Heaviside's Operator Calculus
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Mathematical and Physical Interpretations of Fractional Derivatives and Integrals
R. Hilfer Mathematical and Physical Interpretations of Fractional Derivatives and Integrals Abstract: Brief descriptions of various mathematical and physical interpretations of fractional derivatives and integrals have been collected into this chapter as points of reference and departure for deeper studies. “Mathematical interpre- tation” in the title means a brief description of the basic mathematical idea underlying a precise definition. “Physical interpretation” means a brief descrip- tion of the physical theory underlying an identification of the fractional order with a known physical quantity. Numerous interpretations had to be left out due to page limitations. Only a crude, rough and ready description is given for each interpretation. For precise theorems and proofs an extensive list of references can serve as a starting point. Keywords: fractional derivatives and integrals, Riemann-Liouville integrals, Weyl integrals, Riesz potentials, operational calculus, functional calculus, Mikusinski calculus, Hille-Phillips calculus, Riesz-Dunford calculus, classification, phase transitions, time evolution, anomalous diffusion, continuous time random walks Mathematics Subject Classification 2010: 26A33, 34A08, 35R11 published in: Handbook of Fractional Calculus: Basic Theory, Vol. 1, Ch. 3, pp. 47-86, de Gruyter, Berlin (2019) ISBN 978-3-11-057162-2 R. Hilfer, ICP, Fakultät für Mathematik und Physik, Universität Stuttgart, Allmandring 3, 70569 Stuttgart, Deutschland DOI https://doi.org/10.1515/9783110571622 Interpretations 1 1 Prolegomena -
Determination of the Magnetic Permeability, Electrical Conductivity
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TII.2018.2885406, IEEE Transactions on Industrial Informatics TII-18-2870 1 Determination of the magnetic permeability, electrical conductivity, and thickness of ferrite metallic plates using a multi-frequency electromagnetic sensing system Mingyang Lu, Yuedong Xie, Wenqian Zhu, Anthony Peyton, and Wuliang Yin, Senior Member, IEEE Abstract—In this paper, an inverse method was developed by the sensor are not only dependent on the magnetic which can, in principle, reconstruct arbitrary permeability, permeability of the strip but is also an unwanted function of the conductivity, thickness, and lift-off with a multi-frequency electrical conductivity and thickness of the strip and the electromagnetic sensor from inductance spectroscopic distance between the strip steel and the sensor (lift-off). The measurements. confounding cross-sensitivities to these parameters need to be Both the finite element method and the Dodd & Deeds rejected by the processing algorithms applied to inductance formulation are used to solve the forward problem during the spectra. inversion process. For the inverse solution, a modified Newton– Raphson method was used to adjust each set of parameters In recent years, the eddy current technique (ECT) [2-5] and (permeability, conductivity, thickness, and lift-off) to fit the alternating current potential drop (ACPD) technique [6-8] inductances (measured or simulated) in a least-squared sense were the two primary electromagnetic non-destructive testing because of its known convergence properties. The approximate techniques (NDT) [9-21] on metals’ permeability Jacobian matrix (sensitivity matrix) for each set of the parameter measurements. -
On Fractional Whittaker Equation and Operational Calculus
J. Math. Sci. Univ. Tokyo 20 (2013), 127–146. On Fractional Whittaker Equation and Operational Calculus By M. M. Rodrigues and N. Vieira Abstract. This paper is intended to investigate a fractional dif- ferential Whittaker’s equation of order 2α, with α ∈]0, 1], involving the Riemann-Liouville derivative. We seek a possible solution in terms of power series by using operational approach for the Laplace and Mellin transform. A recurrence relation for coefficients is obtained. The exis- tence and uniqueness of solutions is discussed via Banach fixed point theorem. 1. Introduction The Whittaker functions arise as solutions of the Whittaker differential equation (see [4], Vol.1)). These functions have acquired a significant in- creasing due to its frequent use in applications of mathematics to physical and technical problems [1]. Moreover, they are closely related to the con- fluent hypergeometric functions, which play an important role in various branches of applied mathematics and theoretical physics, for instance, fluid mechanics, electromagnetic diffraction theory, and atomic structure theory. This justifies the continuous effort in studying properties of these functions and in gathering information about them. As far as the authors aware, there were no attempts to study the corresponding fractional Whittaker equation (see below). Fractional differential equations are widely used for modeling anomalous relaxation and diffusion phenomena (see [3], Ch. 5, [6], Ch. 2). A systematic development of the analytic theory of fractional differential equations with variable coefficients can be found, for instance, in the books of Samko, Kilbas and Marichev (see [13], Ch. 3). 2010 Mathematics Subject Classification. Primary 35R11; Secondary 34B30, 42A38, 47H10. -
Units in Electromagnetism (PDF)
Units in electromagnetism Almost all textbooks on electricity and magnetism (including Griffiths’s book) use the same set of units | the so-called rationalized or Giorgi units. These have the advantage of common use. On the other hand there are all sorts of \0"s and \µ0"s to memorize. Could anyone think of a system that doesn't have all this junk to memorize? Yes, Carl Friedrich Gauss could. This problem describes the Gaussian system of units. [In working this problem, keep in mind the distinction between \dimensions" (like length, time, and charge) and \units" (like meters, seconds, and coulombs).] a. In the Gaussian system, the measure of charge is q q~ = p : 4π0 Write down Coulomb's law in the Gaussian system. Show that in this system, the dimensions ofq ~ are [length]3=2[mass]1=2[time]−1: There is no need, in this system, for a unit of charge like the coulomb, which is independent of the units of mass, length, and time. b. The electric field in the Gaussian system is given by F~ E~~ = : q~ How is this measure of electric field (E~~) related to the standard (Giorgi) field (E~ )? What are the dimensions of E~~? c. The magnetic field in the Gaussian system is given by r4π B~~ = B~ : µ0 What are the dimensions of B~~ and how do they compare to the dimensions of E~~? d. In the Giorgi system, the Lorentz force law is F~ = q(E~ + ~v × B~ ): p What is the Lorentz force law expressed in the Gaussian system? Recall that c = 1= 0µ0. -
On the First Electromagnetic Measurement of the Velocity of Light by Wilhelm Weber and Rudolf Kohlrausch
Andre Koch Torres Assis On the First Electromagnetic Measurement of the Velocity of Light by Wilhelm Weber and Rudolf Kohlrausch Abstract The electrostatic, electrodynamic and electromagnetic systems of units utilized during last century by Ampère, Gauss, Weber, Maxwell and all the others are analyzed. It is shown how the constant c was introduced in physics by Weber's force of 1846. It is shown that it has the unit of velocity and is the ratio of the electromagnetic and electrostatic units of charge. Weber and Kohlrausch's experiment of 1855 to determine c is quoted, emphasizing that they were the first to measure this quantity and obtained the same value as that of light velocity in vacuum. It is shown how Kirchhoff in 1857 and Weber (1857-64) independently of one another obtained the fact that an electromagnetic signal propagates at light velocity along a thin wire of negligible resistivity. They obtained the telegraphy equation utilizing Weber’s action at a distance force. This was accomplished before the development of Maxwell’s electromagnetic theory of light and before Heaviside’s work. 1. Introduction In this work the introduction of the constant c in electromagnetism by Wilhelm Weber in 1846 is analyzed. It is the ratio of electromagnetic and electrostatic units of charge, one of the most fundamental constants of nature. The meaning of this constant is discussed, the first measurement performed by Weber and Kohlrausch in 1855, and the derivation of the telegraphy equation by Kirchhoff and Weber in 1857. Initially the basic systems of units utilized during last century for describing electromagnetic quantities is presented, along with a short review of Weber’s electrodynamics. -
SKIFFS: Superconducting Kinetic Inductance Field-Frequency Sensors for Sensitive Magnetometry in Moderate Background Magnetic Fields
SKIFFS: Superconducting Kinetic Inductance Field-Frequency Sensors for sensitive magnetometry in moderate background magnetic fields Cite as: Appl. Phys. Lett. 113, 172601 (2018); https://doi.org/10.1063/1.5049615 Submitted: 24 July 2018 . Accepted: 10 October 2018 . Published Online: 25 October 2018 A. T. Asfaw , E. I. Kleinbaum, T. M. Hazard , A. Gyenis, A. A. Houck, and S. A. Lyon ARTICLES YOU MAY BE INTERESTED IN Multi-frequency spin manipulation using rapidly tunable superconducting coplanar waveguide microresonators Applied Physics Letters 111, 032601 (2017); https://doi.org/10.1063/1.4993930 Publisher's Note: “Anomalous Nernst effect in Ir22Mn78/Co20Fe60B20/MgO layers with perpendicular magnetic anisotropy” [Appl. Phys. Lett. 111, 222401 (2017)] Applied Physics Letters 113, 179901 (2018); https://doi.org/10.1063/1.5018606 Tunneling anomalous Hall effect in a ferroelectric tunnel junction Applied Physics Letters 113, 172405 (2018); https://doi.org/10.1063/1.5051629 Appl. Phys. Lett. 113, 172601 (2018); https://doi.org/10.1063/1.5049615 113, 172601 © 2018 Author(s). APPLIED PHYSICS LETTERS 113, 172601 (2018) SKIFFS: Superconducting Kinetic Inductance Field-Frequency Sensors for sensitive magnetometry in moderate background magnetic fields A. T. Asfaw,a) E. I. Kleinbaum, T. M. Hazard, A. Gyenis, A. A. Houck, and S. A. Lyon Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA (Received 24 July 2018; accepted 10 October 2018; published online 25 October 2018) We describe sensitive magnetometry using lumped-element resonators fabricated from a supercon- ducting thin film of NbTiN. Taking advantage of the large kinetic inductance of the superconduc- tor, we demonstrate a continuous resonance frequency shift of 27 MHz for a change in the magnetic field of 1.8 lT within a perpendicular background field of 60 mT. -
American Mathematical Society TRANSLATIONS Series 2 • Volume 130
American Mathematical Society TRANSLATIONS Series 2 • Volume 130 One-Dimensional Inverse Problems of Mathematical Physics aM „ °m American Mathematical Society One-Dimensional Inverse Problems of Mathematical Physics http://dx.doi.org/10.1090/trans2/130 American Mathematical Society TRANSLATIONS Series 2 • Volume 130 One-Dimensional Inverse Problems of Mathematical Physics By M. M. Lavrent'ev K. G. Reznitskaya V. G. Yakhno do, n American Mathematical Society r, --1 Providence, Rhode Island Translated by J. R. SCHULENBERGER Translation edited by LEV J. LEIFMAN 1980 Mathematics Subject Classification (1985 Revision): Primary 35K05, 35L05, 35R30. Abstract. Problems of determining a variable coefficient and right side for hyperbolic and parabolic equations on the basis of known solutions at fixed points of space for all times are considered in this monograph. Here the desired coefficient of the equation is a function of only one coordinate, while the desired right side is a function only of time. On the basis of solution of direct problems the inverse problems are reduced to nonlinear operator equations for which uniqueness and in some cases also existence questions are investigated. The problems studied have applied importance, since they are models for interpreting data of geophysical prospecting by seismic and electric means. The monograph is of interest to mathematicians concerned with mathematical physics. Bibliography: 75 titles. Library of Congress Cataloging-in-Publication Data Lavrent'ev, M. M. (Mlkhail Milchanovich) One-dimensional inverse problems of mathematical physics. (American Mathematical Society translations; ser. 2, v. 130) Translation of: Odnomernye obratnye zadachi matematicheskoi Bibliography: p. 67. 1. Inverse problems (Differential equations) 2. Mathematical physics. -
Simple Circuit Theory and the Solution of Two Electricity Problems from The
Simple circuit theory and the solution of two electricity problems from the Victorian Age A C Tort ∗ Departamento de F´ısica Te´orica - Instituto de F´ısica Universidade Federal do Rio de Janeiro Caixa Postal 68.528; CEP 21941-972 Rio de Janeiro, Brazil May 22, 2018 Abstract Two problems from the Victorian Age, the subdivision of light and the determination of the leakage point in an undersea telegraphic cable are discussed and suggested as a concrete illustrations of the relationships between textbook physics and the real world. Ohm’s law and simple algebra are the only tools we need to discuss them in the classroom. arXiv:0811.0954v1 [physics.pop-ph] 6 Nov 2008 ∗e-mail: [email protected]. 1 1 Introduction Some time ago, the present author had the opportunity of reading Paul J. Nahin’s [1] fascinating biog- raphy of the Victorian physicist and electrician Oliver Heaviside (1850-1925). Heaviside’s scientific life unrolls against a background of theoretical and technical challenges that the scientific and technological developments fostered by the Industrial Revolution presented to engineers and physicists of those times. It is a time where electromagnetic theory as formulated by James Clerk Maxwell (1831-1879) was un- derstood by only a small group of men, Lodge, FitzGerald and Heaviside, among others, that had the mathematical sophistication and imagination to grasp the meaning and take part in the great Maxwellian synthesis. Almost all of the electrical engineers, or electricians as they were called at the time, considered themselves as “practical men”, which effectively meant that most of them had a working knowledge of the electromagnetic phenomena spiced up with bits of electrical theory, to wit, Ohm’s law and the Joule effect. -
The Concept of Field in the History of Electromagnetism
The concept of field in the history of electromagnetism Giovanni Miano Department of Electrical Engineering University of Naples Federico II ET2011-XXVII Riunione Annuale dei Ricercatori di Elettrotecnica Bologna 16-17 giugno 2011 Celebration of the 150th Birthday of Maxwell’s Equations 150 years ago (on March 1861) a young Maxwell (30 years old) published the first part of the paper On physical lines of force in which he wrote down the equations that, by bringing together the physics of electricity and magnetism, laid the foundations for electromagnetism and modern physics. Statue of Maxwell with its dog Toby. Plaque on E-side of the statue. Edinburgh, George Street. Talk Outline ! A brief survey of the birth of the electromagnetism: a long and intriguing story ! A rapid comparison of Weber’s electrodynamics and Maxwell’s theory: “direct action at distance” and “field theory” General References E. T. Wittaker, Theories of Aether and Electricity, Longam, Green and Co., London, 1910. O. Darrigol, Electrodynamics from Ampère to Einste in, Oxford University Press, 2000. O. M. Bucci, The Genesis of Maxwell’s Equations, in “History of Wireless”, T. K. Sarkar et al. Eds., Wiley-Interscience, 2006. Magnetism and Electricity In 1600 Gilbert published the “De Magnete, Magneticisque Corporibus, et de Magno Magnete Tellure” (On the Magnet and Magnetic Bodies, and on That Great Magnet the Earth). ! The Earth is magnetic ()*+(,-.*, Magnesia ad Sipylum) and this is why a compass points north. ! In a quite large class of bodies (glass, sulphur, …) the friction induces the same effect observed in the amber (!"#$%&'(, Elektron). Gilbert gave to it the name “electricus”. -
Chapter 32 Inductance and Magnetic Materials
Chapter 32 Inductance and Magnetic Materials The appearance of an induced emf in one circuit due to changes in the magnetic field produced by a nearby circuit is called mutual induction. The response of the circuit is characterized by their mutual inductance. Can we find an induced emf due to its own magnetic field changes? Yes! The appearance of an induced emf in a circuit associated with changes in its own magnet field is called self-induction. The corresponding property is called self-inductance. A circuit element, such as a coil, that is designed specifically to have self-inductance is called an inductor. 1 32.1 Inductance Close: As the flux through the coil changes, there is an induced emf that opposites this change. The self induced emf try to prevent the rise in the current. As a result, the current does not reach its final value instantly, but instead rises gradually as in right figure. Open: When the switch is opened, the flux rapidly decreases. This time the self- induced emf tries to maintain the flux. When the current in the windings of an electromagnet is shut off, the self-induced emf can be large enough to produce a spark across the switch contacts. 2 32.1 Inductance (II) Since the self-induction and mutual induction occur simultaneously, both contribute to the flux and to the induced emf in each coil. The flux through coil 1 is the sum of two terms: N1Φ1 = N1(Φ11 + Φ12 ) The net emf induced in coil 1 due to changes in I1 and I2 is d V = −N (Φ + Φ ) emf 1 dt 11 12 3 Self-Inductance It is convenient to express the induced emf in terms of a current rather than the magnetic flux through it. -
An Analytic Exact Form of the Unit Step Function
Mathematics and Statistics 2(7): 235-237, 2014 http://www.hrpub.org DOI: 10.13189/ms.2014.020702 An Analytic Exact Form of the Unit Step Function J. Venetis Section of Mechanics, Faculty of Applied Mathematics and Physical Sciences, National Technical University of Athens *Corresponding Author: [email protected] Copyright © 2014 Horizon Research Publishing All rights reserved. Abstract In this paper, the author obtains an analytic Meanwhile, there are many smooth analytic exact form of the unit step function, which is also known as approximations to the unit step function as it can be seen in Heaviside function and constitutes a fundamental concept of the literature [4,5,6]. Besides, Sullivan et al [7] obtained a the Operational Calculus. Particularly, this function is linear algebraic approximation to this function by means of a equivalently expressed in a closed form as the summation of linear combination of exponential functions. two inverse trigonometric functions. The novelty of this However, the majority of all these approaches lead to work is that the exact representation which is proposed here closed – form representations consisting of non - elementary is not performed in terms of non – elementary special special functions, e.g. Logistic function, Hyperfunction, or functions, e.g. Dirac delta function or Error function and Error function and also most of its algebraic exact forms are also is neither the limit of a function, nor the limit of a expressed in terms generalized integrals or infinitesimal sequence of functions with point wise or uniform terms, something that complicates the related computational convergence. Therefore it may be much more appropriate in procedures. -
HISTORICAL SURVEY SOME PIONEERS of the APPLICATIONS of FRACTIONAL CALCULUS Duarte Valério 1, José Tenreiro Machado 2, Virginia
HISTORICAL SURVEY SOME PIONEERS OF THE APPLICATIONS OF FRACTIONAL CALCULUS Duarte Val´erio 1,Jos´e Tenreiro Machado 2, Virginia Kiryakova 3 Abstract In the last decades fractional calculus (FC) became an area of intensive research and development. This paper goes back and recalls important pio- neers that started to apply FC to scientific and engineering problems during the nineteenth and twentieth centuries. Those we present are, in alphabet- ical order: Niels Abel, Kenneth and Robert Cole, Andrew Gemant, Andrey N. Gerasimov, Oliver Heaviside, Paul L´evy, Rashid Sh. Nigmatullin, Yuri N. Rabotnov, George Scott Blair. MSC 2010 : Primary 26A33; Secondary 01A55, 01A60, 34A08 Key Words and Phrases: fractional calculus, applications, pioneers, Abel, Cole, Gemant, Gerasimov, Heaviside, L´evy, Nigmatullin, Rabotnov, Scott Blair 1. Introduction In 1695 Gottfried Leibniz asked Guillaume l’Hˆopital if the (integer) order of derivatives and integrals could be extended. Was it possible if the order was some irrational, fractional or complex number? “Dream commands life” and this idea motivated many mathematicians, physicists and engineers to develop the concept of fractional calculus (FC). Dur- ing four centuries many famous mathematicians contributed to the theo- retical development of FC. We can list (in alphabetical order) some im- portant researchers since 1695 (see details at [1, 2, 3], and posters at http://www.math.bas.bg/∼fcaa): c 2014 Diogenes Co., Sofia pp. 552–578 , DOI: 10.2478/s13540-014-0185-1 SOME PIONEERS OF THE APPLICATIONS . 553 • Abel, Niels Henrik (5 August 1802 - 6 April 1829), Norwegian math- ematician • Al-Bassam, M. A. (20th century), mathematician of Iraqi origin • Cole, Kenneth (1900 - 1984) and Robert (1914 - 1990), American physicists • Cossar, James (d.