In this paper we numerically investigate the chaotic behaviours of the fractional-order Ikeda delay system. The results show that chaos exists in the fractional-order Ikeda delay system with order less than 1. The low...In this paper we numerically investigate the chaotic behaviours of the fractional-order Ikeda delay system. The results show that chaos exists in the fractional-order Ikeda delay system with order less than 1. The lowest order for chaos to be a, ble to appear in this system is found to be 0.1. Master-slave synchronization of chaotic fractional-order Ikeda delay systems with linear coupling is also studied.展开更多
The velocity field of generalized second order fluid with fractional anomalous diiusion caused by a plate moving impulsively in its own plane is investigated and the anomalous diffusion problems of the stress field an...The velocity field of generalized second order fluid with fractional anomalous diiusion caused by a plate moving impulsively in its own plane is investigated and the anomalous diffusion problems of the stress field and vortex sheet caused by this process are studied. Many previous and classical results can be considered as particular cases of this paper, such as the solutions of the fractional diffusion equations obtained by Wyss; the classical Rayleigh’s time-space similarity solution; the relationship between stress field and velocity field obtained by Bagley and co-worker and Podlubny’s results on the fractional motion equation of a plate. In addition, a lot of significant results also are obtained. For example, the necessary condition for causing the vortex sheet is that the time fractional diffusion index β must be greater than that of generalized second order fluid α; the establiihment of the vorticity distribution function depends on the time history of the velocity profile at a given point, and the time history can be described by the fractional calculus.展开更多
The generalized fractional element networks are presented in this paper. In order to extend the structure of the model solutions to the generalized function space and make it contain more physical meanings, the restri...The generalized fractional element networks are presented in this paper. In order to extend the structure of the model solutions to the generalized function space and make it contain more physical meanings, the restriction on the parameters of the fractional element proposed by Schiessel et al. is eliminated and a 'compatibility equation' is added. The discretization method for solving the inverse Laplace transform is used and developed. The generalized solutions of the model equations are given. At the same time the generalized fractional element network--Zener and Poyinting-Thomson models are discussed in detail. It is shown that all the results obtained previously about the models of single parameter with fractional order and the classical models with integer order can be contained as the special cases of the results of this paper.展开更多
′In this article, we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations. We use the improved （G′/G）-expansion func...′In this article, we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations. We use the improved （G′/G）-expansion function method to calculate the exact solutions to the time- and space-fractional derivative foam drainage equation and the time- and space-fractional derivative nonlinear KdV equation. This method is efficient and powerful for solving wide classes of nonlinear evolution fractional order equations.展开更多
The fractional calculus approach is introduced into the rheological constitutive model of a generalized second grade fluid. A constitutive model with fractional derivative is developed for the generalized second grade...The fractional calculus approach is introduced into the rheological constitutive model of a generalized second grade fluid. A constitutive model with fractional derivative is developed for the generalized second grade fluid. Unsteady Couette flow of the generalized second grade fluid is studied by using the method of the discrete inverse Laplace transform and generalized Mittag-Leffler function. And then an exact solution is obtained for this problem with arbitrary fractional derivative. This provides a new analytical tool for the study of viscoelastic fluid mechanics.展开更多
The fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced.The flow near a wall suddenly set in mo- tion is studied for a non-Newtonian viscoelastic fluid with the frac...The fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced.The flow near a wall suddenly set in mo- tion is studied for a non-Newtonian viscoelastic fluid with the fractional Maxwell model.Exact solutions of velocity and stress are obtained by using the discrete in- verse Laplace transform of the sequential fractional derivatives.It is found that the effect of the fractional orders in the constitutive relationship on the flow field is signif- icant.The results show that for small times there are appreciable viscoelastic effects on the shear stress at the plate,for large times the viscoelastic effects become weak.展开更多
基金Project supported by the National Natural Science Foundation of China （Grant No 60404005）.
文摘In this paper we numerically investigate the chaotic behaviours of the fractional-order Ikeda delay system. The results show that chaos exists in the fractional-order Ikeda delay system with order less than 1. The lowest order for chaos to be a, ble to appear in this system is found to be 0.1. Master-slave synchronization of chaotic fractional-order Ikeda delay systems with linear coupling is also studied.
基金the Doctoral Program Foundation of the Education Ministry of China the National Natural Science Foundation of China (Grant No. 10002003) Foundation for University Key Teacher by the Ministry of Education of China.
文摘The velocity field of generalized second order fluid with fractional anomalous diiusion caused by a plate moving impulsively in its own plane is investigated and the anomalous diffusion problems of the stress field and vortex sheet caused by this process are studied. Many previous and classical results can be considered as particular cases of this paper, such as the solutions of the fractional diffusion equations obtained by Wyss; the classical Rayleigh’s time-space similarity solution; the relationship between stress field and velocity field obtained by Bagley and co-worker and Podlubny’s results on the fractional motion equation of a plate. In addition, a lot of significant results also are obtained. For example, the necessary condition for causing the vortex sheet is that the time fractional diffusion index β must be greater than that of generalized second order fluid α; the establiihment of the vorticity distribution function depends on the time history of the velocity profile at a given point, and the time history can be described by the fractional calculus.
基金the Doctoral Program Foundation of the Ministry of Education of China,the National Natural Science Foundation of China(Grant Nos.10272067 and 10002003)the Foundation for University Key Teacher by the Ministry of Education.
文摘The generalized fractional element networks are presented in this paper. In order to extend the structure of the model solutions to the generalized function space and make it contain more physical meanings, the restriction on the parameters of the fractional element proposed by Schiessel et al. is eliminated and a 'compatibility equation' is added. The discretization method for solving the inverse Laplace transform is used and developed. The generalized solutions of the model equations are given. At the same time the generalized fractional element network--Zener and Poyinting-Thomson models are discussed in detail. It is shown that all the results obtained previously about the models of single parameter with fractional order and the classical models with integer order can be contained as the special cases of the results of this paper.
文摘′In this article, we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations. We use the improved （G′/G）-expansion function method to calculate the exact solutions to the time- and space-fractional derivative foam drainage equation and the time- and space-fractional derivative nonlinear KdV equation. This method is efficient and powerful for solving wide classes of nonlinear evolution fractional order equations.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 10002003), the Foundation for University Key Teacher by the Ministry of Education of China and the JSPS postdoctoral fellowship for foreign researchers.
文摘The fractional calculus approach is introduced into the rheological constitutive model of a generalized second grade fluid. A constitutive model with fractional derivative is developed for the generalized second grade fluid. Unsteady Couette flow of the generalized second grade fluid is studied by using the method of the discrete inverse Laplace transform and generalized Mittag-Leffler function. And then an exact solution is obtained for this problem with arbitrary fractional derivative. This provides a new analytical tool for the study of viscoelastic fluid mechanics.
基金The project supported by the National Natural Science Foundation of China (10002003)Foundation for University Key Teacher by the Ministry of EducationResearch Fund for the Doctoral Program of Higher Education
文摘The fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced.The flow near a wall suddenly set in mo- tion is studied for a non-Newtonian viscoelastic fluid with the fractional Maxwell model.Exact solutions of velocity and stress are obtained by using the discrete in- verse Laplace transform of the sequential fractional derivatives.It is found that the effect of the fractional orders in the constitutive relationship on the flow field is signif- icant.The results show that for small times there are appreciable viscoelastic effects on the shear stress at the plate,for large times the viscoelastic effects become weak.
