On the stability analysis of the time-fractional variable order Klein-Gordon equation and a numerical simulation

Year 2020, , 981 - 992, 30.06.2020

Sinan Deniz

https://doi.org/10.31801/cfsuasmas.450209

Cited By: 2

Abstract

In this paper, the Klein - Gordon equation is generalized using the concept of the variational order derivative. We try to construct the Crank-Nicholson scheme for numerical solutions of the modified Klein- Gordon equation. Stability analysis of the Crank-Nicholson scheme is examined and analyzed to prove the proposed method is stable for solving the time-fractional variable order Klein- Gordon equation. A numerical example is also given for illustration.

Keywords

Klein-Gordon equation, fractional variable order derivative, Crank-Nicholson scheme

References

• Podlubny, I., Fractional differential equations, Academic Press, New York, 1999.
• Caputo, M., Linear models of dissipation whose Q is almost frequency independent, part II, Geophys J. Int., 13(5) (1967), 529-539.
• Bildik, N., Deniz, S., Saad, K.M., A comparative study on solving fractional cubic isothermal auto-catalytic chemical system via new efficient technique, Chaos, Solitons & Fractals, 132 (2020).
• S.G. Samko, Kilbas, A.A., Marichev, O.I., Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach, New York, 1993.
• Atangana, A., On the new fractional derivative and application to nonlinear Fisher's reaction-diffusion equation, Applied Mathematics and Computation, 273 (2016), 948-956.
• Arikoglu, A., Ibrahim O., Solution of fractional differential equations by using differential transform method, Chaos, Solitons & Fractals, 34.5 (2007), 1473-1481.
• Deniz, S., Semi-analytical analysis of Allen-Cahn model with a new fractional derivative, Mathematical Methods in the Applied Sciences, , (2020), https://doi.org/10.1002/mma.5892
• Kilicman, A., Gupta, V. G., Shrama, B., On the solution of fractional Maxwell equations by Sumudu transform, Journal of Mathematics Research, 2 (4) (2010), 147.
• Bildik, N., Deniz, S., A new fractional analysis on the polluted lakes system, Chaos, Solitons & Fractals, 122 (2019), 17-24.
• Saad, K.M., Deniz, S., Baleanu, D., On the New Fractional Analysis of Nagumo Equation, International Journal of Biomathematics, 12 (03) (2019), 1950034.
• Atangana, A., Aydin S., The time-fractional coupled-Korteweg-de-Vries equations, Abstract and Applied Analysis. Vol. 2013., Hindawi Publishing Corporation, (2013).
• Atangana, A., Botha, J., A generalized groundwater flow equation using the concept of variable-order derivative." Boundary Value Problems 2013.1 (2013): 53.
• Atangana, A., Cloot, A.H., Stability and convergence of the space fractional variable-order Schrödinger equation, Advances in Difference Equations, 2013.1 (2013), 80.
• Atangana, A., On the stability and convergence of the time-fractional variable order telegraph equation, Journal of Computational Physics, 293 (2015), 104-114.
• Li, C. P., Zhang, F. R., A survey on the stability of fractional differential equations, The European Physical Journal-Special Topics, 193.1 (2011): 27-47.
• Deniz, S., Semi-analytical investigation of modified Boussinesq-Burger equations, J. BAUN Inst. Sci. Technol., 22, (1) (January 2020), 327-333.
• Bildik, N., Deniz, S., A practical method for analytical evaluation of approximate solutions of Fisher's equations, ITM Web of Conferences, 13 (2017), Article Number: 01001.
• Bildik, N., Deniz, S., New analytic approximate solutions to the generalized regularized long wave equations, Bulletin of the Korean Mathematical Society, 55 (3) (May 2018), 749-762. Bildik, N., Deniz, S., Solving the Burgers' and regularized long wave equations using the new perturbation iteration technique, Numerical Methods for Partial Differential Equations, 34, (5) (2018), 1489-1501.
• Kilicman, A., Eltayeb, H., A note on defining singular integral as distribution and partial differential equations with convolution term, Mathematical and Computer Modelling, 49 (1) (2009), 327-336.
• Deniz, S., Modification of coupled Drinfelâd-Sokolov-Wilson Equation and approximate solutions by optimal perturbation iteration method, Afyon Kocatepe University Journal of Science and Engineering, 20 (1) (February 2020), 3540.
• Agarwal, P., Deniz, S., Jain, S., Alderremy, A.A., Aly, S., A new analysis of a partial differential equation arising in biology and population genetics via semi analytical techniques, Physica A: Statistical Mechanics and its Applications, Volume 542 (15 March 2020), 122769.
• Bildik, N., Deniz, S., New approximate solutions to the nonlinear Klein-Gordon equations using perturbation iteration techniques, Discrete and Continuous Dynamical Systems Series-S, Volume 13 (3) (March 2020), 503-518.
• Golmankhaneh, A.K., Baleanu, D., On nonlinear fractional Klein-Gordon equation, Signal Processing, 91 3 (2011), 446-451.
• Sweilam, N.H., Al-Mekhlafi. S.M., Albalawi, A.O., A novel variable-order fractional nonlinear Klein Gordon model: A numerical approach, Numer Methods Partial Differential Eq., 2019, 1 - 13, https://doi.org/10.1002/num.22367
• Petras, I., Fractional-order nonlinear systems: modeling, analysis and simulation, Springer Science & Business Media, 2011.
• Bagley, R.L., Torvik, P.J., Fractional calculus-A different approach to the analysis of viscoelastically damped structures, AIAA Journal, (ISSN 0001-1452) 21 (1983): 741-748.
• Miller, K.S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
• Meerschaert, M.M., Tadjeran, C., Finite difference approximations for fractional advection dispersion equations, J. Comput. Appl. Math., 172 (2004), 65-77.
• Tadjeran, C., Meerschaert, M.M., Scheffler, H.P., A second order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys. 213 (2006), 205-213.
• Liu, Y., Fang, Z., Li, H., He, S., A mixed finite element method for a time-fractional fourth-order partial differential equation, Appl. Math. Comput., 243 (2014), 703-717.
• Bildik, N., Deniz, S., On the asymptotic stability of some particular differential equations, International Journal of Applied Physics and Mathematics, 5(4) (2015), 252-258.
• Gopalsamy, K.. Stability and oscillations in delay differential equations of population dynamics, Vol. 74. Springer Science & Business Media, 2013.
• Deniz, S., Bildik, N., Sezer, M., A note on stability analysis of Taylor collocation method, Celal Bayar University Journal of Science, 13 (1) (2017), 149-153.