There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

ductor, where the conductivity is sufficiently high, the displacement current density can be neglected. In this case, the conduction current density is given by the product of the electric field and the conductance. One of the aspects of with SE is it attenuates the higher frequency components of a signal. The SE was first verified by Kelvin in 1887. Since then many researchers developed work on the subject and presently a comprehensive physical model, based on the Maxwell equations, is well established. The Maxwell formalism plays a fundamental role in the electromagnetic theory. These equations lead to the derivation of mathematical descriptions useful in many applications in physics and engineering. Maxwell is generally The Maxwell equations involve only the integer-order calculus and, therefore, it is natural that the resulting classical models adopted in electrical engineering reflect this perspective. Recently, a closer look of some phenomof precise models, seem to point out the requirement for a fractional calculus approach. Bearing these ideas in mind, in this study we address the SE and we re-evaluate the results demonstrating its fractional-order nature. Department of Electrical Engineering, Institute of Engineering of Porto Rua Dr. Antonio Bernardino de Almeida, 4200-072 Porto, Portugal; E-mail: jtm,isj,amf@isep.ipp.pt Engineering Systems, Vila-Real, Portugal; E-mail: jboavent@utad.pt Institute of Intelligent Engineering Systems, Budapest Tech, John von Neumann Faculty of Informatics, Budapest, Hungary; E-mail: tar@nik.bmf.hu the high-frequency effects is the skin effect (SE ). The fundamental problem regarded as the 19th century scientist who had the greatest influence on 20th century physics, making contributions to the fundamental models of nature. enas present in electrical systems and the motivation towards the development

Advances in Fractional Calculus

A review on electrochemical double-layer capacitors

This brief studies the effects of fractional dynamics in chaotic systems. In particular, Chua's system is modified to include fractional order elements. By varying the total system order incrementally from 3.6 to 3.7, it is demonstrated that systems of "order" less than three can exhibit chaos as well as other nonlinear behavior. This effectively forces a clarification of the definition of order which can no longer be considered only by the total number of differentiations or by the highest power of the Laplace variable. >

/pdf/chaos-in-a-fractional-order-chua-s-system-3mh2xv70qf.pdf

Chaos in a fractional order Chua's system

Many physical processes appear to exhibit fractional order behavior that may vary with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. This paper develops the concept of variable and distributed order fractional operators. Definitions based on the Riemann-Liouville definitions are introduced and behavior of the operators is studied. Several time domain definitions that assign different arguments to the order q in the Riemann-Liouville definition are introduced. For each of these definitions various characteristics are determined. These include: time invariance of the operator, operator initialization, physical realization, linearity, operational transforms. and memory characteristics of the defining kernels. A measure (m2) for memory retentiveness of the order history is introduced. A generalized linear argument for the order q allows the concept of "tailored" variable order fractional operators whose a, memory may be chosen for a particular application. Memory retentiveness (m2) and order dynamic behavior are investigated and applications are shown. The concept of distributed order operators where the order of the time based operator depends on an additional independent (spatial) variable is also forwarded. Several definitions and their Laplace transforms are developed, analysis methods with these operators are demonstrated, and examples shown. Finally operators of multivariable and distributed order are defined in their various applications are outlined.

/pdf/variable-order-and-distributed-order-fractional-operators-1d2ze0j2xm.pdf

Variable Order and Distributed Order Fractional Operators

Due to the importance of historical effects in fractional-order systems,this paper presents a general fractional-order system and control theorythat includes the time-varying initialization response. Previous studieshave not properly accounted for these historical effects. Theinitialization response, along with the forced response, forfractional-order systems is determined. The scalar fractional-orderimpulse response is determined, and is a generalization of theexponential function. Stability properties of fractional-order systemsare presented in the complex w-plane, which is a transformation of thes-plane. Time responses are discussed with respect to pole positions inthe complex w-plane and frequency response behavior is included. Afractional-order vector space representation, which is a generalizationof the state space concept, is presented including the initializationresponse. Control methods for vector representations of initializedfractional-order systems are shown. Finally, the fractional-orderdifferintegral is generalized to continuous order-distributions whichhave the possibility of including all fractional orders in a transferfunction.

Dynamics and Control of Initialized Fractional-Order Systems

Fractional-order system identification based on continuous order-distributions

This paper provides a formalized basis for initialization in the fractional calculus The intent is to make the fractional calculus readily accessible to engineering and the sciences A modified set of definitions for the fractional calculus is provided which formally include the effects of initialization Conceptualizations of fractional derivatives and integrals are shown Physical examples of the basic elements from electronics are presented along with examples from dynamics, material science, viscoelasticity, filtering, instrumentation, and electrochemistry to indicate the broad application of the theory and to demonstrate the use of the mathematics The fundamental criteria for a generalized calculus established by Ross (1974) are shown to hold for the generalized fractional calculus under appropriate conditions A new generalized form for the Laplace transform of the generalized differintegral is derived The concept of a variable structure (order) differintegral is presented along with initial efforts toward meaningful definitions

/pdf/initialization-conceptualization-and-application-in-the-196aqb6vwm.pdf

Initialization, conceptualization, and application in the generalized (fractional) calculus.

Previous papers have used two important functions for the solution of fractional order differential equations, the Mittag-Leffler functionE(sub q)[at(exp q)](1903a, 1903b, 1905), and the F-function F(sub q)[a,t] of Hartley & Lorenzo (1998). These functions provided direct solution and important understanding for the fundamental linear fractional order differential equation and for the related initial value problem (Hartley and Lorenzo, 1999). This paper examines related functions and their Laplace transforms. Presented for consideration are two generalized functions, the R-function and the G-function, useful in analysis and as a basis for computation in the fractional calculus. The R-function is unique in that it contains all of the derivatives and integrals of the F-function. The R-function also returns itself on qth order differ-integration. An example application of the R-function is provided. A further generalization of the R-function, called the G-function brings in the effects of repeated and partially repeated fractional poles.

/pdf/generalized-functions-for-the-fractional-calculus-51mu4dsfuk.pdf

Tom T. Hartley

Papers

Variable Order and Distributed Order Fractional Operators

Dynamics and Control of Initialized Fractional-Order Systems

Fractional-order system identification based on continuous order-distributions

Initialization, conceptualization, and application in the generalized (fractional) calculus.

Generalized functions for the fractional calculus.