REGULAR FRACTIONAL DIRAC TYPE SYSTEMS

In this paper, we study one dimensional fractional Dirac type systems which includes the right-sided Caputo and the left-sided Riemann-Liouvile fractional derivatives of same order α,α∈(0,1). We investigate the properties of the eigenvalues and the eigenfunctions of this syste

Spectral expansion for singular conformable fractional Sturm-Liouville problem

With this study, the spectral function for singular conformable fractional Sturm-Lioville problem is demonstrated. Further, we establish a Parseval equality and spectral expansion formula by terms of the spectral function

Spectral expansion for singular conformable fractional Sturm-Liouville problem

With this study, the spectral function for singular conformable fractional Sturm-Lioville problem is demonstrated. Further, we establish a Parseval equality and spectral expansion formula by terms of the spectral function

Fractional Calculus - Theory and Applications

In recent years, fractional calculus has led to tremendous progress in various areas of science and mathematics. New definitions of fractional derivatives and integrals have been uncovered, extending their classical definitions in various ways. Moreover, rigorous analysis of the functional properties of these new definitions has been an active area of research in mathematical analysis. Systems considering differential equations with fractional-order operators have been investigated thoroughly from analytical and numerical points of view, and potential applications have been proposed for use in sciences and in technology. The purpose of this Special Issue is to serve as a specialized forum for the dissemination of recent progress in the theory of fractional calculus and its potential applications

I. Complete and orthonormal sets of exponential-type orbitals with noninteger principal quantum numbers

The definition for the Slater-type orbitals is generalized. Transformation between an orthonormal basis function and the Slater-type orbital with non-integer principal quantum numbers is investigated. Analytical expressions for the linear combination coefficients are derived. In order to test the accuracy of the formulas, the numerical Gram-Schmidt procedure is performed for the non-integer Slater-type orbitals. A closed form expression for the orthogonalized Slater-type orbitals is achieved. It is used to generalize complete orthonormal sets of exponential-type orbitals obtained by Guseinov in [Int. J. Quant. Chem. 90, 114 (2002)] to non-integer values of principal quantum numbers. Riemann-Liouville type fractional calculus operators are considered to be use in atomic and molecular physics. It is shown that the relativistic molecular auxiliary functions and their analytical solutions for positive real values of parameters on arbitrary range are the natural Riemann-Liouville type fractional operators

Deep and shallow overturning circulations in the tropical atmosphere

2013 Fall.Includes bibliographical references.This dissertation examines the dynamics of zonally symmetric, deep and shallow overturning circulations in the tropical atmosphere. The dynamics are discussed in the context of idealized analytical solutions of the equatorial β-plane version of the Eliassen meridional circulation equation that arises in balanced models of the Hadley circulation. This elliptic equation for the meridional circulation has been solved analytically by first performing a vertical normal mode transform that converts the partial differential equation into a system of ordinary differential equations for the meridional structures of all the vertical modes. These meridional structure equations can be solved via the Green's function, which can be expressed in terms of parabolic cylinder functions of half-integer order. The analytical solutions take simple forms in two special cases: (1) Forcing by deep diabatic heating that projects only onto the first internal mode in the absence of Ekman pumping; (2) Forcing by Ekman pumping in the absence of any diabatic heating. Case (1) leads to deep overturning circulations, while case (2) leads to shallow overturning circulations. Both circulations show a marked asymmetry between the winter hemisphere and summer hemisphere overturning cells. This asymmetry is due to the basic anisotropy introduced by the spatially varying inertial stability coefficient in the Eliassen meridional circulation equation. A simple physical interpretation is that fluid parcels forced near the equator to overturn by diabatic and frictional processes tend to move much more easily in the horizontal direction because the resistance to horizontal motion (i.e. inertial stability) is so much less than the resistance to vertical motion (i.e., static stability)

A comparison of numerical methods to solve fractional partial differential equations

A comparison of two numerical methods - ¯nite di®erence and Adomian decomposition method (ADM) - to solve a variety of fractional partial dif- ferential equations that occur in ¯nance are investigated. These fractional partial di®erential equations fall into the class of L¶evy models. They are known as the Finite Moment Log Stable (FMLS), CGMY and the extended Koponen (KoBol) models. Convergence criteria for these models under the numerical methods are studied. ADM fails to accurately price a claim writ- ten on these models. However, the ¯nite di®erence scheme works well for the FMLS and KoBol models