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This paper presents an approximate solution of time variable order fractional differential equations with sub-diffusion and super-diffusion. The aim of paper is to solve and analyze this problem by a fully discrete local discontinuous Galerkin scheme. The method is based on local discontinuous Galerkin method in space and a finite difference technique in time. The numerical stability and convergence of the proposed method are investigated then the convergence rate O (h k + 1 + △ t 2 − α (t n)) in the case of sub-diffusion and O (h k + 1 + △ t) in the case of super-diffusion are proven for the presented scheme. Finally, provided numerical examples illustrate efficiency of the method and accuracy of the theory. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to introduce a numerical method for time-space fractional sine-Gordon equation. The fractional derivative on the space and on the time are considered in the sense of Reimann–Liouville (of order 1 ≤ β ≤ 2) and in the sense of Caputo (of variable order 1 ≤ α (t) < 2), respectively. The basic idea is to apply local discontinuous Galerkin method in space and a finite difference method in time. The stability and convergence analysis of the method are presented. Numerical results show that the accuracy and reliability of the proposed method for time-space fractional sine-Gordon equation. [ABSTRACT FROM AUTHOR]