International Scientific Journal

Thermal Science - Online First

online first only

Analytical solutions of nonlinear Klein-Gordon equations using Multistep Modified Reduced Differential Transform Method

This paper explores the approximate analytical solution of nonlinear Klein-Gordon equations (NKGE) by using Multistep Modified Reduced Differential Transform Method (MMRDTM). Through this proposed strategy, the nonlinear term is substituted by associating Adomian polynomials obtained by utilization of a multistep approach. The NKGE solutions can be obtained with a reduced number of computed terms. In addition, the approximate solutions converge rapidly in a wide time region. Three examples are provided to illustrate the effectiveness of the proposed method to obtain solutions for the NKGE. Graphical results are shown to represent the behavior of the solution so as to demonstrate the validity and accuracy of the MMRDTM.
PAPER REVISED: 2018-11-12
PAPER ACCEPTED: 2019-01-16
  1. Wazwaz, A.,The Modified Decomposition Method for Analytic Treatment of Differential Equations, 17 (2006b), 3,pp. 165-176
  2. El-Sayed, S. M., The Decomposition Method for Studying the Klein - Gordon Equation, 18 (2003), pp. 1025-1030
  3. Wazwaz, A., Compactons, Solitons and Periodic Solutions for Some Forms of Nonlinear Klein - Gordon Equations, 28 (2006a), pp. 1005-1013
  4. Servi, S., & Oturanç, G., Reduced Differential Transform Method for Solving Klein Gordon Equations, Proceedings of the World Congress on Engineering, London UK, 2011,Vol.I, pp. 2-6.
  5. Hafez, M. G., et al., Exact Traveling Wave Solutions to the Klein - Gordon Equation using the Novel (G'/G)-Expansion Method, Results In Physics, 4 (2014), pp. 177-184
  6. Venkatesh, S. G., et al., An Efficient Approach for Solving Klein - Gordon Equation Arising in Quantum Field Theory using Wavelets, Computational and Applied Mathematics, (2016)
  7. Agom, E. U., & Ogunfiditimi, F. O.,Exact Solution of Nonlinear Klein-Gordon Equations with Quadratic Nonlinearity by Modified Adomian Decomposition Method, Journal of Mathematical Computational Science (2018), 4,pp. 484-493.
  8. Jameel A.F., et al., Differential Transformation Method for Solving High Order Fuzzy Initial Value Problems, Italian Journal of Pure and Applied Mathematics, 39 (2018),pp. 194-208.
  9. Rao, T. R. R.,Numerical Solution of Sine Gordon Equations Through Reduced Differential Transform Method, Global Journal of Pure and Applied Mathematics, 13 (2017), 7,pp. 3879-3888.
  10. Acan, O., & Keskİn, Y.,Reduced Differential Transform Method for (2+1) Dimensional Type of the Zakharov-Kuznetsov ZK (n, n) Equations, AIP Conference Proceedings, 1648 (2015)
  11. Marasi, H. R., et al., Modified Differential Transform Method for Singular Lane-Emden Equations in Integer and Fractional Order, Journal of Applied and Engineering Mathematics, 5 (2015),1,pp. 124-131.
  12. Benhammouda, B., & Leal, H. V.,A New Multi‑Step Technique with Differential Transform Method for Analytical Solution of Some Nonlinear Variable Delay Differential Equations, SpringerPlus, 5 (2016), 1723
  13. Kang-Le W., & Kang-Jia W., A Modification of the Reduced Differential Transform Method for Fractional Calculus, Thermal Science, 22 (2018), 4,pp. 1871 - 1875.
  14. Hossein J., et al., Reduced Differential Transform and Variational Iteration Methods for 3-D Diffusion Model in Fractal Heat Transfer Within Local Fractional Operators, Thermal Science, 22 (2018),1, pp. S301-S307.
  15. Ray, S. S. ,Numerical Solutions and Solitary Wave Solutions of Fractional KdV Equations using Modified Fractional Reduced Differential Transform Method, Journal of Mathematical Chemistry,51 (2013),8,pp. 2214-2229
  16. El-Zahar, E. R., Applications of Adaptive Multi Step Differential Transform Method to Singular Perturbation Problems Arising in Science and Engineering, Applied Mathematics and Information Sciences, 9 (2015),1, pp. 223-232
  17. Che Haziqah C.H., et al., Analytical Solutions of Nonlinear Schrodinger Equations using Multi-step Modified Reduced Differential Transform Method, International Journal of Advanced Computer Technology, 7 (2018),11,pp. 2939-2944.
  18. Huang, D., et al., Numerical Approximation of Nonlinear Klein-Gordon Equation Using an Element-Free Approach, Mathematical Problems in Engineering (2015).
  19. Kanth, A. S. V. R., & Aruna, K. ,Differential Transform Method for Solving the Linear and Nonlinear Klein -Gordon Equation, Computer Physics Communications, 180 (2009),5, pp.