A New Perspective on The Numerical Solution for Fractional Klein Gordon Equation

Year 2019, Volume: 22 Issue: 2, 443 - 451, 01.06.2019

Berat Karaagac , Yusuf Ucar N. Murat Yagmurlu Alaattin Esen

https://doi.org/10.2339/politeknik.428986

Abstract

In the present manuscript, a new numerical scheme is
presented for solving the time fractional nonlinear Klein-Gordon equation. The
approximate solutions of the fractional equation are based on cubic B-spline
collocation finite element method and L2 algorithm. The fractional derivative
in the given equation is handled in terms of Caputo sense. Using the methods,
fractional differential equation is converted into algebraic equation system
that are appropriate for computer coding. Then, two model problems are
considered and their error norms are calculated to demonstrate the reliability
and efficiency of the proposed method. The newly calculated error norms show
that numerical results are in a good agreement with the exact solutions.

Keywords

Finite element method, collocation, Fractional Klein Gordon equation, Caputo derivative

References

• [1] Kumara D., Seadawy A. R. and Joardare A. K., “Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology”, Chinese Journal of Physics, 56: 75–85, (2018).
• [2] Kumara D., Seadawy A. R. and Joardare A. K., “Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology”, Chinese Journal of Physics, 56: 75–85, (2018).
• [3] Sohail A., Maqbool K. and Ellahi R., “Stability analysis for fractional-order partial differential equations by means of space spectral time Adams-Bashforth Moulton method”, Numerical solution for Partial Differential Equations, 34: 19–29, (2018).
• [4] Singh J., Kumar D., Hammouch Z. And Atangana A.,” A fractional epidemiological model for computer viruses pertaining to a new fractional derivative”, Applied Mathematics and Computation, 316: 504–515,(2018).
• [5] Tasbozan O. and Esen A.,” Quadratic B-Spline Galerkin Method for Numerical Solutions of Fractional Telegraph Equations” Bulletin of Mathematical Sciences and Applications, 18: 23-39, (2017).
• [6] Bulut F., Oruç, Ö. And Esen A.,” Numerical Solutions of Fractional System of Partial Differential Equations By Haar Wavelets” Computer Modeling in Engineering & Sciences, 108 :263-284,(2015).
• [7] Nagy A. M., “Numerical solution of time fractional nonlinear Klein–Gordon equation using Sinc–Chebyshev collocation method”, Applied Mathematics and Computation, 310:139-148,(2017).
• [8] Lim S. C. and S. V. Muniandy. "Stochastic quantization of nonlocal fields." Physics Letters A 324: 396-405(2004).
• [9] Kheiri H., Shahi S. and Mojaver A., “Analytical solutions for the fractional Klein-Gordon equation”, Computational Methods for Differential Equations, 2:99-114,(2014).
• [10] Mohebbi A., Abbaszadeh M. and Dehghan M., “High-Order Difference Scheme for the Solution of Linear Time Fractional Klein–Gordon Equations, Numerical solution for Partial Differential Equations”Numerical solution for Partial Differential Equations, 30: 1234-1253.(2014).
• [11] Lyu P. and Vong S., “A linearized second-order scheme for nonlinear time fractional Klein-Gordon type equations”, Numerical Algorithms, 78:485-511,(2018).
• [12] Khader M. M., Swetlam N. H . and Mahdy A. M. S., “The Chebyshev Collection Method for Solving Fractional Order Klein-Gordon Equation”, Wseas Trans. Math, 13: 31-38,(2014).
• [13] Alqahtani R. T., “Approximate Solution of Non-Linear Fractional Klein-Gordon Equation Using Spectral Collocation Method”, Applied Mathematics, 6: 2175-2181,(2015).
• [14] Li C., and Chen A., "Numerical methods for fractional partial differential equations." International Journal of Computer Mathematics 95: 1048-1099, (2018).
• [15] Ara A., Khan N. A., "Wavelets optimization method for evaluation of fractional partial differential equations: an application to financial modelling." Advances in Difference Equations 2018.1:8. (2018)
• [16] Liu Y. , Khan M., and Yan Y.,"Fourier spectral methods for stochastic space fractional partial differential equations driven by special additive noises." Journal of Computational Analysis and Applications 24: 290-309, (2018).
• [17] Baseri, A., Abbasbandy S., and Babolian E., "A collocation method for fractional diffusion equation in a long time with Chebyshev functions." Applied Mathematics and Computation 322:55-65, (2018)
• [18] Yuste S. B., “Weighted average finite difference methods for fractional diffusion equations”, Journal of Computational Physics, 216:264-274 ,(2006).
• [19] Prenter P. M., “Splines and Variational Methods”, Wiley, (1975).