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A piecewise variational iteration method for Riccati differential equations. (English) Zbl 1189.65164
Summary: We introduce a piecewise variational iteration method for Riccati differential equations, which is a modified variational iteration method (MVIM). 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. However, the solutions obtained using the MVIM give good approximations for a larger interval, rather than a local vicinity of the initial position. Some numerical examples are selected to illustrate the effectiveness and simplicity of the method. Numerical results show that the method does not share the drawback of the conventional VIM and is a satisfactory method for Riccati differential equations.
MSC:
65L99Numerical methods for ODE
34A45Theoretical approximation of solutions of ODE
References:
[1]Reid, W. T.: Riccati differential equations, (1972)
[2]Carinena, J. F.; Marmo, G.; Perelomov, A. M.; Ranada, M. F.: Related operators and exact solutions of Schrödinger equations, International journal of modern physics A 13, 4913-4929 (1998) · Zbl 0927.34065 · doi:10.1142/S0217751X98002298
[3]Scott, M. R.: Invariant imbedding and its applications to ordinary differential equations, (1973)
[4]El-Tawil, M. A.; Bahnasawi, A. A.; Abdel-Naby, A.: Solving Riccati differential equation using adomians decomposition method, Applied mathematics and computation 157, 503-514 (2004) · Zbl 1054.65071 · doi:10.1016/j.amc.2003.08.049
[5]Abbasbandy, S.: Homotopy perturbation method for quadratic Riccati differential equation and comparison with adomians decomposition method, Applied mathematics and computation 172, 485-490 (2006) · Zbl 1088.65063 · doi:10.1016/j.amc.2005.02.014
[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 · doi:10.1016/j.cam.2006.07.012
[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 · doi:10.1016/j.amc.2005.07.035
[8]He, J. H.: Variational iteration method–A kind of nonlinear analytical technique: some examples, International journal of non-linear mechanics 34, 699-708 (1999)
[9]He, J. H.: Variational iteration method for autonomous ordinary differential system, Applied mathematics and computation 114, 115-123 (2000) · Zbl 1027.34009 · doi:10.1016/S0096-3003(99)00104-6
[10]He, J. H.: Some asymptotic methods for strongly nonlinear equations, International journal of modern physics B 20, 1141-1199 (2006) · Zbl 1102.34039 · doi:10.1142/S0217979206033796
[11]He, J. H.; Wu, X. H.: Construction of solitary solution and compaction-like solution by variational iteration method, Chaos, solitons and fractals 29, 108-113 (2006) · Zbl 1147.35338 · doi:10.1016/j.chaos.2005.10.100
[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 · doi:10.1016/j.camwa.2006.12.083
[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 · doi:10.1016/j.cam.2006.07.009
[14]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 · doi:10.1016/j.cam.2006.07.014
[15]Xu, L.: The variational iteration method for fourth order boundary value problems, Chaos, solitons and fractals (2007)
[16]Assas, Laila M. B.: Variational iteration method for solving coupled-KdV equations, Chaos, solitons and fractals (2007)
[17]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 · doi:10.1016/j.cam.2006.07.016
[18]Abdou, M. A.; Soliman, A. A.: Variational iteration method for solving burger’s and coupled burger’s equations, Journal of computational and applied mathematics 181, 245-251 (2005) · Zbl 1072.65127 · doi:10.1016/j.cam.2004.11.032
[19]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, No. 4, 411-418 (2006)
[20]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)
[21]Shou, D. H.; He, J. H.: Application of parameter-expanding method to strongly nonlinear oscillators, International journal of nonlinear sciences and numerical simulation 8, No. 1, 121-124 (2007)
[22]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, No. 1, 65-70 (2006)
[23]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 · doi:10.1016/j.cam.2006.07.017
[24]J. Biazar, H. Ghazvini, He’s variational iteration method for solving hyperbolic differential equations, International Journal of Nonlinear Sciences and Numerical Simulation, 8 (3) 311–314
[25]H. Ozer, 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
[26]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, No. 2, 153-158 (2007)