A NUMERICAL ALGORITHM BASED ON ULTRASPHERICAL WAVELETS FOR SOLUTION OF LINEAR AND NONLINEAR KLEIN-GORDON EQUATIONS

In this paper, Galerkin method based on the Ultraspherical wavelets expansion together with operational matrix of integration is developed to solve linear and nonlinear Klein Gordon (KG) equations with the given initial and boundary conditions. Firstly, we present the ultraspherical wavelets, then the corresponding operational matrix of integration is presented. To transform the given PDE into a system of linear-nonlinear algebraic equations which can be efficiently solved by suitable solvers, we utilize the operational matrix of integration and both properties of Ultraspherical wavelets. The applicability of the method is shown by two test problems and acquired results show that the method is good accuracy and efficiency.

Keywords:

Ultraspherical wavelets, Klein-Gordon equation Galerkin method, operational matrix of integration,

PDF

___

[1] 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), 400-410.
[2] Wazwaz A. M., (2006) The modified decomposition method for analytic treatment of differential equations, Applied Mathematics and Computation, 173(1), 165-176.
[3] Bülbül B., & Sezer M., (2013) A new approach to numerical solution of nonlinear Klein-Gordon equation, Mathematical Problems in Engineering, 2013.
[4] Sadigh B. S., (2011), Numerical Solution Of Klein-Gordon Equation By Using The Adomian’s Decomposition And Variational Iterative Methods, Int. J. Industrial Mathematics,79-89.
[5] Hepson O. E., Korkmaz A., & Dag I., (2018) On the numerical solution of the Klein-Gordon equation by exponential cubic B-spline collocation method, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(1), 412-421.
[6] Bildik N., & Deniz S., (2020) New approximate solutions to the nonlinear Klein-Gordon equations using perturbation iteration techniques, Discrete & Continuous Dynamical Systems-S, 13(3), 503.
[7] Yusufoğlu E., (2008) The variational iteration method for studying the Klein–Gordon equation, Applied Mathematics Letters, 21(7), 669-674.
[8] Han H., & Zhang Z., (2009) An analysis of the finite-difference method for one-dimensional Klein–Gordon equation on unbounded domain, Applied numerical mathematics, 59(7), 1568-1583.
[9] Kaya D., & El-Sayed S. M., (2004) A numerical solution of the Klein–Gordon equation and convergence of the decomposition method, Applied mathematics and computation, 156(2), 341-353.
[10] 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), 278-290.
[11] Wang T. M., & 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), 2444-2448.
[12] Secer A., & Ozdemir N., (2019) An effective computational approach based on Gegenbauer wavelets for solving the time-fractional Kdv-Burgers-Kuramoto equation, Advances in Difference Equations, 2019(1), 386.
[13] Ozdemir N., Secer A., & Bayram M., (2019) The Gegenbauer Wavelets-Based Computational Methods for the Coupled System of Burgers’ Equations with Time-Fractional Derivative, Mathematics, 7(6), 486.
[14] Seçer A., Altun S., & Bayram M., (2019) Legendre Wavelet Operational Matrix Method for Solving Fractional Differential Equations in Some Special Conditions, Thermal Science, 23(Suppl. 1), 203-2014.
[15] Secer A., & Bakir Y., (2019) Chebyshev wavelet collocation method for Ginzburg-Landau equation, Thermal Science, 23(Suppl. 1), 57-65.
[16] Heydari M.H., Maalek Ghaini, F.M., Hooshmandasl M.R., (2014) Legendre wavelets method for numerical solution of time-fractional heat equation, Wavelet Linear Algebra, 1, 19–31.
[17] Heydari M.H., Hooshmandasl M.R., Ghaini F.M., (2014) A new approach of the Chebyshev wavelets method for partial differential equations with boundary conditions of the telegraph type, Appl. Math. Model., 38, 1597–1606.
[18] Secer A., & Ozdemirn N., (2019) Modified Laguerre wavelet based Galerkin method for fractional and fractional-order delay differential equations, Thermal Science, 23(Suppl. 1), 13-21.
[19] Dehestani H., Ordokhani Y., & Razzaghi M., (2019) On the applicability of Genocchi wavelet method for different kinds of fractional‐order differential equations with delay, Numerical Linear Algebra with Applications, 26(5), e2259.
[20] Zheng X, Wei Z., (2015) Discontinuous Legendre Wavelet Galerkin Method for One-Dimensional Advection-Diffusion Equation, Springer P. Math. Stat., 6(09), 1581.
[21] Giordano C, Laforgia A., (2013) On the Bernstein-type inequalities for ultraspherical polynomials, J. comput. Appl. Math, 153(1-2), 243-248.
[22] Chi-Hsu W., (1983) On the generalization of block pulse operational matrices for fractional and operational calculus, J. Frankl. Inst., 315(2), 91-102.
[23] Yin F., Song J., Cao X., Lu F., (2013) Couple of the variational iteration method and Legendre wavelets for nonlinear partial differential equations, J. Appl. Math..
[24] Maleknejad K., Khodabin M., Rostami M., (2012) Numerical solution of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions, Math. Comput. Model, 55(3-4), 791-800.
[25] Maleknejad K., Khodabin M., Rostami, M., (2012) A numerical method for solving m-dimensional stochastic Itô–Volterra integral equations by stochastic operational matrix, Comput. Math. Appl., 63(1), 133-143.
[26] Canuto C., Hussaini M.Y., Quarteroni A., Thomas Jr., (2012) A. Spectral methods in fluid dynamics, Springer Science & Business Media.
[27] Odibat Z., & Momani S., (2009) The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics, Computers & Mathematics with Applications, 58(11-12), 2199-2208.
[28] Abbasbandy S., (2007) Numerical solution of non‐linear Klein–Gordon equations by variational iteration method, International Journal for Numerical Methods in Engineering, 70(7), 876-881.

Sigma Mühendislik ve Fen Bilimleri Dergisi-Cover

ISSN: 1304-7191
Yayın Aralığı: Yılda 4 Sayı
Yayıncı: Yıldız Teknik Üniversitesi

Arşiv