VIM-Pad´e technique for solving nonlinear and delay initial value problems

Document Type : Research Paper


1 Department of Mathematics, Guilan Science and Research Branch, Islamic Azad University, Rasht, Iran.

2 Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran.

3 Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran.

4 Faculty of Finance Sciences, Kharazmi University, Tehran, Iran.


In this work, we employ a combination of variational iteration method (VIM) and Pad´e approximation method, called the VIM-Pad´e technique, to solve some nonlinear initial value problems and a delay differential equation (DDE). Some examples are provided to illustrate the capability and reliability of the technique. The obtained results by using the VIM are compared to the results of this technique. This comparison shows that VIM-Pad´e technique is more effective than VIM and yields faster convergence compared to the VIM.


[1] T. A. Abassy, M. El-Tawil, and H. Kamel, The Solution of KdV and mKdV equations using Adomian Pad´e approximation, Internat. J. Nonlinear Sci. Numer. Simulation, 5(4) (2004), 327-339.
[2] S. Abbasbandy, The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys. Lett. A, 360 (2006), 109-113.
[3] S. Abbasbandy, The application of the homotopy analysis method to nonlinear equations arising in heat transfer, Phys. Lett. A, 360 (2006), 109-113.
[4] S. Abbasbandy, The application of homotopy analysis method to solve a generalized HirotaSatsuma coupled KdV equation, Phys. Lett. A, 361 (2007), 478-483.
[5] S. Abbasbandy, Soliton solutions for the 5th-order KdV equation with the homotopy analysis method, Nonlinear Dyn., 51 (2008), 83-87.
[6] F. Abidi and K. Omrani, The homotopy analysis method for solving the Fornberg-Whitham equation and comparison with Adomians decomposition method, Comput. Math. Appl., 59 (2010). 2743-2750.
[7] G. Adomian, A review of the decomposition method and some recent results for nonlinear equations, Comput. Math. Appl., 21 (1991), 101-127.
[8] E. Alizadeh, K. Sedighi, M. Farhadi, and H. R. Ebrahimi-Kebria, Analytical approximate solution of the cooling problem by Adomian decomposition method, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 462-472.
[9] M. L. H. O. Arino and E. A. Dads, Delay Differential Equations and Applications , IOS Press/Springer, Amsterdam, 2002.
[10] G. A. Jr. Baker. Essentials of Pad´e Approximants, Academic Press, 1975.
[11] R. Bellman and K. L. Cooke, Differential-difference Equations , Academic Press, New York, 1963.
[12] A. Beiranvand, K. Ivaz, Solving The Stefan Problem with Kinetics, Computational Methods for Differential Equations, 2(1) (2014), 37-49.
[13] J. Biazar and H. Ghazvini, Hes variational iteration method for solving hyperbolic differential equations, Int. J. Nonlinear Sci. Numer. Simul., 8 (3) (2007), 311-314.
[14] J. Biazar, P. Gholamin, and K. Hosseini, Variational iteration method for solving Fokker-Planck equation, J. Franklin Inst., 347 (2010), 1137-1147.
[15] J. Biazar, R. Ansari, K. Hosseini, and P. Gholamin, Obtaining DAlemberts wave formula from variational iteration and homotopy perturbation methods, Mathematical Sciences, In press.
[16] G. A. Bocharov and F. A. Rihan, Numerical modelling in biosciences using delay differential equations, Journal of Computational and Applied Mathematics, 125(1-2) (2000), 183-199.
[17] J. Cheng, S. J. Liao, R. N. Mohapatra, and K. Vajravelu, Series solutions of nano boundary layer flows by means of the homotopy analysis method, J. Math. Anal. Appl., 343 (2008), 233-245.
[18] M. Dehghan, J. Manafian, and A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numerical Methods for Partial Differential Equations Journal, 26 (2010).
[19] M. Dehghan, J. Manafian, and A. Saadatmandi, Application of semianalytic methods for the FitzhughNagumo equation, which models the transmission of nerve impulses, Mathematical Methods in the Applied Sciences, 33 (2010), 1384-98.
[20] Y. Eugene, Application of the decomposition method to the solution of the reaction-convectiondiffusion equation, Appl. Math. Comput., 56 (1993), 1-27.
[21] F. Geng, Y. Lina, and M. Cui, A piecewise variational iteration method for Riccati differential equations, Computers and Mathematics with Applications., 58 (2009), 2518-2522.
[22] T. Hayat, M. Khan, and S. Asghar, Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid, Acta. Mech., 168 (2004), 213-232.
[23] J. H. He, Variational iteration method for delay differential equations, Commun. Nonlinear Sci. Numer. Simul., 2 (4) (1997), 235-236.
[24] J. H. He, Variational iteration method- a kind of non-linear analytical technique: some examples, Int. J. Nonlinear Mech., 34 (1999), 699-708.
[25] E. Hesameddini and A. Rahimi, A new numerical scheme for solving systems of integrodifferential equations, Computational Methods for Differential Equations, 1(2) (2013), 108-119.
[26] M. Inc, On exact solution of Laplace equation with Dirichlet and Neumann boundary conditions by the homotopy analysis method, Phys. Lett. A, 365 (2007), 412-415.
[27] H. Jafari, A. Golbabai, S. Seifi, and K. Sayevand, Homotopy analysis method for solving multiterm linear and nonlinear diffusion-wave equations of fractional order, Comput. Math. Appl., 59 (2010), 1337-1344.
[28] S. J. Liao and K. F. Cheung, Homotopy analysis of nonlinear progressive waves in deep water, J. Eng. Math., 45 (2003), 105-116.
[29] S. A. Kechil and I. Hashim, Non-perturbative solution of free-convective boundary-layer equation by Adomian decomposition method, Phys. Lett. A, 363 (2007) 110-114.
[30] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, 1993.
[31] M. Moghimizand and M. T. Ahmadian, Application of homotopy analysis method in studying dynamic pull-in instability of Microsystems, Mech. Research Commun., 36 (2009), 851-858.
[32] A. Molabahrami and F. Khani, The homotopy analysis method to solve the Burgers-Huxley equation, Nonlinear Anal: Real World Appl.,10 (2009), 589-600.
[33] R. D. Richtmyer and K. W. Morton. Difference Methods for Initial-Value Problems. Inter. Science Publishers, New York, 1967.
[34] X. Shang, P. Wu, and X. Shao, An efficient method for solving Emden-Fowler equations, J. Franklin Institute, 346 (2009), 889-897.
[35] L. Song and H. Zhang, Application of homotopy analysis method to fractional KdV-BurgersKuramoto equation, Phys. Lett. A, 367 (2007), 88-94.
[36] Z. Wang, L. Zoub, and H. Zhang, Applying homotopy analysis method for solving differentialdifference equation, Phys. Lett. A, 369 (2007), 77-84.
[37] M. Wazewska-Czyzewska and A. Lasota, Mathematical models of the red cell system (Polish), Mat. Stosow, 6 (1976), 25-40.
[38] A. M. Wazwaz, A new approach to the nonlinear advection problem: An application of the decomposition method, Appl. Math. Comput., 72 (1995), 175-181.
[39] A. Wazwaz and A. Gorguis, An analytic study of Fishers equation by using Adomian decomposition method, Appl. Math. Comput., 154 (2004), 609-620.
[40] P. Yang, Y. Chenand, and Zhi-BinLi, ADM-Pad´e technique for the nonlinear lattice equations, Applied Mathematics and Computation, 210 (2009),362-375.
[41] A. Yildirim and T. Ozis, Solutions of singular IVPs of Lane-Emden type by the variational iteration method, Nonlinear Anal., 70 (2009), 2480-2484.
Volume 9, Issue 3
July 2021
Pages 749-761
  • Receive Date: 04 September 2019
  • Revise Date: 29 March 2020
  • Accept Date: 05 April 2020
  • First Publish Date: 01 July 2021