Ram K. Saxena

Mittag-Leffler Functions and Their Applications

Motivated essentially by the success of the applications of the Mittag-Leffler functions in many areas nof science and engineering, the authors present, in a unified manner, a detailed account or rather a brief nsurvey of the Mittag-Leffler function, generalized Mittag-Leffler functions, Mittag-Leffler type functions, nand their interesting and useful properties. Applications of G. M. Mittag-Leffler functions in certain areas of nphysical and applied sciences are also demonstrated. During the last two decades this function has come ninto prominence after about nine decades of its discovery by a Swedish Mathematician Mittag-Leffler, ndue to the vast potential of its applications in solving the problems of physical, biological, engineering, and nearth sciences, and so forth. In this survey paper, nearly all types of Mittag-Leffler type functions existing in the nliterature are presented. An attempt is made to present nearly an exhaustive list of references concerning nthe Mittag-Leffler functions to make the reader familiar with the present trend of research in Mittag-Leffler ntype functions and their applications.

On fractional kinetic equations

The subject of this paper is to derive the solution of generalized fractional kinetic equations. The results are obtained in a compact form containing the Mittag-Leffler function, which naturally occurs whenever one is dealing with fractional integral equations. The results derived in this paper provide an extension of a result given by Haubold and Mathai in a recent paper (Haubold and Mathai, 2000).

Further solutions of fractional reaction-diffusion equations in terms of the H-function

This paper deals with the investigation of the solution of an unified fractional reaction-diffusion equation associated with the Caputo derivative as the time-derivative and Riesz-Feller fractional derivative as the space-derivative. The solution is derived by the application of the Laplace and Fourier transforms in closed form in terms of the H-function. The results derived are of general nature and include the results investigated earlier by many authors, notably by Mainardi et al. (2001, 2005) for the fundamental solution of the space-time fractional diffusion equation, and Saxena et al. (2006a, b) for fractional reaction- diffusion equations. The advantage of using Riesz-Feller derivative lies in the fact that the solution of the fractional reaction-diffusion equation containing this derivative includes the fundamental solution for space-time fractional diffusion, which itself is a generalization of neutral fractional diffusion, space-fractional diffusion, and time-fractional diffusion. These specialized types of diffusion can be interpreted as spatial probability density functions evolving in time and are expressible in terms of the H-functions in compact form.

On a generalized hypergeometric distribution

In this article we introduce a general family of statistical probability distributions from which almost all the classical probability distributions are obtained as special cases. The distribution function, the characteristic function, the distribution of the sample mean of a simple random sample from a statistical population designated by this general probability function, distributions of some order statistics and the distribution of the ratio of two independent stochastic variables having the probability functions in this family of probability distributions, are investigated. Some special and interesting properties enjoyed by this general distribution are also pointed out. A number of statistical distributions is studied by many authors from time to time because of the practical applications of these special distributions. Some general classes of statistical distributions, such as the general exponential family, the general exponential type family, the series distribution type, the general gamma distribution, are studied by KHATRI (1959), PATIL (1961), STACY (1962), MATHAI (1966) and others. Consider the probability distribution with the probability density function,

Analysis of Solar Neutrino Data from Super-Kamiokande I and II

Department of Mathematics and Statistics, Jai Narain Vyas University, Jodhpur 342004, India;E-Mail: ram.saxena@yahoo.com* Author to whom correspondence should be addressed; E-Mail: hans.haubold@unvienna.org;Tel.: +43-676-4252050; Fax: +43-1-26060-5830.Received: 5 December 2013; in revised form: 16 December 2013 / Accepted: 27 February 2014 /Published: 10 March 2014Abstract: We are going back to the roots of the original solar neutrino problem: theanalysis of data from solar neutrino experiments. The application of standard deviationanalysis (SDA) and diffusion entropy analysis (DEA) to the Super-Kamiokande I and IIdata reveals that they represent a non-Gaussian signal. The Hurst exponent is different fromthe scaling exponent of the probability density function, and both the Hurst exponent andscaling exponent of the probability density function of the Super-Kamiokande data deviateconsiderably from the value of 0.5, which indicates that the statistics of the underlyingphenomenon is anomalous. To develop a road to the possible interpretation of this finding,we utilize Mathai’s pathway model and consider fractional reaction and fractional diffusionas possible explanations of the non-Gaussian content of the Super-Kamiokande data.Keywords: solar neutrinos; Super-Kamiokande; data analysis; standard deviation;diffusion entropy; entropic pathway model, fractional reaction; fractional diffusion;thermonuclear functionsWe are going back to the roots of the original solar neutrino problem: analysis of data from solar neutrino experiments. The application of standard deviation analysis (SDA) and diffusion entropy analysis (DEA) to the SuperKamiokande I and II data reveals that they represent a non-Gaussian signal. The Hurst exponent is different from the scaling exponent of the probability density function and both Hurst exponent and scaling exponent of the probability density function of the SuperKamiokande data deviate considerably from the value of 0.5 which indicates that the statistics of the underlying phenomenon is anomalous. To develop a road to the possible interpretation of this finding we utilize Mathais pathway model and consider fractional reaction and fractional diffusion as possible explanations of the non-Gaussian content of the SuperKamiokande data.

An extended general Hurwitz-Lerch zeta function as a Mathieu (a, λ)-series

Abstract It is shown that an integral representation for the extension of a general Hurwitz–Lerch zeta function recently obtained by Garg etxa0al. (2008)xa0 [5] is a special case of the closed form integral expression for the Mathieu ( a , λ ) -series given by Pogany (2005)xa0 [1] . As an immediate consequence of the derived results, new integral expressions and related bilateral bounding inequalities are investigated.

Computational Solutions of Distributed Order Reaction-Diffusion Systems Associated with Riemann-Liouville Derivatives

This article is in continuation of the authors research attempts to derive computational solutions of an unified reaction-diffusion equation of distributed order associated with Caputo derivatives as the time-derivative and Riesz-Feller derivative as space derivative. This article presents computational solutions of distributed order fractional reaction-diffusion equations associated with Riemann-Liouville derivatives of fractional orders as the time-derivatives and Riesz-Feller fractional derivatives as the space derivatives. The method followed in deriving the solution is that of joint Laplace and Fourier transforms. The solution is derived in a closed and computational form in terms of the familiar Mittag-Leffler function. It provides an elegant extension of results available in the literature. The results obtained are presented in the form of two theorems. Some results associated specifically with fractional Riesz derivatives are also derived as special cases of the most general result. It will be seen that in case of distributed order fractional reaction-diffusion, the solution comes in a compact and closed form in terms of a generalization of the Kampe de Feriet hypergeometric series in two variables. The convergence of the double series occurring in the solution is also given.

Applications in Statistics

Special functions are used in almost all areas of statistics. Statistical densities are basically elementary special functions or product of such functions. Hence, the theory of special functions is directly applicable to statistical distribution theory. While studying generalized densities, structural properties of densities, Bayesian inference, distributions of test statistics, characterization of densities and related studies of probability theory, stochastic processes and time series problems, and special functions and generalized special functions in the categories of Meijer’s G-functions and H-functions come in naturally.

Some mathieu-type series for the I-function occuring in the fokker-planck equation

Closed form expressions are obtained for a family of convergent Mathieu type a{series and its alternating variants, whose terms contain an I{function which is a generalization of the Fox’s H{function. The results derived are of general character and provide an elegant generalization for the closed form expressions of these series associated with the H{function by Pog any [9], for Fox{Wright functions by Pog any and Srivastava [10] and for pFq and Meijer’s G{function by Pog any and Tomovski [13], and others.

Analytical solution of space-time fractional telegraph-type equations involving Hilfer and Hadamard derivatives

ABSTRACT In this paper we consider space-time fractional telegraph equations, where the time derivatives are intended in the sense of Hilfer and Hadamard while the space-fractional derivatives are meant in the sense of Riesz-Feller. We provide the Fourier transforms of the solutions of some Cauchy problems for these fractional equations. Probabilistic interpretations of some specific cases are also provided.