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.
Highlights
In recent years, fractional calculus and especially fractional di¤erential equations (FDEs) have been extensively used for many di¤erent ...elds of mathematical physics such as relaxation processes,control theory of dynamical systems, viscoelasticity, di¤usion and so on [1,2,3,4,5]
The Klein - Gordon equation is generalized using the concept of the variational order derivative
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
Summary
Fractional calculus and especially fractional di¤erential equations (FDEs) have been extensively used for many di¤erent ...elds of mathematical physics such as relaxation processes,control theory of dynamical systems, viscoelasticity, di¤usion and so on [1,2,3,4,5]. Klein-Gordon equation, fractional variable order derivative, CrankNicholson scheme, stability. C 2020 Ankara University C om munications Faculty of Sciences U niversity of A nkara-Series A 1 M athem atics and Statistics As it is well known, partial di¤erential equations are encountered frequently in many ...elds of applied physics [16,17,18,19,20,21,22,23]. In [25], Sweilam et al has constructed a new and e¤ective numerical scheme, namely weighted average nonstandard ...nite di¤erence method, for analyzing the time variable-order fractional of nonlinear Klein-Gordon equation and so on. We investigate the stability of the linear time-fractional variable order Klein-Gordon equation: Dtt(x;t)y(x; t) yxx(x; t) + y(x; t) = 0; 1 < (x; t) 2; > 0;.
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