Study On The Solutions Of Some Fractional Differential Equation Boundary Value Problems

In the study of fractional differential equation,the problems of existence,uniqueness and numerical solution of boundary value problems are common problems.With the development and progress in recent years,there have been new advances in the study of the fixed point theory of fractional differential equation,abstract spaces,delay differential equation and linear differential equation.In this context,we study the existence and uniqueness of solutions for two classes of fractional differential equation boundary value problems,and the numerical solutions for a class of fractional differential equation boundary value problems.The first,we consider a class of nonlinear fractional differential equation boundary value problems with Caputo derivatives in Banach spaces.The nonlinear term is a functional equation with the first derivative of unknown function.By converting the differential equation boundary value problem to an equivalent integral equation.First,by converting the differential equation into an equivalent integral equation,the integral operator is obtained,and then it is proved that the operator is completely continuous,finally,the existence and uniqueness of the solution of the boundary value problem are obtained by means of the norm form of the cone extension-contraction fixed-point theorem and the Leray-Schauder fixed-point Theorem,and the Banach contraction mapping fixed-point theorem is used to obtain the uniqueness of the solution.Secondary,we study a class of nonlinear fractional differential equation boundary value problems with Riemann-Liouville derivatives in ordered Banach spaces.The nonlinear term is a functional equation with the first derivative of unknown function.First,the integral expression of the solution of the boundary value problem is obtained by means of integral transformation,and the integral operator is obtained,It is proved that the operator is a condensing map,then,by using the Leray-Schauder fixed-point theorem and Kuratowskii non-compact measure theory,the existence and uniqueness of positive solutions for the boundary value problem are obtained.Finally,we consider a class of linear fractional boundary value problems with delay and Caputo's derivative,which is called the differential equation problem.By using the property of alternative Legendre polynomials(Alps),the numerical approximation of the delay differential equation is carried out,and the delay differential equation is transformed into an algebraic system.Finally,a numerical example is given to verify the accuracy of the method.The results show that the method is more accurate.There are1 figures,2 tables and 65 references in this paper.