Abstract
In this paper, we present the solution of the Klein--Gordon equation. Klein--Gordon equation is the relativistic version of the Schrödinger equation, which is used to describe spinless particles. The He’s variational iteration method (VIM) is implemented to give approximate and analytical solutions for this equation. The variational iteration method is based on the incorporation of a general Lagrange multiplier in the construction of correction functional for the equation. Application of variational iteration technique to this problem shows rapid convergence of the sequence constructed by this method to the exact solution. Moreover, this technique reduces the volume of calculations by avoiding discretization of the variables, linearization or small perturbations.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Sakurai, J.J.: Advanced Quantum Mechanics. Addison-Wesley, New York (1967)
Jiménez, S., Vázquez, L.: Analysis of four numerical schemes for a nonlinear Klein—Gordon equation. Appl. Math. Comput. 35(1), 61–94 (1990)
Lynch, M.A.M.: Large amplitude instability in finite difference approximations to the Klein—Gordon equation. Appl. Numer. Math. 31(2), 173–182 (1999)
Lee, I.J.: Numerical solution for nonlinear Klein—Gordon equation by collocation method with respect to spectral method. J. Korean Math. Soc. 32(3), 541–551 (1995)
Wong, Y.S., Chang, Q., Gong, L.: An initial-boundary value problem of a nonlinear Klein—Gordon equation. Appl. Math. Comput. 84(1), 77–93 (1997)
Fang, D., Zhong, S.: Global solutions for nonlinear Klein—Gordon equations in infinite homogeneous wave guides. J. Differ. Equ. 231, 212–234 (2006)
Metcalfe, J., Sogge, C.D., Stewart, A.: Nonlinear hyperbolic equations in infinite homogeneous wave guides. Comm. Partial Differ. Equ. 30(4–6), 643–661 (2005)
Klainerman, S.: Global existence of small amplitude solutions to nonlinear Klein—Gordon equations in four space-time dimensions. Comm. Pure Appl. Math. 38, 631–641 (1985)
Sirendaoreji, S.: Auxiliary equation method and new solutions of Klein—Gordon equations. Chaos, Solitons Fractals 31, 943–950 (2007)
Kevrekidis, P.G., Konotop, V.V.: Compactons in discrete nonlinear Klein—Gordon models. Math. Comput. Simul. 62, 79–89 (2003)
Khalifa, M.E., Elgamal, M.: A numerical solution to Klein—Gordon equation with Dirichlet boundary condition. Appl. Math. Comput. 160, 451–475 (2005)
Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics. Appl. Math. Sci. 68, Springer-Verlag (1988)
Wang, Q., Cheng, D.: Numerical solution of damped nonlinear Klein—Gordon equations using variational method and finite element approach. Appl. Math. Comput. 162, 381–401 (2005)
He, J.H.: A new approach to nonlinear partial differential equations. Commun. Nonlinear Sci. Numer. Simul. 2(4), 230–235 (1997)
He, J.H.: Approximate solution of nonlinear differential equations with convolution product nonlinearities. Comput. Methods Appl. Mech. Eng. 167, 57–68 (1998)
Abdou, M.A., Soliman, A.A.: Variational iteration method for solving Burgers' and coupled Burgers' equations. J. Comput. Appl. Math. 181, 245–251 (2005)
Khuri, S.A.: A new approach to Bratu's problem. Appl. Math. Comput. 147(1), 131–136 (2004)
He, J.H., Wu, X.H.: Construction of solitary solution and compacton-like solution by variational iteration method. Chaos, Solitons Fractals 29, 108–113 (2006)
Moghimi, M., Hejazi, F.S.A.: Variational iteration method for solving generalized Burgers—Fisher and Burgers equations. Chaos, Solitons Fractals (in press)
Soliman, A.A.: Numerical simulation of the generalized regularized long wave equation by He's variational iteration method. Math. Comput. Simul. 70, 119–124 (2005)
Abdou, M.A., Soliman, A.A.: New applications of variational iteration method. Physica D 211, 1–8 (2005)
Soliman A.A., Abdou, M.A. : Numerical solutions of nonlinear evolution equations using variational iteration method. J. Comput. Appl. Math. (in press) A1
Dehghan, M.: The solution of a nonclassic problem for one-dimensional hyperbolic equation using the decomposition procedure. Int. J. Comput. Math. 81, 979–989 (2004)
Wazwaz, A.M.: The variational iteration method for rational solutions for KdV, K(2, 2), Burgers, and cubic Boussinesq equations. J. Comput. Appl. Math. (in press) A1
Sweilam, N.H.: Harmonic wave generation in non linear thermoelasticity by variational iteration method and Adomian's method. J. Comput. Appl. Math. (in press) A1
Tatari, M., Dehghan, M.: Solution of problems in calculus of variations via He's variational iteration method. Phys. Lett. A (accepted) A1
Dehghan, M., Tatari, M.: Identifying an unknown function in a parabolic equation with overspecified data via He's variational iteration method. Chaos, Solitons Fractals (in press) A1
He, J.H.: Variational iteration method for delay differential equations. Commun. Nonlinear Sci. Numer. Simul. 2(4), 235–236 (1997)
He, J.H.: Variational iteration method for autonomous ordinary differential systems. Appl. Math. Comput. 114, 115–123 (2000)
He, J.H.: Approximate analytical solution of Blasius' equation. Commun. Nonlinear Sci. Numer. Simul. 4(1), 75–78 (1999)
Tatari, M., Dehghan, M.: On the convergence of He's variational iteration method. J. Comput. Appl. Math. (in press) A1
Abassy, T.A., El-Tawil, M.A., El Zoheiryb, H.: Solving nonlinear partial differential equations using the modified variational iteration Pade technique. J. Comput. Appl. Math. (in press) A1
Abassy, T.A., El-Tawil, M.A., El Zoheiry, H.: Toward a modified variational iteration method. J. Comput. Appl. Math. (in press) A1
He, J.H.: Some asymptotic methods for strongly nonlinear equations. Int. J. Modern Phys. B 20(10), 1141–1199 (2006)
Dehghan, M.: Finite difference procedueres for solving a problem arising in modeling and design of certain optoelectronic devices. Math. Comput. Simul. 71, 16–30 (2006)
Nayfeh, A.H. : Introduction to Perturbation Techniques. John Wiley, New York (1981)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shakeri, F., Dehghan, M. Numerical solution of the Klein–Gordon equation via He’s variational iteration method. Nonlinear Dyn 51, 89–97 (2008). https://doi.org/10.1007/s11071-006-9194-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-006-9194-x