General Convergence Analysis for the Perturbation Iteration Technique

Year 2017, Volume: 6 , 1 - 9, 30.06.2017

Necdet Bildik

Abstract

In this study, we propose a different approach of the newly developed perturbation iteration method to analyze its convergence properties when solving nonlinear equations. Our main goal is to give some theorems which prove that this technique is convergent under some special conditions. Error estimate is also provided as a result of related theorems. A few interesting problems are investigated to illustrate our arguments.

Keywords

Perturbation iteration method, nonlinear differential equation, convergence, series expansion

References

• Aksoy, Y., Pakdemirli, M., New perturbation–iteration solutions for Bratu-type equations, Computers & Mathematics with Applications, 59(8)(2010), 2802–2808.
• Aksoy, Y. et al., New perturbation-iteration solutions for nonlinear heat transfer equations, International Journal of Numerical Methods for Heat & Fluid Flow, 22(7)(2012), 814–828.
• Barari, A., et al. Application of homotopy perturbation method and variational iteration method to nonlinear oscillator differential equations, Acta Applicandae Mathematicae, 104(2)(2008), 161–171.
• Bayram, M., et al., Approximate solutions some nonlinear evolutions equations by using the reduced differential transform method, International Journal of Applied Mathematical Research, 1(3)(2012), 288–302.
• Bildik, N., Deniz, S., Applications of Taylor collocation method and Lambert W function to the systems of delay differential equations, Turk. J. Math. Comput. Sci., Article ID 20130033, 13 pages, 2013.
• Bildik, N., Deniz, S., Comparison of solutions of systems of delay differential equations using Taylor collocation method, Lambert W function and variational iteration method, Scientia Iranica. Transaction D, Computer Science & Engineering and Electrical Engineering, 22(3)(2015), 1052–1058.
• Bildik, N., Deniz, S., Modified Adomian decomposition method for solving Riccati differential equations, Review of the Air Force Academy, 3(30)(2015), doi: 10.19062/1842-9238.2015.14.3.3.
• Bildik, N., Deniz, S., On the asymptotic stability of some particular differential equations, International Journal of Applied Physics and Mathematics, 5(4)(2015), 252–258.
• Bildik, N., Deniz, S., The use of Sumudu Decomposition Method for solving Predator-Prey Systems, Mathematical Sciences Letters, 5(3)(2016), 285–289.
• Bildik, N., Deniz, S., Modification of Perturbation-Iteration Method to solve different types of nonlinear dierential equations, AIP Conf. Proc., 1798, 020027 (2017); doi: 10.1063/1.4972619.
• Bildik, N., Tosun, M., Deniz,S., Euler matrix method for solving complex differential equations with variable coecients in rectangular domains, International Journal of Applied Physics and Mathematics, 7(1)(2017), 69–78.
• Bildik, N., Deniz, S., A new ecient method for solving delay differential equations and a comparison with other methods, The European Physical Journal Plus, 132(51)(2017). DOI: 10.1140/epjp/i2017-11344-9.
• Boyce, W. E., DiPrima, R. C., Charles, W. H., Elementary Differential Equations and Boundary Value Problems, Vol. 9, New York: Wiley, 1969.
• Deniz, S., Bildik, N., Comparison of Adomian decomposition method and Taylor matrix method in solving dierent kinds of partial differential equations, International Journal of Modelling and Optimization, 4(4)(2014), 292–298.
• Deniz, S., Bildik, N., Applications of optimal perturbation iteration method for solving nonlinear dierential equations”, AIP Conf. Proc., 1798, 020046 (2017); doi: 10.1063/1.4972638.
• Dolapçı, T. et al., New perturbation iteration solutions for Fredholm and Volterra integral equations, Journal of Applied Mathematics, 2013(2013).
• Evans, D., Bulut, H., A new approach to the gas dynamics equation: An application of the decomposition method, International Journal of Computer Mathematics, 79(7)(2002), 817–822.
• Fuy, W. B. A comparison of numerical and analytical methods for the solution of a Riccati equation, International Journal of Mathematical Education in Science and Technology, 20(3)(1989), 421–427.
• Gupta, A. K., Ray, S. S., Comparison between homotopy perturbation method and optimal homotopy asymptotic method for the soliton solutions of Boussinesq–Burger equations, Computers & Fluids, 103(2014), 34–41.
• Gupta, A. K., Ray, S. S., The comparison of two reliable methods for accurate solution of time-fractional Kaup–Kupershmidt equation arising in capillary gravity waves, Mathematical Methods in the Applied Sciences, 39(3)(2016), 583–592.
• He, J. H., Variational iteration method–a kind of non-linear analytical technique: some examples, International Journal of Non-Linear Mechanics, 34(4)(1999), 699–708.
• He, J. H., Homotopy perturbation method for bifurcation of nonlinear problems, International Journal of Nonlinear Sciences and Numerical Simulation, 6(2)(2005), 207–208.
• Hosseini, M. M., Nasabzadeh, H., On the convergence of Adomian decomposition method, Applied mathematics and computation, 182(1)(2006), 536–543.
• Odibat, Z. M., A study on the convergence of variational iteration method, Mathematical and Computer Modelling, 51(9)(2010), 1181–1192.
• Öziş, T., Ağırseven, D., He’s homotopy perturbation method for solving heat-like and wave-like equations with variable coeffcients, Physics Letters A, 372(38)(2008), 5944–5950.
• Şenol, M., et al., Perturbation-Iteration Method for First-Order Differential Equations and Systems, Abstract and Applied Analysis, Vol. 2013. Hindawi Publishing Corporation, 2013.
• Tatari, M., Dehghan, M., On the convergence of He’s variational iteration method, Journal of Computational and Applied Mathematics, 207(1)(2007), 121–128.
• Turkyilmazoglu, M., Convergence of the homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, 12(1-8)(2011), 9–14.
• Vasile, M. et al., An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate, Applied Mathematics Letters, 22(2)(2009), 245–251.
• Vasile, M., Heris¸anu, N., Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer, International Communications in Heat and Mass Transfer, 35(6)(2008), 710–715.