Modification of the Hpm by Using Optimal Newton Interpolation Polynomial for Quadratic Riccati Differential Equation
- Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University, Mashhad, Iran.
- Department of Control, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran.
In this work, an efficient modification of the homotopy analysis method by using optimal Newton interpolation polynomials is given for the approximate solutions of the Riccati differential equations. This presented method can be applied to linear and nonlinear models. Examples show that the method is effective.
- quadratic Riccati differential equation
- modification of the HPM
- Newton interpolation.
S. Abbasbandy, A new application of He’s variational iteration method for quadratic Riccati differential equation by using adomian’s polynomials, Comput. Appl. Math., 207 (2007), 59-63.
W. T. Reid, Riccati differential equations , Mathematics in science and engineering, Academic Press, New York, 86. , (),
S. Abbasbandy, Iterated He’s homotopy perturbation method for quadratic Riccati differential equation, Appl. Math. Comput. , 175 (2006), 581-589.
J. Biazar, H. Aminikhah, Study of convergence of homotopy perturbation method for systems of partial differential, Computers Mathematics with Applications, 58 (2009), 2221-2230.
J. Biazar, H. Ghazvini, Exact solutions for non-linear Schrodinger equations by He’s homotopy perturbation method, Physics Letters A, 366 (2007), 79-84.
SJ. Liao, Beyond perturbation: introduction to the homotopy analysis method, CRC Press, Champan and Hall, Boca Raton (2003)
H. Aminikhah, M. Hemmatnezhad, An effcient method for quadratic Riccati differential equation, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 835-839.
A. Ghorbani, S. Momani, An effective variational iteration algorithm for solving Riccati differential equations, Appl. Math. Lett., 23 (2010), 922-927.
H. Fatoorehchi, H. Abolghasemi, Improving the differential transform method: A novel technique to obtain the differential transforms of nonlinearities by the Adomian polynomials, Applied Mathematical Modeling. , 37 (2013), 6008-6017.
S. Liao, Comparison between the homotopy analysis method and homotopy perturbation method, Applied Mathematics and Computation, 169 (2005), 1186-1194.
R. Abazari, Solution of Riccati types matrix differential equations using matrix differential transform method, Appl. Math. Informatics. , 27, 5-6 (2009), 1133-1143.
N. Alam khan, M. Jamil, A. Ara, Approximate Solutions to Time-Fractional Schrodinger Equation via Homotopy Analysis Method, ISRN Mathematical Physics, 197068 (2012), 1-11.
M. Lakestani, M. Dehghan, A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials, Computer Physics Communications, 181 (2010), 957-966.