Oscillation Of Fractional Dynamic Equations And Its Applications

Oscillation theory which as an important branch of the qualitative theory of differential equations, describes the oscillatory behaviour of the solutions around the x axis. And it has important value in the actual production and daily life. For example, the hull model, which is floating in the water, its rocking behavior can be described by the oscillation theory of a class of delay differential equations. For another example, in the field of economics, the hysteresis phenomenon between the production and consumption, and the vagaries of commodity prices can both be explained by the oscillatory behavior of the solutions of certain functional differential equations. On the industry, the mechanical oscillation and electro-magnetic induction all contain the theories of oscillation. In recently, oscillation theory of differential equations is widely used in control, ecology, economics, biology, life sciences, engineering,etc. So it has been received great consideration.Fractional differential equation, which means the differential equations containing fractional order derivatives, has more applicability than integer order differential equations in certain circumstances. So it has been used in physics, biology, communications engineering and other fields. From more and more attentions, large amounts of theories of fractional differential equations have been established by the researchers. But, affected by the special characteristic of fractional order derivative, the qualitative theory of fractional differential equations still remain much research area. Many research problem need to be resolved.Therefore, in this paper, we study the oscillation of fractional differential equations,overcome the difficulty that fractional derivative does not have the same good properties as integer ones, and get the oscillation criteria for the fractional differential equations. Also in this paper, from the classical oscillation theorem, we discuss the hot problem, the oscillation for dynamic equations on time scales, and get many good results.This paper is devoted to the study of the oscillation of solutions of nonlinear fractional differential equations, neutral fractional differential equations and fractional differential equations with constant coefficients. Otherwise, it also contains the oscillation of second order dynamic equations, third order Emden-Fowler type dynamic equations and higher order dynamic equations on time scales. Many valuable results have been obtained.In chapter one, we introduce research background, definitions of fractional deritive and present situation of the oscillation theory of fractional differential equations.In chapter two, we investigate the oscillation of solutions of nonlinear fractional differential equations with Riemann-Liouville fractional derivative. We use the Riccati transformation technique to obtain some sufficient conditions.In chapter three, we study the oscillatory behavior of solutions of neutral fractional differential equations with modified Riemann-Liouville fractional derivative. By the new comparison theorems, some oscillation criteria have been established in this paper.In chapter four, we discuss the oscillation for fractional delay differential equations with constant coefficients. Using Laplace transform and characteristic equations, we obtain a sufficient and necessary condition for the oscillation of the equations.In chapter five, we consider the oscillation of solutions of dynamic equations on time scale. By using Hille and Nehari type oscillation criteria, the sufficient conditions for the oscillation of two different dynamic equations have been established.In chapter six, by Kwong-Wong’s integration, some oscillation criteria have been established for a kind of second order dynamic equations.In chapter seven, we summarize the main results and the innovations in this paper.Finally, we prospect some future research work based on this paper.