M. M. Hosseini, “A comparative study of numerical methods for solving quadratic 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 Publishers 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 ,” Applied Mathematics and Computation, vol. 133, no. 2-3, pp. 643–653, 2002.