Abstract
In this paper, an efficient nonstandard finite difference scheme for the numerical solution of chaotic fractional-order Chen system is developed. In the new method, an appropriate nonlocal framework in conjunction with the Grünwald-Letnikov approximation are applied for the discretization of fractional differential system. By constructing the discretization with the nonstandard finite difference scheme, high resolution of the system can be obtained, and the numerical instabilities of the nonlinear fractional-order Chen chaotic system can be also addressed to some extent. In addition, a new fractional derivative of the Caputo type is employed in the context of fractional-order Chen system to further decrease the computational complexity in the long-term treatment of fractional model. Numerical simulations demonstrate the applicability, accuracy and efficiency of the developed method.
Highlights
During the last decades fractional calculus has become a powerful tool to describe the dynamics of complex systems in numerous branches of science and engineering [1]–[6]
This means that the interval of integration may be very large with the increase of time t, and as a result, the computational complexity of these fractional derivatives and the resulting fractional differential equations may be prohibitive for systems of global dynamics
CONSTRUCTION OF NONSTANDARD FINITE DIFFERENCE SCHEME FOR CHAOTIC FRACTIONAL-ORDER CHEN SYSTEM we develop a straightforward method based on the nonstandard finite difference scheme and the GrünwaldLetnikov approximation to obtain numerical solutions of the fractional-order Chen system
Summary
During the last decades fractional calculus has become a powerful tool to describe the dynamics of complex systems in numerous branches of science and engineering [1]–[6]. The main contribution of the present study is the efficient construction of a nonstandard finite discretization scheme in conjunction with a new definition of fractional derivative for solving chaotic fractional-order Chen system. Since the length of the interval of the new derivative can be adaptively chosen depending on the specific problem, the computational burden for the solution of the chaotic system can be significantly decreased, and as a result, the developed method is more efficient than the most used predictor-corrector method in capturing chaotic behavior of the fractional-order Chen system. Comparisons of the developed method with the predictor-corrector method are made
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