Bertram Ross

Bio: Bertram Ross is an academic researcher. The author has contributed to research in topics: Fractional calculus & Legendre polynomials. The author has an hindex of 1, co-authored 1 publications receiving 7229 citations.

An Introduction to the Fractional Calculus and Fractional Differential Equations

Abstract: Historical Survey The Modern Approach The Riemann-Liouville Fractional Integral The Riemann-Liouville Fractional Calculus Fractional Differential Equations Further Results Associated with Fractional Differential Equations The Weyl Fractional Calculus Some Historical Arguments.

Fractional-order systems and PI/sup /spl lambda//D/sup /spl mu//-controllers

Abstract: Dynamic systems of an arbitrary real order (fractional-order systems) are considered. The concept of a fractional-order PI/sup /spl lambda//D/sup /spl mu//-controller, involving fractional-order integrator and fractional-order differentiator, is proposed. The Laplace transform formula for a new function of the Mittag-Leffler-type made it possible to obtain explicit analytical expressions for the unit-step and unit-impulse response of a linear fractional-order system with fractional-order controller for both open- and closed-loops. An example demonstrating the use of the obtained formulas and the advantages of the proposed PI/sup /spl lambda//D/sup /spl mu//-controllers is given.

The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics

Abstract: Fractional dynamics has experienced a firm upswing during the past few years, having been forged into a mature framework in the theory of stochastic processes. A large number of research papers developing fractional dynamics further, or applying it to various systems have appeared since our first review article on the fractional Fokker–Planck equation (Metzler R and Klafter J 2000a, Phys. Rep. 339 1–77). It therefore appears timely to put these new works in a cohesive perspective. In this review we cover both the theoretical modelling of sub- and superdiffusive processes, placing emphasis on superdiffusion, and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function. We also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems.

A new Definition of Fractional Derivative without Singular Kernel

Abstract: In the paper, we present a new definition of fractional deriva tive with a smooth kernel which takes on two different representations for the temporal and spatial variable. The first works on the time variables; thus it is suitable to use th e Laplace transform. The second definition is related to the spatial va riables, by a non-local fractional derivative, for which it is more convenient to work with the Fourier transform. The interest for this new approach with a regular kernel was born from the prospect that there is a class of non-local systems, which have the ability to descri be the material heterogeneities and the fluctuations of diff erent scales, which cannot be well described by classical local theories or by fractional models with singular kernel.