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Year 2016, Volume: 6 Issue: 1, 115 - 125, 01.06.2016

Abstract

References

  • Greiner, W., (2000), Relativistic Quantum Mechanics-Wave Equations, 3rd edition, Springer-Verlag, Berlin.
  • Dehghan, M. and Shokri, A., (2009), Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions, Journal of Computational and Applied Mathematics, 230 (2), pp 400-410.
  • Biswas, A., Yildirim, A., Hayat, T., Aldossary, O.M. and Sassaman, R., (2012), Soliton perturbation theory for the generalized Klein-Gordon equation with full nonlinearity, Proceedings of the Romanian Academy A, 13 (1), pp. 32-41.
  • Biswas, A., Zony, C. and Zerrad, E., (2008), Soliton perturbation theory for the quadratic nonlinear
  • Klein-Gordon equation, Applied Mathematics and Computation, 203 (1), pp. 153-156. Biswas, A., Ebadi, G., Fessak, M., Johnpillai, A.G., Johnson, S., Krishnan, E.V. and Yildirim, A., (2012), Solutions of the perturbed Klein-Gordon equations, Iranian Journal of Science and Technology A, 36 (4), pp. 431-452.
  • Sassaman, R. and Biswas, A., (2009), Soliton perturbation theory for phi-four model and nonlinear
  • Klein-Gordon equations, Communications in Nonlinear Science and Numerical Simulation, 14 (8), pp. 3249.
  • Sassaman, R. and Biswas, A., (2009), Topological and non-topological solitons of the generalized
  • Klein-Gordon equations, Applied Mathematics and Computation, 215 (1), pp. 212-220. Sassaman, R., Heidari, A. and Biswas, A., (2010), Topological and non-topological solitons of nonlinear
  • Klein-Gordon equations by He’s semi-inverse variational principle, Journal of the Franklin Institute, (7), pp. 1148-1157.
  • Sassaman, R., Heidari, A., Majid, F., Zerrad, E. and Biswas, A., (2010), Topological and non- topological solitons of the generalized Klein-Gordon equations in 1+2 dimensions, Dynamics of Con- tinuous, Discrete & Impulsive Systems A, 17 (2), pp. 275-286.
  • Sassaman, R. and Biswas, A., (2010), Topological and non-topological solitons of the Klein-Gordon equations in 1+2 dimensions, Nonlinear Dynamics, 61 (1-2), pp. 23-28.
  • Wazwaz, A., (2006), The modified decomposition method for analytic treatment of differential equa- tions, Applied Mathematics and Computation, 173 (1), pp. 165-176.
  • Duncan, D.B., (1997), Symplectic finite difference approximations of the nonlinear Klein-Gordon equation, SIAM Journal on Numerical Analysis, 34 (5), pp. 1742-1760.
  • Abbasbandy, S., (2007), Numerical solution of non-linear Klein-Gordon equations by variational iter- ation method, International Journal for Numerical Methods in Engineering, 70 (7), pp. 876-881.
  • Yusufoglu, E., (2008), The variational iteration method for studying the Klein-Gordon equation
  • Applied Mathematics Letters, 21 (7), pp. 669-674. Wang, Q. and Cheng, D., (2005), Numerical solution of damped nonlinear Klein-Gordon equations using variational method and finite element approach, Applied Mathematics and Computation, 162 (1), pp. 381-401.
  • Rashidinia, J., Ghasemi, M. and Jalilian, R., (2010), Numerical solution of the nonlinear Klein-Gordon equation, Journal of Computational and Applied Mathematics, 233 (8), pp. 1866-1878.
  • Han, H. and Zhang, Z., (2009), An analysis of the finite-difference method for one-dimensional Klein
  • Gordon equation on unbounded domain, Applied Numerical Mathematics, 59 (7), pp. 1568-1583.
  • Kaya, D. and El-Sayed, S.M., (2004), A numerical solution of the Klein-Gordon equation and conver- gence of the decomposition method, Applied Mathematics and Computation, 156 (2), pp. 341-353.
  • Ebaid, A., (2009), Exact solutions for the generalized Klein-Gordon equation via a transformation and Exp-function method and comparison with Adomian’s method, Journal of Computational and Applied Mathematics, 223 (1), pp. 278-290.
  • Wang, T.M. and Zhu, J.M., (2009), New explicit solutions of the Klein-Gordon equation using the variational iteration method combined with the Exp-function method, Computers & Mathematics with Applications, 58 (11-12), pp. 2448.
  • Odibat, Z. and Momani, S., (2007), A reliable treatment of homotopy perturbation method for Klein
  • Gordon equations, Physics Letters A, 365 (5-6), pp. 351-357. Wazwaz, A.M., (2005) Compactons, solitons and periodic solutions for variants of the KdV and the KP equations, Applied Mathematics and Computation, 161 (2), pp. 561-575.
  • Liu, S., Fu, Z. and Liu, S., (2005), Periodic solutions for a class of coupled nonlinear partial differential equations, Physics Letters A, 336 (2-3), pp. 175-179.
  • Song, M., Liu, Z., Zerrad, E. and Biswas, A., (2013), Singular soliton solution and bifurcation analysis of Klein-Gordon equation with power law nonlinearity, Frontiers of Mathematics in China, 8 (1), pp. 201.
  • Song, M., Liu, Z., Eerrad, E. and Biswas, A., (2013), Singular solitons and bifurcation analysis of quadratic nonlinear klein-gordon equation, Applied Mathematics and Information Sciences, 7 (4), pp. 1340.
  • Sassaman, R. and Biswas, A., (2011), Soliton solution of the generalized klein-gordon equation by semi-inverse variational principle, Mathematics in Engineering Science and Aerospace, 2 (1), pp. 99
  • Sassaman, R. and Biswas, A., (2011), 1-soliton solution of the perturbed klein-gordon equation
  • Physics Express, 1 (1), pp. 9-14. Sassaman, R., Edwards, M., Majid, F. and Biswas, A., (2010), 1-soliton solution of the coupled nonlinear klein-gordon equations, Studies in Mathematical Sciences, 1 (1), pp. 30-37.
  • Khalique, C.M. and Biswas, A., (2010), Analysis of non-linear Klein-Gordon equations using Lie symmetry, Applied Mathematics Letters, 23 (11), pp. 1397-1400.
  • Biswas, A., Khalique, C.M. and Adem, A.R., (2012), Traveling wave solutions of the nonlinear dis- persive Klein-Gordon equations, Journal of King Saud University, 24 (4), pp. 339-342.
  • Aksoy, Y. and Pakdemirli, M., (2010), New perturbation-iteration solutions for Bratu-type equations,Computers and Mathematics with Applications, 59 (8), pp. 2802-2808.

SOLVING LINEAR AND NONLINEAR KLEIN-GORDON EQUATIONS BY NEW PERTURBATION ITERATION TRANSFORM METHOD

Year 2016, Volume: 6 Issue: 1, 115 - 125, 01.06.2016

Abstract

We present an effective algorithm to solve the Linear and Nonlinear KleinGordon equation, which is based on the Perturbation Iteration Transform Method PITM . The Klein-Gordon equation is the name given to the equation of motion of a quantum scalar or pseudo scalar field, a field whose quanta are spin-less particles. It describes the quantum amplitude for finding a point particle in various places, the relativistic wave function, but the particle propagates both forwards and backwards in time. The Perturbation Iteration Transform Method PITM is a combined form of the Laplace Transform Method and Perturbation Iteration Algorithm. The method provides the solution in the form of a rapidly convergent series. Some numerical examples are used to illustrate the preciseness and effectiveness of the proposed method. The results show that the PITM is very efficient, simple and can be applied to other nonlinear problems.

References

  • Greiner, W., (2000), Relativistic Quantum Mechanics-Wave Equations, 3rd edition, Springer-Verlag, Berlin.
  • Dehghan, M. and Shokri, A., (2009), Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions, Journal of Computational and Applied Mathematics, 230 (2), pp 400-410.
  • Biswas, A., Yildirim, A., Hayat, T., Aldossary, O.M. and Sassaman, R., (2012), Soliton perturbation theory for the generalized Klein-Gordon equation with full nonlinearity, Proceedings of the Romanian Academy A, 13 (1), pp. 32-41.
  • Biswas, A., Zony, C. and Zerrad, E., (2008), Soliton perturbation theory for the quadratic nonlinear
  • Klein-Gordon equation, Applied Mathematics and Computation, 203 (1), pp. 153-156. Biswas, A., Ebadi, G., Fessak, M., Johnpillai, A.G., Johnson, S., Krishnan, E.V. and Yildirim, A., (2012), Solutions of the perturbed Klein-Gordon equations, Iranian Journal of Science and Technology A, 36 (4), pp. 431-452.
  • Sassaman, R. and Biswas, A., (2009), Soliton perturbation theory for phi-four model and nonlinear
  • Klein-Gordon equations, Communications in Nonlinear Science and Numerical Simulation, 14 (8), pp. 3249.
  • Sassaman, R. and Biswas, A., (2009), Topological and non-topological solitons of the generalized
  • Klein-Gordon equations, Applied Mathematics and Computation, 215 (1), pp. 212-220. Sassaman, R., Heidari, A. and Biswas, A., (2010), Topological and non-topological solitons of nonlinear
  • Klein-Gordon equations by He’s semi-inverse variational principle, Journal of the Franklin Institute, (7), pp. 1148-1157.
  • Sassaman, R., Heidari, A., Majid, F., Zerrad, E. and Biswas, A., (2010), Topological and non- topological solitons of the generalized Klein-Gordon equations in 1+2 dimensions, Dynamics of Con- tinuous, Discrete & Impulsive Systems A, 17 (2), pp. 275-286.
  • Sassaman, R. and Biswas, A., (2010), Topological and non-topological solitons of the Klein-Gordon equations in 1+2 dimensions, Nonlinear Dynamics, 61 (1-2), pp. 23-28.
  • Wazwaz, A., (2006), The modified decomposition method for analytic treatment of differential equa- tions, Applied Mathematics and Computation, 173 (1), pp. 165-176.
  • Duncan, D.B., (1997), Symplectic finite difference approximations of the nonlinear Klein-Gordon equation, SIAM Journal on Numerical Analysis, 34 (5), pp. 1742-1760.
  • Abbasbandy, S., (2007), Numerical solution of non-linear Klein-Gordon equations by variational iter- ation method, International Journal for Numerical Methods in Engineering, 70 (7), pp. 876-881.
  • Yusufoglu, E., (2008), The variational iteration method for studying the Klein-Gordon equation
  • Applied Mathematics Letters, 21 (7), pp. 669-674. Wang, Q. and Cheng, D., (2005), Numerical solution of damped nonlinear Klein-Gordon equations using variational method and finite element approach, Applied Mathematics and Computation, 162 (1), pp. 381-401.
  • Rashidinia, J., Ghasemi, M. and Jalilian, R., (2010), Numerical solution of the nonlinear Klein-Gordon equation, Journal of Computational and Applied Mathematics, 233 (8), pp. 1866-1878.
  • Han, H. and Zhang, Z., (2009), An analysis of the finite-difference method for one-dimensional Klein
  • Gordon equation on unbounded domain, Applied Numerical Mathematics, 59 (7), pp. 1568-1583.
  • Kaya, D. and El-Sayed, S.M., (2004), A numerical solution of the Klein-Gordon equation and conver- gence of the decomposition method, Applied Mathematics and Computation, 156 (2), pp. 341-353.
  • Ebaid, A., (2009), Exact solutions for the generalized Klein-Gordon equation via a transformation and Exp-function method and comparison with Adomian’s method, Journal of Computational and Applied Mathematics, 223 (1), pp. 278-290.
  • Wang, T.M. and Zhu, J.M., (2009), New explicit solutions of the Klein-Gordon equation using the variational iteration method combined with the Exp-function method, Computers & Mathematics with Applications, 58 (11-12), pp. 2448.
  • Odibat, Z. and Momani, S., (2007), A reliable treatment of homotopy perturbation method for Klein
  • Gordon equations, Physics Letters A, 365 (5-6), pp. 351-357. Wazwaz, A.M., (2005) Compactons, solitons and periodic solutions for variants of the KdV and the KP equations, Applied Mathematics and Computation, 161 (2), pp. 561-575.
  • Liu, S., Fu, Z. and Liu, S., (2005), Periodic solutions for a class of coupled nonlinear partial differential equations, Physics Letters A, 336 (2-3), pp. 175-179.
  • Song, M., Liu, Z., Zerrad, E. and Biswas, A., (2013), Singular soliton solution and bifurcation analysis of Klein-Gordon equation with power law nonlinearity, Frontiers of Mathematics in China, 8 (1), pp. 201.
  • Song, M., Liu, Z., Eerrad, E. and Biswas, A., (2013), Singular solitons and bifurcation analysis of quadratic nonlinear klein-gordon equation, Applied Mathematics and Information Sciences, 7 (4), pp. 1340.
  • Sassaman, R. and Biswas, A., (2011), Soliton solution of the generalized klein-gordon equation by semi-inverse variational principle, Mathematics in Engineering Science and Aerospace, 2 (1), pp. 99
  • Sassaman, R. and Biswas, A., (2011), 1-soliton solution of the perturbed klein-gordon equation
  • Physics Express, 1 (1), pp. 9-14. Sassaman, R., Edwards, M., Majid, F. and Biswas, A., (2010), 1-soliton solution of the coupled nonlinear klein-gordon equations, Studies in Mathematical Sciences, 1 (1), pp. 30-37.
  • Khalique, C.M. and Biswas, A., (2010), Analysis of non-linear Klein-Gordon equations using Lie symmetry, Applied Mathematics Letters, 23 (11), pp. 1397-1400.
  • Biswas, A., Khalique, C.M. and Adem, A.R., (2012), Traveling wave solutions of the nonlinear dis- persive Klein-Gordon equations, Journal of King Saud University, 24 (4), pp. 339-342.
  • Aksoy, Y. and Pakdemirli, M., (2010), New perturbation-iteration solutions for Bratu-type equations,Computers and Mathematics with Applications, 59 (8), pp. 2802-2808.
There are 34 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

M. Khalid This is me

M. Sultana This is me

F. Zaidi This is me

U. Arshad This is me

Publication Date June 1, 2016
Published in Issue Year 2016 Volume: 6 Issue: 1

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