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THE NUMERICAL SOLUTION OF THE TIME-FRACTIONAL NON-LINEAR KLEIN-GORDON EQUATION VIA SPECTRAL COLLOCATION METHOD

ABSTRACT
In this paper, we consider the numerical solution of the time-fractional non-linear Klein-Gordon equation. We propose a spectral collocation method in both temporal and spatial discretizations with a spectral expansion of Jacobi interpolation polynomial for this equation. A rigorous error analysis is provided for the spectral methods to show both the errors of approximate solutions and the errors of approximate derivatives of the solutions decaying exponentially in infinity-norm and weighted L2-norm. Numerical tests are carried out to confirm the theoretical results.
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PAPER SUBMITTED: 2018-08-24
PAPER REVISED: 2018-09-05
PAPER ACCEPTED: 2019-02-13
PUBLISHED ONLINE: 2019-05-26
DOI REFERENCE: https://doi.org/10.2298/TSCI180824220Y
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2019, VOLUME 23, ISSUE Issue 3, PAGES [1529 - 1537]
REFERENCES
  1. Hilfe R. Applications of Fractional Calculus in Physics. World Scientific, Singapore, 1999.
  2. Yang Y., et al., Numerical Solutions for Solving Time Fractional Fokker-Planck Equations Based on Spectral Collocation Methods, Journal of Computational and Applied Mathematics, 339, pp.389-404, 2018
  3. Yang Y., Jacobi Spectral Galerkin Methods for Fractional Integro-Differential Equations, Calcolo,52 (2015) 519-542
  4. Hariharan G. Wavelet Method for a Class of Fractional Klein-Gordon Equations. Journal of Computational and Nonlinear Dynamics, 2, 2013, 8, pp.1-6
  5. Yusufoglu E. The Variational Iteration Method for Studying the Klein-Gordon Equation. Applied Mathematics Letters, 21, 2008, pp. 669-674
  6. Yang Y., et al., Spectral Collocation Method for the Time-Fractional Diffusion-Wave Equation and Convergence Analysis, Computers and Mathematics with Applications, 73, 2017, 6, pp. 1218-1232
  7. Yang Y., et al., Spectral Collocation Methods for Nonlinear Volterra Integro-Differential Equations with Weakly Singular Kernels, Bulletin of the Malaysian Mathematical Sciences Society.42, 2019, 1, pp. 297-314
  8. Yang Y., et al., Numerical Simulation of Time Fractional Cable Equations and Convergence Analysis, Numerical Methods for Partial Differential Equations, 34, 2018, 5, pp. 1556-1576
  9. Mastroianni G., et al., Optimal Systems of Nodes for Lagrange Interpolation on Bounded Intervals: Asurvey. Journal of Computational and Applied Mathematics, 134, 2001, 1, pp. 325-341
  10. Henry D. Geometric Theory of Semilinear Parabolic Equations. Springer-Verlag, 1989
  11. Nevai P. Mean Convergence of Lagrange Interpolation: III. Transactions of the American Mathematical Society, 282,1984, pp. 669-698
  12. Yang Y., et al.,Convergence Analysis of the Jacobi Spectral Collocation Method for Fractional Integro-Differential Equations, Acta. Math. Sci. 34B (2014) 673-690

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence