Study On Fractional Calculus Operators With Mittag-Leffler Type Functions

Fractional order calculus is one of the most important branches of modern mathematical analysis,and it is also widely used in the field of engineering.In describing complex phenomena in the real world,fractional order differential equations play a crucial role,and fractional order models can better model the real world problems,and the Mittag-Leffler function,as a generalization of exponential functions,plays an important role in the application of fractional order differential equations.In this paper,we study the generalized fractional order operator,the Mikusiński operator calculus method for solving fractional order differential equations,the properties of the sevenparameter Mittag-Leffler function,and the anomalous diffusion model for monotone increasing functions.The specific research is divided into the following areas:In Chapter 1,the background and status of the development of fractional order calculus and Mittag-Leffler type functions are introduced,and the basic knowledge to be used in later chapters is given.In Chapter 2,we devote to extending this useful but unpopular method of solving differential equations by algebraic forms of the Mikusinski operational calculus and applying it to solving differential equations containing the(k,ψ)-Riemann-Liouville operator.The form of the solution of this equation is extended and an image of the solution is given.The solution of this equation can be expressed in terms of the MittagLeffler function.The work in this chapter generalizes to some extent the corresponding results in the Fahad-Fernandez[2021]literature.In Chapter 3,we prove properties about the seven-parameter Mittag-Leffler function proposed by Andric et al.in the literature Andric-Farid[2018],such as its level expressions,and expressions for relations with several special functions,and applies them to generalize Opial-type inequalities.In Chapter 4,we base on the Marchaud-type fractional order derivative operator,give a formula for the Riesz-type fractional derivative,and an anomalous diffusion model for monotone increasing functions.In Chapter 5,we summarize and outlook the results obtained in this thesis.There are 4 figures and 86 references in this thesis.