Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Variational Iteration Method for Generalized Pantograph Equation with Convergence Analysis

Discontinuity, Nonlinearity, and Complexity 3(2) (2014) 109--121 | DOI:10.5890/DNC.2014.06.001

Mohsen Alipour$^{1}$ , Dumitru Baleanu$^{2}$,$^{3}$,$^{4}$ , Kobra Karimi$^{5}$, Sunil Kumar$^{6}$

$^{1}$ Faculty of Basic Science, Babol University of Technology, P.O. Box 47148-71167, Babol, Iran

$^{2}$ Department of Mathematics, Cankaya University, Ogretmenler Cad. 14, Balgat, 06530 Ankara, Turkey

$^{3}$ Institute of Space Sciences, P.O. Box MG 23, Magurele, 077125 Bucharest, Romania

$^{4}$ Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia

$^{5}$ Department of Mathematics, Buin Zahra Technical University, P.O. Box 34517-45346, Buin Zahra, Qazvin, Iran

$^{6}$ Department of Mathematics, National Institute of Technology, Jamshedpur, 831014, Jharkhand, India

Download Full Text PDF



In this paper, we solve generalized pantograph equation by changing the problem to a system of ordinary equations and using the variational iteration method. We discuss convergence of the proposed method to the exact solution. Finally, illustrative examples are given to demonstrate the efficiency of the method.


  1. [1]  He, J.H. (1997), A new approach to nonlinear partial differential equations, Communications in Nonlinear Science and Numerical Simulation, 2 (4), 230-235.
  2. [2]  He, J.H. (1998), A varitional approach to nonlinear problems and its application, Mech. Applic., 20 (1), 30-31.
  3. [3]  He, J.H. and Wu, X. H. (2007), Variation interaction method: New development and applications, Computers & Mathematics with Applications, 54, 881-894.
  4. [4]  He, J.H. (1998), Approximate Solution of nonlinear differential equationwith convolution product nonlinearities, Computer Methods in Applied Mechanics and Engineering, 167, 69-73.
  5. [5]  Rafei, M., Ganji, D.D., Daniali, H. and Pashaei, H. (1992), The variational iteration method for nonlinear oscillators with discontinutities, Journal of Sound and Vibration, 30 (6), 1338-1360.
  6. [6]  Saadatmandi, A. and Dehghan, M. (2009), Variational iteration method for solving generalized pantograph equation, Computers & Mathematics with Applications, 58, 2190-2196.
  7. [7]  Yu, Z.H. (2008), Variational iteration method for solving the multi-pantograph delay equation, Physical Letter A, 372, 6475-6479.
  8. [8]  Abdou, M.A. and Soliman, A.A. (2005), Variational iteration method for Solving Burger's and. Coupled Burger's equations, Journal of Computational and Applied Mathematics, 181, 245-251.
  9. [9]  Wang, S.A. and He, J.H. (2007), Variational iteration method for solving integro-differential equations, Physical Letter A, 367, 188-191.
  10. [10]  Bo, T.L., Xie, L. and Zheng, X.J. (2007), Numerical approach to wind ripple in desert, IInternational Journal of Nonlinear Sciences and Numerical Simulation, 8 (2), 223-228.
  11. [11]  Dehghan, M. and Shakeri, F. (2007), Numerical solution of a biological population model using He's Variational method, Computers and Mathematics with Applications, 54, 1197-1209.
  12. [12]  Abbasbandy, S. (2008), Numerical method for non-linear wave and diffusion equations by the Variational iteration method, International Journal for Numerical Methods in Engineering, 73, 1836-1843.
  13. [13]  Abbasbandy, S. and Shivanian, E. (2008), Application of the Variational Iteration Method for System of Nonlinear Volterra's Integro-Differential Equations, Zeitschrift für Naturforschung, 63a, 538-542.
  14. [14]  Dehghan, M. and Tatari, M. (2008), Identifying an unknown function in a parabolic equation over specified data via He's Variational iteration method, Chaos, Solitons & Fractals, 36, 157-166.
  15. [15]  Dehghan, M. and Shakeri, F. (2008), Application of He's variational iteration method for solving the Cauchy reactiondiffusion problem, Journal of Computational and Applied Mathematics, 214, 435-446.
  16. [16]  Dehghan, M. and Shakeri, F. (2008), Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New Astronomy, 13, 53-59.
  17. [17]  Abbasbandy, S. (2007), Numerical solutions of nonlinear Klein-Gordon equation by variational iteration method, International Journal for Numerical Methods in Engineering, 70, 876-881.
  18. [18]  Abbasbandy, S. (2007), A new application of He's variational iteration method for quadratic Riccati-differential equation by using Adomian's polynomials, Journal of Computational and Applied Mathematics, 207 (1), 59-63.