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Chaotic systems arise everywhere in control theory and nonlinear vibration. This paper uses a high-precision numerical approach for capturing chaotic attractors of the fractional EI Ni ñ o chaotic systems. The numerical results indicate that some of the chaotic attractors of fractional-order systems are topologically equivalent to those discovered in the integer-order systems, and some unusual attractors of fractional EI Ni ñ o chaotic systems are found by using the present method. [ABSTRACT FROM AUTHOR]

Due to the non-locality of fractional derivative, the analytical solution and good approximate solution of fractional partial differential equations are usually difficult to get. Reproducing kernel space is a perfect space in studying this type of equations, however the numerical results of equations by using the traditional reproducing kernel method (RKM) isn't very good. Based on this problem, we present the piecewise technique in the reproducing kernel space to solve this type of equations. The focus of this paper is to verify the stability and high accuracy of the present method by comparing the absolute error with traditional RKM and study the effect on absolute error for different values of α. Furthermore, we can study the distribution of entire space at a particular time period. Three numerical experiments are provided to verify the efficiency and stability of the proposed method. Meanwhile, it is tested by experiments that the change of the value of α has little effect on its accuracy. [ABSTRACT FROM AUTHOR]

In previous works, we have devoted to using the reproducing kernel methods solving integer order differential equations, based on the review of previous works, in this paper, we mainly present a method for solving a class of higher order fractional differential equations with general boundary value problems by using Taylor formula into reproducing kernel space. Its analytical solution is represented in the form of series. The analytical solution and approximate solution obtained by this method is given and it is uniformly converge to the exact solution and its corresponding derivatives. The numerical examples are studied to demonstrate the accuracy of the present method. [ABSTRACT FROM PUBLISHER]