Numerical solution of the Klein-Gordon equation via He’s variational iteration method. (English) Zbl 1179.81064

Summary: 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.


81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
49S05 Variational principles of physics
81T80 Simulation and numerical modelling (quantum field theory) (MSC2010)
Full Text: DOI


[1] Sakurai, J.J.: Advanced Quantum Mechanics. Addison-Wesley, New York (1967)
[2] Jiménez, S., Vázquez, L.: Analysis of four numerical schemes for a nonlinear Klein–Gordon equation. Appl. Math. Comput. 35(1), 61–94 (1990) · Zbl 0697.65090 · doi:10.1016/0096-3003(90)90091-G
[3] Lynch, M.A.M.: Large amplitude instability in finite difference approximations to the Klein–Gordon equation. Appl. Numer. Math. 31(2), 173–182 (1999) · Zbl 0937.65098 · doi:10.1016/S0168-9274(98)00128-7
[4] 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) · Zbl 0841.65088
[5] 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) · Zbl 0884.65091 · doi:10.1016/S0096-3003(96)00065-3
[6] Fang, D., Zhong, S.: Global solutions for nonlinear Klein–Gordon equations in infinite homogeneous wave guides. J. Differ. Equ. 231, 212–234 (2006) · Zbl 1106.58020 · doi:10.1016/j.jde.2006.07.028
[7] 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) · Zbl 1078.35076 · doi:10.1081/PDE-200059267
[8] 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) · Zbl 0597.35100 · doi:10.1002/cpa.3160380512
[9] Sirendaoreji, S.: Auxiliary equation method and new solutions of Klein–Gordon equations. Chaos, Solitons Fractals 31, 943–950 (2007) · Zbl 1143.35341 · doi:10.1016/j.chaos.2005.10.048
[10] Kevrekidis, P.G., Konotop, V.V.: Compactons in discrete nonlinear Klein–Gordon models. Math. Comput. Simul. 62, 79–89 (2003) · Zbl 1022.39019 · doi:10.1016/S0378-4754(02)00184-2
[11] Khalifa, M.E., Elgamal, M.: A numerical solution to Klein–Gordon equation with Dirichlet boundary condition. Appl. Math. Comput. 160, 451–475 (2005) · Zbl 1126.65090 · doi:10.1016/j.amc.2003.11.014
[12] Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics. Appl. Math. Sci. 68, Springer-Verlag (1988) · Zbl 0662.35001
[13] 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) · Zbl 1063.65107 · doi:10.1016/j.amc.2003.12.102
[14] He, J.H.: A new approach to nonlinear partial differential equations. Commun. Nonlinear Sci. Numer. Simul. 2(4), 230–235 (1997) · doi:10.1016/S1007-5704(97)90007-1
[15] He, J.H.: Approximate solution of nonlinear differential equations with convolution product nonlinearities. Comput. Methods Appl. Mech. Eng. 167, 57–68 (1998) · Zbl 0942.76077 · doi:10.1016/S0045-7825(98)00108-X
[16] Abdou, M.A., Soliman, A.A.: Variational iteration method for solving Burgers’ and coupled Burgers’ equations. J. Comput. Appl. Math. 181, 245–251 (2005) · Zbl 1072.65127 · doi:10.1016/j.cam.2004.11.032
[17] Khuri, S.A.: A new approach to Bratu’s problem. Appl. Math. Comput. 147(1), 131–136 (2004) · Zbl 1032.65084 · doi:10.1016/S0096-3003(02)00656-2
[18] 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) · Zbl 1147.35338 · doi:10.1016/j.chaos.2005.10.100
[19] Moghimi, M., Hejazi, F.S.A.: Variational iteration method for solving generalized Burgers–Fisher and Burgers equations. Chaos, Solitons Fractals (in press) · Zbl 1138.35398
[20] 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) · Zbl 1152.65467 · doi:10.1016/j.matcom.2005.06.002
[21] Abdou, M.A., Soliman, A.A.: New applications of variational iteration method. Physica D 211, 1–8 (2005) · Zbl 1084.35539 · doi:10.1016/j.physd.2005.08.002
[22] Soliman A.A., Abdou, M.A. : Numerical solutions of nonlinear evolution equations using variational iteration method. J. Comput. Appl. Math. (in press) A1 · Zbl 1120.65111
[23] 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) · Zbl 1056.65099 · doi:10.1080/00207160410001712297
[24] 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 · Zbl 1119.65102
[25] 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 · Zbl 1115.74028
[26] Tatari, M., Dehghan, M.: Solution of problems in calculus of variations via He’s variational iteration method. Phys. Lett. A (accepted) A1 · Zbl 1197.65112
[27] 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 · Zbl 1152.35390
[28] He, J.H.: Variational iteration method for delay differential equations. Commun. Nonlinear Sci. Numer. Simul. 2(4), 235–236 (1997) · Zbl 0924.34063 · doi:10.1016/S1007-5704(97)90008-3
[29] He, J.H.: Variational iteration method for autonomous ordinary differential systems. Appl. Math. Comput. 114, 115–123 (2000) · Zbl 1027.34009 · doi:10.1016/S0096-3003(99)00104-6
[30] He, J.H.: Approximate analytical solution of Blasius’ equation. Commun. Nonlinear Sci. Numer. Simul. 4(1), 75–78 (1999) · Zbl 0932.34005 · doi:10.1016/S1007-5704(99)90063-1
[31] Tatari, M., Dehghan, M.: On the convergence of He’s variational iteration method. J. Comput. Appl. Math. (in press) A1 · Zbl 1120.65112
[32] 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
[33] Abassy, T.A., El-Tawil, M.A., El Zoheiry, H.: Toward a modified variational iteration method. J. Comput. Appl. Math. (in press) A1 · Zbl 1119.65096
[34] He, J.H.: Some asymptotic methods for strongly nonlinear equations. Int. J. Modern Phys. B 20(10), 1141–1199 (2006) · Zbl 1102.34039 · doi:10.1142/S0217979206033796
[35] 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) · Zbl 1089.65085 · doi:10.1016/j.matcom.2005.10.001
[36] Nayfeh, A.H. : Introduction to Perturbation Techniques. John Wiley, New York (1981) · Zbl 0449.34001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.