Some Applications Of Fractional Calculus To The Researches On Quantum Mechanics And Non-Newtonian Fluid Mechanics

This paper is composed of five chapters, which are independent and correlative to one another. In chapterl, i.e. prologue, the fractional calculus and some special functions are introduced. They are the basic math tools needed in this paper. In section§1.1, the fractional calculus and its development history, current status are introduced concisely, the definitions and the main properties of the Riemann-Liouville fractional integral operator _oD_t^-β, differential operator_oD_t^-β(0?^αand Riesz/Weyl fractional derivative _∞D_x^μ(μ>0) are given. Andthe Laplace transforms of fractional integral and derivative operators are discussed. In section§1.2, the definitions and some important formulae of the generalizedMittag-Leffler function E_α,β(z) are given. In section§1.3, the definition of seriesexpression and some basic properties of H-Fox function are outlined. The special cases of Fox function and its Fourier sine and cosine transforms are discussed. H-Fox function is a powerful tool for the solving of the fractional differential equations. This chapter is the basis for the remainder.In chapter 2, we investigate some applications of fractional calculus in quantum mechanics. Some physical applications of the space fractional Schrodinger equation are considered. In§2.2, using the properties of H-Fox function, we solve the time-dependent fractional Schrodinger equation for a free particle and give the wave function of itIn §2.3 the energy levels and the normalized wave functions of a particle in an infinite square potential wellare given as followsrespectively. In §2.4 according to the time-independent fractional Schrodinger equation we consider the motion of a particle from a rectangular potential walland give the reflection coefficientand the transmission coefficientWhen the particle is passing over a rectangular potential well instead of a potentialbarrier, the above theory still holds, in which . In the last section, the fractional time-independent Schrodinger equationand the equivalent Lippmann-Scheinger functionare considered, and the Green's function of it is givenIn chapter 3, by analogy with generalized Oldroyd-B constitutive relationshipwe introduce a modified phenomenological Darcy's law with fractional derivative as followsBased on a modified Darcy's law for a viscoelastic fluid, the circular motion of a generalized Oldroyd-B fluid with fractional derivative model through a porous mediumis investigated. Using the discrete Laplace transform of the sequential fractional derivatives and Hankel transform, exact solutions of velocity and temperature fields are obtained in terms of generalized Mittag-Leffler function In special, whenη= 0, the above result can be simplified to the solution of vortex velocity field for a clear generalized Oldroyd-B fluid (nonporous case), and to be the solution for ordinary Oldroyd-B fluid through porous medium, whenα=1,β=1.Whenα=0,λ→0,η= 0, the solution of vortex velocity field for generalizedsecond grade fluid can be obtained, which is the same as the result obtaied by Shen and includes some past classical results as special.In chapter 4, the fractional calculus approach has also been taken into account in the Darcy's law and the constitutive relationship of fluid model. Based on a modified Darcy's law for a viscoelastic fluid, Stokes' first problem is used to solve a generalized Oldroyd-B fluid problem in a porous half spaceBy using the Laplace transform method, an exact solution of Stokes' first problem in the porous half space is obtained as follows As special cases, we can obtain the Stokes' first problems for the generalized second grade fluid, for the fractional Maxwell fluid and for the ordinary Oldroyd-B fluid in porous medium, respectively.In chapter 5, the unsteady Couette flow of generalized Oldroyd-B fluid with fractional derivative is studied. The dimensionless equations are:By using the Laplace transform, Weber transform and the generalized Mittag-Leffler function, we obtain the exact solutionSome previous results, such as the solution of unsteady Couette flow of generalized Maxwell fluid and of generalized second grade fluid, and the classical result of Newtonian viscous fluid all can be regarded as special cases of this solution.