On convergence and error analysis of the parametric iteration method

Authors

  • S. A. Saeed Alavi Department of Mathematics, Faculty of Basic Sciences, Payam-e-Noor University of Tehran, Tehran, Iran.
  • Aghileh Heydari Department of Mathematics, Faculty of Basic Sciences, Payam-e-Noor University of Tehran, Tehran, Iran.
  • Farhad Khellat

DOI:

https://doi.org/10.14419/ijamr.v4i1.3985

Published:

2015-01-28

Keywords:

He’s Variational Iteration Method, Parametric Iteration Method, Convergence, Error Bound.

Abstract

Parametric iteration method falls under the category of the analytic approximate methods for solving various kinds of nonlinear differential equations. Its convergence only for some special problems has been proved. However in this paper, an analysis of error is presented, then due to it, the convergence of method for general problems is proved. To assess the performance of the claimed error bound and also the convergence of the method, numerical experiments are presented performed in MATLAB 2012b.

References

[1] A. Ghorbani, “Toward a New Analytical Method for Solving Nonlinear Fractional Differential Equationsâ€, Comput. Meth. Appl. Mech. Engrg. Vol.197, (2008), pp: 4173-4179. http://dx.doi.org/10.1016/j.cma.2008.04.015.

[2] J. H. He, “Variational iteration method – a kind of non–linear analytical technique: some examplesâ€, Int. J. Non–Linear Mech. Vol.34, (1999), pp: 699-708. http://dx.doi.org/10.1016/S0020-7462(98)00048-1.

[3] S. Yang, A. Xiao, H. Sua, “Convergence of the variational iteration method for solving multi-order fractional differential equationsâ€, Computers and Mathematics with Applications, Vol.60, (2010), pp: 2871–2879. http://dx.doi.org/10.1016/j.camwa.2010.09.044.

[4] S. Yang, A. Xiao, “convergence of variational iteration method for solving multi-delay differential equationsâ€, computers & mathematics with applications, Vol.61, No.8, (2011), pp: 2148-2151.

[5] E. YusufoÄŸlu, “Two convergence theorems of variational iteration method for ordinary differential equationsâ€, Appl. Math. Lett. (2011), http://dx.doi.org/10.1016/j.aml.2011.02.005.

[6] D. Khojasteh,â€convergence of variational iteration method for solving linear systems of ODE’s with constant coefficientsâ€, computers & mathematics with applications, Vol.56, No.8,(2008),pp: 2027-2033.

[7] Z. M. Odibat, “A study on the convergence of variational iteration method,†Mathematical and Computer Modelling Vol.51, (2010), pp: 1181_1192.

[8] J. ‎Saberi-Nadjafi‎, ‎A‎. ‎Ghorbani‎, “Piecewise-truncated parametric iteration method‎: ‎a promising‎ analytical method for solving Abel differential equations"‎, ‎Z‎. ‎Naturforsch‎. Vol.‎65a, (2010), pp: 529-539‎.

[9] A. Ghorbani and J. Saberi-Nadjafi, “A Piecewise-Spectral Parametric Iteration Method for Solving the Nonlinear Chaotic Genesio Systemâ€, Mathematical and Computer Modeling, Vol.54, (2011), pp: 131-139. http://dx.doi.org/10.1016/j.mcm.2011.01.044.

[10] A. Ghorbani, M. Gachpazan, J. Saberi-Nadjafi, “A modified parametric iteration method for solving nonlinear second order BVPsâ€, Comput. Appl. Math. Vol.30, No.3, (2011), pp: 499-515. http://dx.doi.org/10.1590/S1807-03022011000300002.

[11] A. S. Alavi, A. Heydari, “Parametric Iteration Method for Solving Linear Optimal Control Problemsâ€, Applied Mathematics, Vol.3, (2012), pp: 1059-1064. http://dx.doi.org/10.4236/am.2012.39155.

C. K. Chui and G. Chen, “Linear Systems and Optimal Controlâ€, Springer-Verlag, Berlin, Heidelberg, (1989), pp:76-80 http://dx.doi.org/10.1007/978-3-642-61312-8.

View Full Article:

Additional Files