×

A modified variational iteration method for solving Riccati differential equations. (English) Zbl 1205.65229

Summary: We introduce a modified variational iteration method (MVIM) for solving Riccati differential equations. The solutions of Riccati differential equations obtained using the traditional variational iteration method (VIM) give good approximations only in the neighborhood of the initial position. The main advantage of the present method is that it can enlarge the convergence region of iterative approximate solutions. Hence, the solutions obtained using the MVIM give good approximations for a larger interval, rather than a local vicinity of the initial position. Numerical results show that the method is simple and effective.

MSC:

65L99 Numerical methods for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Reid, W.T., Riccati differential equations, (1972), Academic Press New York · Zbl 0209.11601
[2] Carinena, J.F.; Marmo, G.; Perelomov, A.M.; Ranada, M.F., Related operators and exact solutions of schrodinger equations, International journal of modern physics A, 13, 4913-4929, (1998) · Zbl 0927.34065
[3] Scott, M.R., Invariant imbedding and its applications to ordinary differential equations, (1973), Addison-Wesley
[4] El-Tawil, M.A.; Bahnasawi, A.A.; Abdel-Naby, A., Solving Riccati differential equation using adomian’s decomposition method, Applied mathematics and computation, 157, 503-514, (2004) · Zbl 1054.65071
[5] Abbasbandy, S., Homotopy perturbation method for quadratic Riccati differential equation and comparison with adomian’s decomposition method, Applied mathematics and computation, 172, 485-490, (2006) · Zbl 1088.65063
[6] Abbasbandy, S., 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, 59-63, (2007) · Zbl 1120.65083
[7] Abbasbandy, S., Iterated he’s homotopy perturbation method for quadratic Riccati differential equation, Applied mathematics and computation, 175, 581-589, (2006) · Zbl 1089.65072
[8] Geng, F.Z.; Lin, Y.Z.; Cui, M.G., A piecewise variational iteration method for Riccati differential equations, Computers and mathematics with applications, 58, 2518-2522, (2009) · Zbl 1189.65164
[9] He, J.H., Variational iteration method—a kind of non-linear analytical technique: some examples, International journal of non-linear mechanics, 34, 699-708, (1999) · Zbl 1342.34005
[10] He, J.H., Variational iteration method for autonomous ordinary differential system, Applied mathematics and computation, 114, 115-123, (2000) · Zbl 1027.34009
[11] He, J.H., Some asymptotic methods for strongly nonlinear equations, International journal of modern physics B, 20, 1141-1199, (2006) · Zbl 1102.34039
[12] He, J.H.; Wu, X.H., Variational iteration method: new development and applications, Computers and mathematics with applications, 54, 881-894, (2007) · Zbl 1141.65372
[13] He, J.H., Variational iteration method—some recent results and new interpretations, Journal of computational and applied mathematics, 207, 3-17, (2007) · Zbl 1119.65049
[14] He, J.H.; Wu, G.C.; Austin, F., The variational iteration method which should be followed, Nonlinear science letters A, 1, 1, 1-30, (2010)
[15] Lu, J.F., Variational iteration method for solving two-point boundary value problems, Journal of computational and applied mathematics, 207, 92-95, (2007) · Zbl 1119.65068
[16] Soliman, A.A.; Abdou, M.A., Numerical solutions of nonlinear evolution equations using variational iteration method, Journal of computational and applied mathematics, 207, 111-120, (2007) · Zbl 1120.65111
[17] Abdou, M.A.; Soliman, A.A., Variational iteration method for solving Burgers and coupled Burgers equations, Journal of computational and applied mathematics, 181, 245-251, (2005) · Zbl 1072.65127
[18] Ganji, D.D.; Sadighi, A., Application of he’s homotopy-perturbation method to nonlinear coupled systems of reaction – diffusion equations, International journal of nonlinear sciences and numerical simulation, 7, 4, 411-418, (2006)
[19] Odibat, Z.M.; Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, International journal of nonlinear sciences and numerical simulation, 7, 27-34, (2006) · Zbl 1401.65087
[20] Shou, D.H.; He, J.H., Application of parameter-expanding method to strongly nonlinear oscillators, International journal of nonlinear sciences and numerical simulation, 8, 1, 121-124, (2007)
[21] Bildik, N.; Konuralp, A., The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations, International journal of nonlinear sciences and numerical simulation, 7, 1, 65-70, (2006) · Zbl 1401.35010
[22] Tatari, M.; Dehghan, M., On the convergence of he’s variational iteration method, Journal of computational and applied mathematics, 207, 121-128, (2007) · Zbl 1120.65112
[23] Biazar, J.; Ghazvini, H., He’s variational iteration method for solving hyperbolic differential equations, International journal of nonlinear sciences and numerical simulation, 8, 3, 311-314, (2007) · Zbl 1193.65144
[24] Ozer, H., Application of the variational iteration method to the boundary value problems with jump discontinuities arising in solid mechanics, International journal of nonlinear sciences and numerical simulation, 8, 4, 513-518, (2007)
[25] Yusufoglu, E., Variational iteration method for construction of some compact and noncompact structures of klein – gordon equations, International journal of nonlinear sciences and numerical simulation, 8, 2, 153-158, (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.