Dynamic Behaviors Of Solutions For Several Classes Of Fractional Differential Equations

Fractional order differential equations are a class of differential equations that contain fractional derivatives.Compared to integer order differential equations,fractional order differential equations can more accurately simulate physical processes and dynamic system processes in nature.Due to its widespread application in natural sciences and social engineering,it has received increasing attention and attention in recent decades.It is well known that differential equations with the p-Laplacian operator can be used to describe some nonlinear phenomena,such as nonlinear heat transfer and nonlinear diffusion,etc.Therefore,studying differential equations with the p-Laplacian operator has certain research significance.When studying the properties of solutions to fractional differential equations,most scholars mainly use fixed point theorems,variational methods,and spectral theory to study the existence of solutions,but there is relatively little research on simulation and analysis of existence and stability.We know that drawing the approximate solution’s image through numerical simulation can intuitively understand the dynamic behavior of the solution.Therefore,this article not only studies the existence and stability of solutions but also draws the approximate solution’s image through numerical simulation.The specific research content is as follows:Firstly,a class of Caputo type fractional coupled equations with integral boundary conditions were studied.By using Leray-Schauder degree theory,the existence of solutions to these equations was obtained,and further discussion on the Hyers-Ulam stability of the solutions was presented along with sufficient conditions for stability.As validation,examples and numerical simulations were used to illustrate the main results.Notably,this chapter not only investigated Hyers-Ulam stability but also combined numerical solutions with boundary value problems to provide images of approximate solutions.Secondly,a class of parameterized Caputo type fractional coupled equations were studied.The existence of at least one solution and three solutions to the fractional differential equation were investigated using monotone iteration and upper-lower solution methods.The convergence of the iterative sequence was proved by constructing a compact map sequence,and the error of the iterative method was obtained.Finally,the Hyers-Ulam stability of the equation was discussed.This chapter’s interesting feature is that approximate solutions are drawn through iterative methods and validated by numerical simulations for the Hyers-Ulam stability of the coupled system.Finally,the existence and uniqueness of solutions to a class of fractional differential equations with the p-Laplacian operator were studied.The Green function and its properties of the boundary value problem were obtained,and the existence and uniqueness of solutions to the boundary value problem were proved using fixed point theorem.Finally,the conclusions’ rationality was validated using iterative method numerical approximations.This chapter’s distinctive feature is the study of fractional differential equations with the p-Laplacian operator.