Research On The Properties Of Solutions On The Weighted Fractional Differential Equations

Caputo-Fabrizio fractional derivative is a new fractional derivative without singular kernel,which has better properties than the Caputo derivative.The fractional calculus theory is the focus of applied mathematics’s research,and has been widely used in many fields such as physics,chemistry,economics,engineering,control theory,etc.The study of the properties of the solutions of differential equations has important theoretical value and practical significance.In this paper,we will study the weighted Caputo-Fabrizio fractional derivative.On the basis of weighted Caputo-Fabrizio fractional derivative of 0 to 1,we give the definition of its high-order derivative,and further define the low order and high order weighted kCaputo-Fabrizio fractional derivative.Through the study of its differential and integral properties and the construction of operators,the differential equation is transformed into an equivalent integral equation,and the existence and stability of the solutions of the relevant fractional differential equations are explored by using the fixed point theorems.The main contents are as follows:First,we define the high-order weighted Caputo-Fabrizio fractional derivative,and explore the properties of its differential and integral,then using the Schaefer’s fixed point theorem,α-ψ-contraction theorem to study the existence of solutions to boundary value problems of fractional differential equations.Moreover,the conclusions are verified in relevant applications.Secondly,we define the weighted k-Caputo-Fabrizio fractional derivative.By dint of the Banach contractive mapping principle,Krasnoselskii’s fixed point theorem and Leray-Schauder’s nonlinear substitution theorem,we study the existence of solutions of pantograph fractional differential equations.In addition,we also study the Hlam-UyersRasias stability and Hlam-Uyers stability of nonlinear boundary value problems.Finally,on the basis of the previous research,a more general definition of weighted fractional derivatives of higher order k-Caputo-Fabrizio is given,and we also illustrate a series of images of weighted Caputo-Fabrizio fractional derivatives.By transforming the differential equation into an equivalent operator equation,using D.O’ Regan fixed point theorem,we obtain the existence of solutions for nonlinear integro-differential equations.At the same time,the Hyers-Ulam-Rassias stability and Hyers-Ulam stability are also derived.