Qualitative Analysis Of Fractional Partial Differential Equations And Dynamical Systems On Time Scales

As everyone knows,for most integer order differential equations and fractional order differential equations it is very difficult to find its general solution,sometimes it is impossible.Thus the mathematical workers only from the equation itself to analyze some properties,such as existence,boundedness,oscillation,asymptotic stability and so on.The study on such problems promote the development of the qualitative theory of equations.The theory of fractional calculus(including fractional differential equation,fractional integral equation,fractional integral differential equations and special functions of mathematical physics equation as well as a new branch of mathematics,in fluid mechanics,rheology,diffusion system,power system control theory and other fields have important applications.Because the theoretical research of fractional differential equation has just started in many aspects,the vibration theory of fractional differential equation is still imperfect.This thesis is concerned with the oscillation of fractional differential equations,time fractional partial differential equation and a class of Volterra-Fredholm type power integral inequality and other issues.The results obtained extend and improve many related results reported in the literature.The main results are described as follows:In Chapter I,we give a brief overview of the oscillation of solutions of fractional differential equation,a class of time fractional differential equations,a Volterra-Fredholm type integral inequality on time scale,and introduces the main work of this paper.In Chapter II,according to the Riemann-Liouville fractional differential integral definition,by using the generalized Riccati technique and integral averaging technique and the differential inequality theory,we discuss the oscillation of solutions for a class of fractional partial differential equations with damping,the results extend and improve the corresponding conclusions in the literature.In Chapter III,we study the oscillation of a class of time fractional partial differential equations with damping,obtain the related conditions of oscillatory solutions under some new criteria,and some new results are obtained.In the Chapter IV,some explicit bounds on solutions to a class of new power nonlinear Volterra-Fredholm type dynamic integral inequalities on time scales are established,which can be used as effective tools in the study of certain dynamic equations.Application examples are also given.In the Chapter V,we use a modified nonlinear dynamic inequality on time scales to study the boundedness of a class of third-order and 9)-th order nonlinear dynamic equations on time scales.The results generalize and improve the corresponding conclusions in the literature.The last part,the future research tendency is prospected.