×

zbMATH — the first resource for mathematics

A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials. (English) Zbl 1120.65083
Summary: The quadratic Riccati differential equation is solved by J.-H. He’s variational iteration method [Int. J. Non-Linear Mech. 34, No. 4, 699–708 (1999; Zbl 1342.34005)] considering Adomian’s polynomials. Comparisons are made between the Adomian’s decomposition method, J.-H. He’s homotopy perturbation method [Appl. Math. Comput. 151, No. 1, 287–292 (2004; Zbl 1039.65052)] and the exact solution. In this application, we do not have secular terms, and if \(\lambda\) , the Lagrange multiplier, is equal to \(-1\), then the Adomian’s decomposition method is obtained. The results reveal that the proposed method is very effective and simple and can be applied for other nonlinear problems.

MSC:
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abbasbandy, S., Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian’s decomposition method, Appl. math. comput., 172, 485-490, (2006) · Zbl 1088.65063
[2] Abbasbandy, S., Iterated He’s homotopy perturbation method for quadratic Riccati differential equation, Appl. math. comput., 175, 581-589, (2006) · Zbl 1089.65072
[3] Abdou, M.A.; Soliman, A.A., New applications of variational iteration method, Physica D, 211, 1-8, (2005) · Zbl 1084.35539
[4] Abdou, M.A.; Soliman, A.A., Variational iteration method for solving burger’s and coupled Burger’s equations, J. comput. appl. math., 181, 245-251, (2005) · Zbl 1072.65127
[5] Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Academic Publishers Dordrecht · Zbl 0802.65122
[6] Bulut, H.; Evans, D.J., On the solution of the Riccati equation by the decomposition method, Int. J. comput. math., 79, 103-109, (2002) · Zbl 0995.65073
[7] El-Tawil, M.A.; Bahnasawi, A.A.; Abdel-Naby, A., Solving Riccati differential equation using Adomian’s decomposition method, Appl. math. comput., 157, 503-514, (2004) · Zbl 1054.65071
[8] He, J.H., A new approach to nonlinear partial differential equations, Comm. nonlinear sci. numer. simul., 2, 230-235, (1997)
[9] He, J.H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. methods appl. mech. eng., 167, 57-68, (1998) · Zbl 0942.76077
[10] He, J.H., Approximate solution of nonlinear differential equations with convolution product nonlinearities, Comput. methods appl. mech. eng., 167, 69-73, (1998) · Zbl 0932.65143
[11] He, J.H., Variational iteration method—a kind of non-linear analytical technique: some examples, Internat. J. nonlinear mech., 34, 699-708, (1999) · Zbl 1342.34005
[12] He, J.H., Variational iteration method for autonomous ordinary differential systems, Appl. math. comput., 114, 115-123, (2000) · Zbl 1027.34009
[13] He, J.H., A coupling method of homotopy technique and perturbation technique for nonlinear problems, Internat. J. nonlinear mech., 35, 1, 37-43, (2000) · Zbl 1068.74618
[14] He, J.H., Variational principle for some nonlinear partial differential equations with variable coefficients, Chaos, solitons and fractals, 19, 847-851, (2004) · Zbl 1135.35303
[15] He, J.H., The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl. math. comput., 151, 287-292, (2004) · Zbl 1039.65052
[16] He, J.H., Some asymptotic methods for strongly nonlinear equations, Internat. J. modern phys. B, 20, 1141-1199, (2006) · Zbl 1102.34039
[17] J.H. He, Non-Perturbative Methods for Strongly Nonlinear Problems, dissertation.de-Verlag im Internet GmbH, Berlin, 2006.
[18] Liao, S.J., Beyond perturbation: introduction to the homotopy analysis method, (2003), Chapman & Hall, London, CRC Press Boca Raton, FL
[19] Momani, S.; Abuasad, S., Application of He’s variational iteration method to Helmholtz equation, Chaos, solitons and fractals, 27, 1119-1123, (2006) · Zbl 1086.65113
[20] Odibat, Z.M.; Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, Internat. J. nonlinear sci. numer. simul., 7, 27-34, (2006)
[21] Wazwaz, A., A new algorithm for calculating Adomian polynomials for nonlinear operators, Appl. math. comput., 111, 53-69, (2000) · Zbl 1023.65108
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.