Abstract
This study introduced a generalized operational matrix using Clique polynomials of a complete graph and proposed the latest approach to solve the nonlinear Klein–Gordon (KG) equation. KG equations describe many real physical phenomena in fluid dynamics, electrical engineering, biogenetics, tribology. By using the properties of the operational matrix, we transform the nonlinear KG equation into a system of algebraic equations. Unknown coefficients to be determined by Newton’s method. The present technique is applied to four problems, and the obtained results are compared with another method in the literature. Also, we discussed some theorems on convergence analysis and continuous property.
Similar content being viewed by others
Change history
04 August 2023
The original article was updated for removal of unnecessary hyphens present in the text.
Abbreviations
- \(\alpha\) :
-
Real number
- \(P\left( y \right)\) :
-
A nonlinear term in Eq. (1.1)
- \(y\left( {x,t} \right)\) :
-
The wave displacement at \(x\) and \(t\)
- \(A\left( x \right)\) :
-
Wave kinks
- \(V\left( G \right)\) :
-
The vertex set of \(G\)
- \(E\left( G \right)\) :
-
Edge set of \(G\)
- \(S\left( {G;x} \right)\) :
-
Clique polynomials of a graph \(G\)
- \(K_{n}\) :
-
Complete graph
References
Rashidinia, J., Mohammadi, R.: Tension spline approach for the numerical solution of nonlinear Klein-Gordon equation. Comput. Phys. Commun. 181, 78–91 (2010)
Raza, N., Rashid Butt, A., Javid, A.: Approximate Solution of Nonlinear Klein-Gordon Equation Using Sobolev Gradients, Hindawi Publishing Corporation. J. Funct. Sp. Vol. 2016, Article ID 1391594, 01–07
Fukang, Y., Tian, T., Song, J., Zhu, M.: Spectral methods using Legendre wavelets for nonlinear Klein/Sine-Gordon equations. J. Comput. Appl. Math. 275, 321–334 (2015)
Rashidinia, J., Ghasemia, M., Jalilian, R.: Numerical solution of the nonlinear Klein/Gordon equation. J. Comput. Appl. Math. 233, 1866–1878 (2010)
Khuri, S.A., Sayfy, A.: A spline collocation approach for the numerical solution of a generalized nonlinear Klein-Gordon equation. Appl. Math. Comput. 216, 1047–1056 (2010)
Chowdhury, M.S.H., Hashim, I.: Application of Homotopy perturbation method to Klein–Gordon and sine-Gordon equations. Chaos, Solitons Fractals 39, 1928–1935 (2009)
Rashidinia, J., Jokar, M.: Numerical solution of nonlinear klein-gordon equation using polynomial wavelets. Adv. Intell. Syst. Comput. 441, 199–214 (2016)
Dehghan, M., Shokri, A.: Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions. J. Comput. Appl. Math. 230, 400–410 (2009)
El-Sayed, S.M.: The decomposition method for studying the Klein-Gordon equation. Chaos Solitons Fractals 18(5), 1025–1030 (2003)
Kanth, A.R., Aruna, K.: Differential transform method for solving the linear nonlinear Klein-Gordon equation. Comput. Phys. Commun. 180, 708–711 (2009)
Shiralashetti, S.C., Kumbinarasaiah, S.: Laguerre wavelets collocation method for the numerical solution of the Benjamina Bona Mohany, equations. J. Taibah Univ. Sci. 13(1), 9–15 (2019)
Shiralashetti, S.C., Kumbinarasaiah, S.: CAS wavelets analytic solution and Genocchi polynomials numerical solutions for the integral and integrodifferential equations. J. Interdiscip. Math. 22(3), 201–218 (2019)
Shiralashetti, S.C., Kumbinarasaiah, S.: Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems. Alexandria Eng. J. 57(4), 2591–2600 (2018)
Shiralashetti, S.C., Kumbinarasaiah, S.: Theoretical study on continuous polynomial wavelet bases through wavelet series collocation method for nonlinear lane-Emden type equations. Appl. Math. Comput. 315, 591–602 (2017)
Mohammadi, A., Aghazadeh, N., Rezapour, S.: Haar wavelet collocation method for solving singular and nonlinear fractional time dependent Emden-Fowler equations with initial and boundary conditions. Math. Sci. 13, 255–265 (2019)
Heydari, M.H., Hooshmandasl, M.R., Malek Ghaini, F.M., Fereidouni, F.: Two dimensional Legendre wavelets for solving fractional Poisson equation with Dirichlet boundary conditions. Eng. Anal. Boundary Elem. 37, 1331–1338 (2013)
Diudea, M. V., Gutman, I., Lorentz, J.: Molecular Topology 1999.
Harary, F.: Graph Theory, Addison Wesley, Reading 1969.
Hoede, C., Li, X.: Clique polynomials, and independent set polynomials of graphs. Discrete Math. 125, 219–228 (1994)
Ganji, R.M., Jafari, H., Baleanu, D.: A new approach for solving multivariable orders differential equations with Mittag Leffler kernel. Chaos Solitons Fractals 130, 109405 (2020)
Srinivasa, K., Rezazadeh, H.: Numerical solution for the fractional order one dimensional telegraph equation via wavelet technique. Int. J. Nonlinear Sci. Numer. Simulation (2020). https://doi.org/10.1515/ijnsns-2019-0300
Ganji, R.M., Jafari, H.: Numerical solution of variable order integro differential equations. Adv. Math. Models Appl. 4(1), 64–69 (2019)
Ganji, R.M., Jafari, H.: A new approach for solving nonlinear Volterra integrodifferential equations with Mittag-Leffler kernel. Proc. Inst. Math. Mech. 46(1), 144–158 (2020)
Ganji, R. M., Jafari, H., Nemati, S.: A new approach for solving integro differential equations of variable order, J. Comput. Appl. Math. 112946 (2020).
Shiralashetti, S.C., Kumbinarasaiah, S.: Cardinal B-spline wavelet based numerical method for the solution of generalized Burgers-Huxley equation. Int. J. Appl. Comput. Math. 4, 73 (2018)
Shiralashetti, S.C., Kumbinarasaiah, S.: New generalized operational matrix of integration to solve nonlinear singular boundary value problems using Hermite wavelets. Arab J. Basic Appl. Sci. 26(1), 385–396 (2019)
Shiralashetti, S.C., Kumbinarasaiah, S.: Laguerre wavelets exact parseval frame based numerical method for the solution of system of differential equations. Int. J. Appl. Comput. Math. 6, 101 (2020)
Srinivasa, K., Rezazadeh, H., Adel, W.: Numerical investigation based on Laguerre wavelet for solving the hunter saxton equation. Int. J. Appl. Comput. Math. 6, 139 (2020)
Kumbinarasaiah, S.: A new approach for the numerical solution for the nonlinear Klein-Gordon equation. SeMA 77, 435–456 (2020)
Islam, M.S., Akbar, M.A., Khan, K.: Analytical solutions of nonlinear Klein-Gordon equation using the improved F expansion method. Opt. Quant. Electron. 50, 224 (2018)
Kurulay, M.: Solving the fractional nonlinear Klein-Gordon equation by means of the Homotopy analysis method. Advances in Difference Equations 2012, 187 (2012)
Chaudry, M.K., Anjan, B.: Analysis of nonlinear Klein-Gordon equations using Lie symmetry. Appl. Math. Lett. 23(11), 1397–1400 (2010)
Necdet, B., Sinan, D.: New approximate solutions to the nonlinear Klein-Gordon equations using perturbation iteration techniques. Discrete Contin. Dyn. Syst. S 13(3), 503–518 (2020)
Deniz, S., Konuralp, A., De la Sen, M.: Optimal perturbation iteration method for solving fractional model of damped Burgers’ equation. Symmetry 12, 958 (2012)
Duan, J.S., Rach, R., Bulean, D., Wazwaz, A.M.: A review of the Adomian decomposition method and its applications to fractional differential equations. Commun. Fract. Calc. 3(2), 73–99 (2012)
Bildik, N., Deniz, S.: A new efficient method for solving delay differential equations and a comparison with other methods. Eur. Phys. J. Plus 132, 51 (2017)
Deniz, S., Bildik, N.: A new analytical technique for solving Lane - Emden type equations arising in astrophysics. Bull. Belg. Math. Soc. Simon Stevin 24(2), 305–320 (2017)
Khaled, M.S., Deniz, S., Dumi̇tru, B.: On a new modified fractional analysis of Nagumo equation. Int. J. Biomath. 12(3), 1950034 (2019)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Kumbinarasaiah, S., Ramane, H.S., Pise, K.S. et al. Numerical Solution for Nonlinear Klein–Gordon Equation via Operational Matrix by Clique Polynomial of Complete Graphs. Int. J. Appl. Comput. Math 7, 12 (2021). https://doi.org/10.1007/s40819-020-00943-x
Accepted:
Published:
DOI: https://doi.org/10.1007/s40819-020-00943-x