Existence Of Solutions For Some Fractional Differential Systems On Metric Graphs

Fractional calculus is a theory about arbitrary differential and integral.Compared with integer differential equations,fractional differential equations have wider application background and more complex properties.For example,in physics,fractional differential equations can be used to describe nonlocal phenomena,such as diffusion,transport and fluctuation.In biology,fractional differential equations can be used to describe the complexity and nonlinear behavior of biological systems,so it is of great significance to study the theory and method of fractional differential equations for a deeper understanding of natural and social phenomena.Graph theory has a number of uses in physics,chemistry,network theory,biological molecular structure,and so on.The reason is because graph theory provides a natural structure,and the resulting mathematical model is applicable to almost all natural and social science fields.Second,graph theory can succinctly describe the complex notion of "object-relation" in different fields.As consequently,the ideas and methods of graph theory have become increasingly essential in a wide range of disciplines.As a result,studying fractional differential equations on graphs has applications in both research and business.This thesis primarily investigates the existence and stability of solutions for various types of fractional differential system boundary value problems on metric graphs,and it simulates the examples with Matlab to derive the approximate solution and iterative process of the system.The complete text is divided into five chapters,with the following content arrangement:In Chapter 1,it introduces the study background and current situation of solving fractional differential equations on the graph,in addition to the fundamental concepts and related theories related to this paper.In Chapter 2,it look at a Caputo-Hadamard fractional differential system on a star graph with two sides.The existence of solutions for the system is obtained by constructing iterative sequences and using the technique of upper and lower solutions.It contributes to the research theory of differential equations on graphs,particularly when the nonlinear term includes fractional derivative and the fractional derivative has integer order,allowing it to be used in a wider sense.In Chapter 3,it analyze the fractional differential system with p-Laplacian operator on an edged star graph.Secondly,by conversion,the analogous fractional differential system defined on is produced.The existence and uniqueness of the system’s solution are determined using the fixed point theorem.Moreover,the Ulam-Hyers stability of the system solution is explored.Lastly,simulation is utilized to determine the iterative process and approximate solution for two cases with distinct backdrop maps(star chart and formaldehyde graph).By combining the boundary value issue of the differential system with numerical simulation,the shape of the approximate solution of the system may be viewed more intuitively in this chapter.On each edge of the star graph,a differential equation model is built,which may be used to many fields such as physics,chemical engineering,and so on.In Chapter 4,it study the fractional differential systems of coupled sequences on glucose graphs.The molecular structure of glucose is first modeled,and then the existence and uniqueness of the solution are determined using the Banach shrinkage mapping principle and Krasnosel’skii’s fixed point theorem,and several forms of Ulam-Hyers stability of the system are examined.Ultimately,an example is shown,and simulation is used to acquire the iterative process and an approximate answer.The origin of each edge is not fixed,especially when simulating the molecular structure of glucose,and when the direction of travel along the edge is modified,the origin changes.Therefore,no specific transformation is needed to adjust the length of each edge in the study,and one of the two vertices of the corresponding edge can be selected as the origin to construct the system.The results enrich the related research work of differential equations on the graph.The 5th chapter is the summary of the research content and the prospect of the future work.