Abstract
In this paper, we compare optimal homotopy asymptotic method and perturbation-iteration method to solve random nonlinear differential equations. Both of these methods are known to be new and very powerful for solving differential equations. We give some numerical examples to prove these claims. These illustrations are also used to check the convergence of the proposed methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adomian G (1988) A review of the decomposition method in applied mathematics. J Math Anal Appl 135(2):501–544
Aksoy Y, Pakdemirli M (2010) New perturbation–iteration solutions for Bratu-type equations. Comput Math Appl 59(8):2802–2808
Aksoy Y et al (2012) New perturbation-iteration solutions for nonlinear heat transfer equations. Int J Numer Methods Heat Fluid Flow 22(7):814–828
Ali J et al (2010) The solution of multipoint boundary value problems by the optimal homotopy asymptotic method. Comput Math Appl 59(6):2000–2006
Atangana A, Botha JF (2012) Analytical solution of the groundwater flow equation obtained via homotopy decomposition method. J Earth Sci Clim Chang 3:115. doi:10.4172/2157-7617.1000115
Atangana A, Belhaouari SB (2013) Solving partial differential equation with space-and time-fractional derivatives via homotopy decomposition method. Math Probl Eng 2013:9. doi:10.1155/2013/318590
Bayram M et al (2012) Approximate solutions some nonlinear evolutions equations by using the reduced differential transform method. Int J Appl Math Res 1(3):288–302
Bildik N, Deniz S (2015a) Comparison of solutions of systems of delay differential equations using Taylor collocation method, Lambert w function and variational iteration method. Sci Iran Trans D Comput Sci Eng Electr 22(3):1052
Bildik N, Deniz S (2015b) Implementation of Taylor collocation and Adomian decomposition method for systems of ordinary differential equations. AIP Publ 1648:370002
Bildik N, Konuralp A (2006) The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations. Int J Nonlinear Sci Numer Simul 7(1):65–70
Bildik N et al (2006) Solution of different type of the partial differential equation by differential transform method and Adomian’s decomposition method. Appl Math Comp 172(1):551–567
Bulut H, Ergüt M, Evans D (2003) The numerical solution of multidimensional partial differential equations by the decomposition method. Int J Comp Math 80(9):1189–1198
Dolapçı IT, Şenol M, Pakdemirli M (2013) New perturbation iteration solutions for Fredholm and Volterra integral equations. J Appl Math 2013:5. doi:10.1155/2013/682537
Fu WB (1989) A comparison of numerical and analytical methods for the solution of a Riccati equation. Int J Math Educ Sci Technol 20(3):421–427
Gupta AK, Ray SS (2014) On the solutions of fractional burgers-fisher and generalized fisher’s equations using two reliable methods. Int J Math Math Sci 2014:16. doi:10.1155/2014/682910
Gupta AK, Ray SS (2014b) Comparison between homotopy perturbation method and optimal homotopy asymptotic method for the soliton solutions of Boussinesq-Burger equations. Comput Fluids 103:34–41
Gupta AK, Ray SS (2015) The comparison of two reliable methods for accurate solution of time‐fractional Kaup–Kupershmidt equation arising in capillary gravity waves. Math Methods Appl Sci 39(3):583–592. doi:10.1002/mma.3503
Gupta AK, Ray SS (2015b) An investigation with Hermite Wavelets for accurate solution of Fractional Jaulent-Miodek equation associated with energy-dependent Schrödinger potential. Appl Math Comput 270:458–471
He Ji-Huan (1999) Variational iteration method—a kind of non-linear analytical technique: some examples. Int J Non Linear Mech 34(4):699–708
He J-H (2005) Homotopy perturbation method for bifurcation of nonlinear problems. Int J Nonlinear Sci Numer Simul 6(2):207–208
Herisanu N, Marinca V, Madescu Gh (2015) An analytical approach to non-linear dynamical model of a permanent magnet synchronous generator. Wind Energy 18(9):1657–1670
Herisanu N, Marinca V (2012) Optimal homotopy perturbation method for a non-conservative dynamical system of a rotating electrical machine. Zeitschrift für Naturforschung A 67(8-9):509–516
Iqbal S et al (2010) Some solutions of the linear and nonlinear Klein-Gordon equations using the optimal homotopy asymptotic method. Appl Math Comput 216(10):2898–2909
Marinca V, Herişanu N (2008) Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer. Int Commun Heat Mass Transfer 35(6):710–715
Marinca V, Herişanu N, Nemeş I (2008) Optimal homotopy asymptotic method with application to thin film flow. Cent Eur J Phys 6(3):648–653
Marinca V et al (2009) An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate. Appl Math Lett 22(2):245–251
Öziş T, Ağırseven D (2008) He’s homotopy perturbation method for solving heat-like and wave-like equations with variable coefficients. Phys Lett A 372(38):5944–5950
Ray SS, Gupta AK (2015) A numerical investigation of time-fractional modified Fornberg-Whitham equation for analyzing the behavior of water waves. Appl Math Comput 266:135–148
Şenol, Mehmet et al (2013) Perturbation-iteration method for first-order differential equations and systems. Abstr Appl Anal 2013:6. doi:10.1155/2013/704137
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bildik, N., Deniz, S. Comparative Study between Optimal Homotopy Asymptotic Method and Perturbation-Iteration Technique for Different Types of Nonlinear Equations. Iran J Sci Technol Trans Sci 42, 647–654 (2018). https://doi.org/10.1007/s40995-016-0039-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40995-016-0039-2