Study On Implicit Fractional Differential Equations And Coupled Systems

Fractional calculus is the promotion and development of integral order calculus,and its theory was gradually established by Letnikov,Riemann,Liouville's efforts.At present,it is widely used in control theory,viscoelastic theory,fluid mechanics,electronic chemistry and other fields;it is of great research significance.In this paper,two kinds of implicit fractional differential equations and a coupled system are investigated.The main results of this paper are as follows:?1?The initial value problem of the nonlinear implicit fractional differential equation involving Caputo fractional derivative of the form is considered,where0???1,f:[0,T]×R×R?R is continuous.Using fixed point theorem,we investigate the existence and interval of existence of solutions,uniqueness.By the Gronwall type inequality of fractional integral form,we present the estimate on solutions,continuous dependence and uniqueness of the initial values on solutions,continuous dependence on parameters and nonlinear terms involved in the equations.Further,we discusse-approximate solution to the implicit fractional differential equations.?2?The initial value problem of a coupled system of nonlinear implicit fractional differential equation involving Caputo fractional derivative of the form is discussed,where 0???1,f:[0,T]×R×R?R is continuous.Using fixed point theorem,we investigate the existence of solutions,uniqueness.By the vector form of the Gronwall type integral inequality,we obtain the estimate on solutions,continuous dependence and uniqueness of the initial values on solutions,continuous dependence on parameters and nonlinear terms involved in the coupled system.Further,we discuss thee-approximate solution of the coupled system.?3?The periodic boundary value problem of implicit fractional differential equation involving Riemann-Liouville fractional derivative of the form is investigated,where ??0, ??R,E_?,? ????1/?2?????, f:(0:1]×R²?R is continuous.Firstly,we give an equivalent equation of the original problem.By the Green function of this periodic boundary value problem and fixed point theorem,some existence and uniqueness results are obtained.