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This paper introduces a numerical approach for the fractional-order Rössler chaotic systems and gives error analysis. The effectiveness of the present method is determined by comparing the numerical results of the high-precision difference scheme and predictor-correctors scheme. Some complex dynamical behaviors for the fractional-order Rössler chaotic systems are shown. [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]

Today, most of the real physical world problems can be best modelled with fractional telegraph equation. Besides modelling, the solution techniques and their reliability are the most important. Therefore, high accuracy solutions are always needed. As we all know, reproducing kernel method (RKM) has been successfully presented for solving various ordinary differential equations. However, the numerical results are not perfectly satisfactory when we directly use the traditional RKM for solving fractional partial differential equation. The aim of this paper is to fill this gap. In this paper, a new method is provided for solving fractional telegraph equation in the reproducing kernel space by piecewise technique, which can obtain more accurate solution than traditional method. Three experiments are given to demonstrate the effectiveness of the present method. [ABSTRACT FROM AUTHOR]