Existence And Simulation Of Solutions For Some Class Of Fractional Differential Equations Boundary Value Problems

Fractional differential equations are widely used in material models,aerospace engineering,biological population models and other fields.Compared with integer differential equations,fractionalr differential equations are often more accurate than integer order differential equations when describing some physical and chemical phenomena.At present,most scholars mainly use fixed point theory,variational method and spectral theory to study the existence and multi-solution of fractional differential equation solutions,but at the same time,the simulation results of their solutions are relatively few.It is well known that,the existence of solutions to differential equations is given and the solution is simulated,so that the behavior of the solutions can be understood intuitively.Therefore,based on the study of the existence of several kinds of BVP,we construct a new iterative series,simulate the solution,and obtain the iterative process simulation diagram and approximation solution.Firstly,the existence of solutions for a class of m-point FDE which the nonlinear term involves derivative are studied.First,the corresponding Green function is obtained by calculation,and the properties of the Green function are studied,then a suitable cone is constructed.By defining the norm in cone and using the properties of the norm to control the derivative term and then controlling the nonlinear term,and finally using various fixed point theorems to obtain the sufficiency conditions for the existence of at least one,two,three and n solutions.Lastly,a few examples are given to verify the applicability of the conclusions,and a new iterative relationship is constructed,the examples are simulated,and the iterative process and approximate solutions are given.Secondly,the existence and multiplicity of positive solutions to a class of nonlinear p-Laplacian FDE with derivatives are studied.First,by using the properties of the p operator and the given boundary value conditions,the Green functions of the BVP are obtained.By using Guo-Krasnosel’skii’s fixed point theorem and Leggett-Williams fixed point theorem,the existence of one or three positive solutions to the BVP are obtained.As a verification,two examples are given,and the examples are verified by iterative method.It is worth mentioning that the differential equation which is studied in this paper not only has the p-Laplacian operator,but also the nonlinear term contains the Riemann-Liouville type fractional differential,which makes the research problem more complicated.Finally,the BVP for a class of nonlinear integro-differential equations are studied.Compared with the general differential equation,the nonlinear term of this kind of problem includes the integral term,which is more difficult to study.The form of the solution of the BVP is obtained,and the existence and uniqueness of the BVP solution are obtained by using Krasnosel’skii’s fixed point theorem and the Banach compression mapping principle.As application,two examples are given,and an iterative simulation is used to illustrate the correctness of the conclusion.