Some Applications Of Fractional Calculus To Viscoelastic Fluid Mechanics And Quantum Mechanics

This paper is composed of five chapters, which are independent and correlative to one another. In chapter 1, i.e. introduction, the history and applications of fractional calculus are introduced. In section§1.1 and§1.2, the development history and recent applications of the fractional calculus are introduced concisely, the definitions and the main properties of the Riemann-Liouville fractional integral operator ₀D_t^-α(O 0D_t^β(0 < R(β) < 1) and local fractional derivative D^qf(y) are given, and the Laplace transforms of fractional integral and derivative operators are discussed. In section§1.3, the definitions and some important formulae of the generalized Mittag-Leffler function E_α,β(z) are given. In section§1.4, the definition, series expression, asymptotic behavior and some basic properties of H-Fox function H_p,q^m,n are given. The special cases of the Fox function are discussed, such as the generalized Mittag-Leffler function E_α,β{z) and H_1,2^1,1{z). H-Fox function is a powerful tool for the solving of the fractional differential equations. In section§1.5, the developments and applications of fractional calculus in various complex system are discussed, respectively. This chapter is the basis for the following chapters of this thesis.In chapter 2, the unsteady Couette flow of a generalized Maxwell fluid(GMF) with fractional derivatives, which can be expressed in terms of scalar form,(?) + K^α₀D_tα(?) =μβγ|., (18)is studied. With the help of integral transforms, i.e., Laplace transform and Weber transform, and generalized Mittag-Leffler function, the exact solution is obtained,where A(λ_i,t) is expressed as,Then we discussed the special case, a—1, the model represents the standard (integer order) Maxwell fluid, and the solution should be expressed aswhereIn chapter 3, we deal with with the unsteady axial Couette flow of fractional second grade fluid (FSGF) and fractional Maxwell fluid (FMF) between two infinitely long concentric circular cylinders. In section§3.3 we get the solutions of the model for FSGF and FMF, which can be represented asandrespectively, where In section§3.4 we discuss the case when the motion velocity of the inner cylinder is not constant 1 but equal to f(t) = kt, and we get the following results:In section§3.5, we analyze the effects of the fractional derivative on the models by use of the numerical results and find that the oscillation exists in the velocity field of FMF. In chapter 4, the generalized fractional Schrodinger equation with space and time fractional derivatives is constructed asIn§4.3, the probability current density is obtained,In section§4.4 and section§4.5, the equations for free particle and square potential well are solved, and the solutions can be expressed in terms of Mittag-Leffler function, respectively. At the end of§4.5, we prove the energy levels for potential well to be time dependent,With the help of Mellin transform and other integral transform, we get the Green's function of generalized fractional Schrodinger equation for free-particle in section§4.6,In section§4.7, we discuss the relationship between the cases of the generalized fractional Schrodinger equation and the ones of the standard quantum.In chapter 5, with the help of Dirac's bra and ket notations, we introduce a new method to solve the fractional diffusion euqation,and get the Green's function of above equation,Then we use this method to solve the equation of time fractional diffusion,and get its solution which is expressed asIn section§5.4, we discuss the applications of this method in classical diffusion equations.