E. M. Wright

Bio: E. M. Wright is an academic researcher. The author has contributed to research in topics: Bessel function & Cylindrical harmonics. The author has an hindex of 1, co-authored 1 publications receiving 175 citations.

Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics

Abstract: We review some applications of fractional calculus developed by the author (partly in collaboration with others) to treat some basic problems in continuum and statistical mechanics. The problems in continuum mechanics concern mathematical modelling of viscoelastic bodies (Sect. 1), and unsteady motion of a particle in a viscous fluid, i.e. the Basset problem (Sect. 2). In the former analysis fractional calculus leads us to introduce intermediate models of viscoelasticity which generalize the classical spring-dashpot models. The latter analysis induces us to introduce a hydrodynamic model suitable to revisit in Sect. 3 the classical theory of the Brownian motion, which is a relevant topic in statistical mechanics. By the tools of fractional calculus we explain the long tails in the velocity correlation and in the displacement variance. In Sect. 4 we consider the fractional diffusion-wave equation, which is obtained from the classical diffusion equation by replacing the first-order time derivative by a fractional derivative of order $0< \beta <2$. Led by our analysis we express the fundamental solutions (the Green functions) in terms of two interrelated auxiliary functions in the similarity variable, which turn out to be of Wright type (see Appendix), and to distinguish slow-diffusion processes ($0 < \beta < 1$) from intermediate processes ($1 < \beta < 2$).

Multi-index Mittag-Leffler Functions

Abstract: Consider the function defined for \(\alpha _{1},\ \alpha _{2} \in \mathbb{R}\) (α 1 2 +α 2 2 ≠ 0) and \(\beta _{1},\beta _{2} \in \mathbb{C}\) by the series $$\displaystyle{ E_{\alpha _{1},\beta _{1};\alpha _{2},\beta _{2}}(z) \equiv \sum _{k=0}^{\infty } \frac{z^{k}} {\varGamma (\alpha _{1}k +\beta _{1})\varGamma (\alpha _{2}k +\beta _{2})}\ \ (z \in \mathbb{C}). }$$ (6.1.1) Such a function with positive α 1 > 0, α 2 > 0 and real \(\beta _{1},\beta _{2} \in \mathbb{R}\) was introduced by Dzherbashian [Dzh60].

Mittag-Leffler Functions and Their Applications

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

Fractional evolution equations in Banach spaces

Abstract: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers.

Mittag-Leffler Functions and Their Applications

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