Existence And Numerical Realization Of Solutions For Boundary Value Problems Of Fractional Differential Equations

Fractional calculus is a further extension of classical integer calculus,but it is not a completely parallel generalization.It is found that the fractional differential equations can accurately describe some physical phenomena and dynamic processes in nature because its order can be changed continuously.At present,it has been involved in physics,biology,chemistry,power systems,engineering,medicine,finance and other fields and has made remarkable achievements,quickly become a hot research topic of concern.At present,the existence of solutions of fractional boundary value problems is studied mainly by the theory of nonlinear analysis.It is very important to solve its numerical solution in practical application.Compared with integer-order differential equations,it has few classical numerical methods that can be used directly.In addition,the order of the solution is not an integer,and the structure of the solution is special,so it is difficult to directly solve the solution by computer.If the nonlinear term contains fractional derivative,it is not easy to realize its numerical solution.In this thesis,a clever method is used to break through these limitations,and the numerical solution is obtained,and the graph of the solution is drawn.In this thesis,the existence of solutions for a class of fractional boundary value problems with nonlinear terms containing explicit fractional derivatives is studied.By using Schauder's fixed point theorem and Krasnosel'skii fixed point theorem,a sufficient condition for the existence of at least one or two positive solutions of the studied problem is established.By using generalized Avery-Henderson fixed point theorem and Avery-Peterson fixed point theorem,A sufficient condition for the existence of at least 3,n or 2n-1 positive solutions of this boundary value problem is established,and four simulation examples are given.Secondly,the realization of numerical solution of boundary value problem of fractional differential equation is studied.A new efficient iterative method is constructed to solve the boundary value problem.The monotone iteration sequence is constructed by using the defined compact operator,and the convergence of the monotone iteration sequence is proved,and its limit is the numerical solution.Furthermore,the uniqueness of the positive solution of the problem is obtained by the principle of compression mapping.In the MATLAB environment,a new algorithm combining Gauss-Kronrod of classical numerical integration with cubic spline interpolation method is used to obtain the numerical solution of the boundary value problem successfully.In particular,the numerical solution graphs of the simulation examples mentioned above are given by using this iterative method.Finally,the numerical methods for upper and lower solutions of a class of fractional boundary value problems are discussed.First,the upper and lower solution method combined with monotone iteration technique is used to prove the existence of the solution of the boundary value problem,and the sufficient conditions for the existence of the solution are established.The most distinctive feature of the upper and lower solution value method proposed in this thesis is that it can construct two iterative sequences of upper and lower solutions by monotone iteration method,and obtain two iterative sequences by using the previously proposed new algorithm function under MATLAB software,whose limit value is the numerical solution of the fractional boundary value problem.Finally,a simulation example is given,the numerical solution is given in the form of figure,and the absolute error table is given to observe the accuracy and convergence speed of the convergence solution.