Finite Series Representation of the Inverse Mittag-Leffler Function

The inverse Mittag-Leffler function Eα,β-1z is valuable in determining the value of the argument of a Mittag-Leffler function given the value of the function and it is not an easy problem. A finite series representation of the inverse Mittag-Leffler function has been found for a range of the parameters α and β; specifically, 0<α<1/2 for β=1 and for β=2. This finite series representation of the inverse Mittag-Leffler function greatly expedites its evaluation and has been illustrated with a number of examples. This represents a significant advancement in the understanding of Mittag-Leffler functions

Time Fractional Schrodinger Equation Revisited

The time fractional Schrodinger equation (TFSE) for a nonrelativistic particle is derived on the basis of the Feynman path integral method by extending it initially to the case of a “free particle” obeying fractional dynamics, obtained by replacing the integer order derivatives with respect to time by those of fractional order. The equations of motion contain quantities which have “fractional” dimensions, chosen such that the “energy” has the correct dimension . The action is defined as a fractional time integral of the Lagrangian, and a “fractional Planck constant” is introduced. The TFSE corresponds to a “subdiffusion” equation with an imaginary fractional diffusion constant and reproduces the regular Schrodinger equation in the limit of integer order. The present work corrects a number of errors in Naber’s work. The correct continuity equation for the probability density is derived and a Green function solution for the case of a “free particle” is obtained. The total probability for a “free” particle is shown to go to zero in the limit of infinite time, in contrast with Naber’s result of a total probability greater than unity. A generalization to the case of a particle moving in a potential is also given

Approximate Solutions to Fractional Subdiffusion Equations: The heat-balance integral method

The work presents integral solutions of the fractional subdiffusion equation by an integral method, as an alternative approach to the solutions employing hypergeometric functions. The integral solution suggests a preliminary defined profile with unknown coefficients and the concept of penetration (boundary layer). The prescribed profile satisfies the boundary conditions imposed by the boundary layer that allows its coefficients to be expressed through its depth as unique parameter. The integral approach to the fractional subdiffusion equation suggests a replacement of the real distribution function by the approximate profile. The solution was performed with Riemann -Liouville time-fractional derivative since the integral approach avoids the definition of the initial value of the time-derivative required by the Laplace transformed equations and leading to a transition to Caputo derivatives. The method is demonstrated by solutions to two simple fractional subdiffusion equations (Dirichlet problems): 1) Time-Fractional Diffusion Equation, and 2) Time-Fractional Drift Equation, both of them having fundamental solutions expressed through the M-Write function. The solutions demonstrate some basic issues of the suggested integral approach, among them: a) Choice of the profile, b) Integration problem emerging when the distribution (profile) is replaced by a prescribed one with unknown coefficients; c) Optimization of the profile in view to minimize the average error of approximations; d) Numerical results allowing comparisons to the known solutions expressed to the M-Write function and error estimations.Comment: 15 pages, 7 figures, 3 table

Revisiting the ‘Evidence of Meters’

A ṛk mantra, the basic unit of ṛgveda, is characterized by a chandas, the poetic meter. It is not at all like the poetry of ancient folk tales of Europe. The viniyoga of the mantra in yajña plays a crucial role in the structural organization and chronology of the ṛgvedasaṃhitā. It is shown that the view of some scholars discussing the chronology of meters (chandas) using Western terminology in terms of dimeters and trimeters is not fruitful in Vedic studies. The various meters used in ṛgveda play important roles in the performance of somayajña, construction of altar in agnicayana and of the symbolic altar chandaściti. A much deeper significance of cosmic principle may be appreciated

Quantum and correlation corrections to the Thomas-Fermi potential

The paper studies the correction to the Thomas-Fermi potential arising from the exchange, inhomogeneity, and correlation effects. An alternative derivation of the Baraff equation is given, based on simple physical considerations. Starting from the Baraff equation and making use of the simple analytical formula of Tietz for the Thomas-Fermi potential, analytical expressions for the correction to the Thomas-Fermi potential are given (a) within the radius of the atom applying the Gellmann-Brueckner high-density formula for the correlation energy of an electron gas and (b) outside this range wherein the correlation energy can be approximated by the Wigner low-density formula. It is found that the main contribution to the solution of the Baraff equation arises from the inhomogeneity, screening, and the exchange terms