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# Topic: Fractional calculus

###### In the News (Sun 26 May 19)

 Fractional calculus - Wikipedia, the free encyclopedia Notice here that fraction is then a misnomer for the exponent, since it need not be rational, but the term fractional calculus has become traditional. As far as the existence of such a theory is concerned, the foundations of the subject were laid by Liouville in a paper from 1832. The fractional derivative of a function to order a is often now defined by means of the Fourier or Mellin integral transforms. en.wikipedia.org /wiki/Fractional_calculus   (945 words)

 Differintegral - Wikipedia, the free encyclopedia In mathematics, the differintegral is the combined differentiation/integration operator used in fractional calculus. The operator does not define a seperate function, but is a notation style for taking both the fractional derivative and the fractional integral of the same expression. See the page on fractional calculus for the general context. en.wikipedia.org /wiki/Initialized_fractional_calculus   (347 words)

 Fractional Calculus   (Site not responding. Last check: 2007-10-21) fractional calculus, fractional integral, fractional derivatives, fractional differential equations. The subject is as old as the calculus of differentiation and goes back to times when Leibniz, Gauss, and Newton invented this kind of calculation. The story on the fractional calculus continued with contributions from Fourier, Abel, Lacroix, Leibniz, Grunwald and Letnikov. www.tuke.sk /petras/FC.html   (342 words)

 Applications of Fractional Calculus   (Site not responding. Last check: 2007-10-21) Fractional calculus concerns the generalization of differentiation and integration to non-integer (fractional) orders. Although fractional calculus is a natural generalization of calculus, and although its mathematical history is equally long, it has, until recently, played a negligible role in physics. Many applications of fractional calculus amount to replacing the time derivative in an evolution equation with a derivative of fractional order. www.ica1.uni-stuttgart.de /Jahresberichte/00/node50.html   (259 words)

 Initialized Fractional Calculus The fractional calculus (which admits of integrals and derivatives of non-integer order) dates back almost to the origin of the better-known ordinary (integer-order) calculus, but thus far has been treated more as a mathematical curiosity than as a scientific and engineering tool. The application of the fractional calculus to scientific and engineering problems has been inhibited by difficulties that arise from the basic definitions given heretofore for integrals and derivatives of arbitrary order. Going beyond viscoelasticity, the initialized fractional calculus can be applied to problems that arise in a variety of scientific, engineering, and purely mathematical disciplines, including creep, percolation, material science, viscous fluid behavior, heat transfer, batteries, electromagnetics, control, communications, filtering, and chaotic systems. www.nasatech.com /Briefs/Oct02/LEW17139.html   (586 words)

 Fractional Paradigm in Electrodynamics Engheta, "On Fractional Calculus and Fractional Multipoles in Electromagnetism," We are interested in bringing the concept of fractional operators, fractional calculus and the theory of electrodynamics together, and to develop an area in electromagnetics which we have named We have applied the concept/tools of fractional calculus in certain problems in electromagnetic theory, and have obtained interesting results that demonstrate some salient physical features and mathematical properties of these operators with applications in radiation and scattering problems. www.ee.upenn.edu /~engheta/Fractional_Paradigm_in_EM.htm   (314 words)

 Igor Podlubny :: Home > Fractional calculus > What is it?   (Site not responding. Last check: 2007-10-21) Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary (non-integer) order. The subject is as old as the differential calculus, and goes back to times when Leibniz and Newton invented differential calculus. Plenary lecture at the Conference of the Slovak Mathematical Society, Jasná, November 23, 2002, in Slovak. www.tuke.sk /podlubny/fc.html   (221 words)

 Hydrogeology Today - Volume 3, Number 2 An example of the latter, talked about more in depth in the next article, is the fractional advection-dispersion equation approach, where the intricacies of heterogeneity and its effect on solute transport are accounted for by a dispersion tensor with a fractional power. Fractional calculus is not a new idea, with roots tracing back to at least 1695. In practical terms, the fractional advection-dispersion equation has both forward and backward memory such that the a fractional derivative incorporates global effects into what would ordinarily be a local derivative. www.hydrogeologic.com /Newsletters/v3n2/v3n2.htm   (1688 words)

 [No title]   (Site not responding. Last check: 2007-10-21) The author has developed a full chain of operational rules, mapping properties and convolutional structure of the generalized (m-tuple) fractional integrals and the corresponding derivatives.\par Historical background and the theme of the book is contained in the Introduction. Some other applications of the generalized fractional calculus: Abel's integral equation, theory of univalent functions and generalized Laplace type transforms are treated in the Chapter 5. Fractional integration operators involving Fox's \$H^{m,0}_{m,m}\$-function are studied here in different functional spaces. zmath.library.ualberta.ca /cgi-bin/zmen/ZMATH/en/quick.html?first=1&maxdocs=3&type=tex&an=0882.26003&format=complete   (360 words)

 [No title] The formal relationship between heavy-tailed delay distributions, hyperbolically decaying of the packet delay auto-covariance function and fractional differential equation is shown. The new interpretation of fractional calculus opens up the new area of using this well-developed mathematical tool in order to understand the local and global characteristics of the packet traffic behaviour. We have shown, that dynamics of the packet passing in the virtual connection in TCP/IP networks is described by the equations in fractional derivative and corresponds to processes with long-range-dependency. www.neva.ru /conf/art/art8.html   (1458 words)

 Fractional Calculus   (Site not responding. Last check: 2007-10-21) Fractional modal decomposition of a boundary-controlled-and-observed infinite-dimensional linear system, by D. Matignon, 5 pages presented at Mathematical Theory of Networks and Systems symposium. Fractional integrodifferential control of the Euler-Bernoulli beam by G. Montseny, J. Fractional Differential Systems: Models, Methods and Applications by D. Matignon and G. Montseny (editors), ESAIM Proceedings, vol. www.tsi.enst.fr /~matignon/research.html   (1114 words)

 Prof. S.L. Kalla: Recent Publications   (Site not responding. Last check: 2007-10-21) SHYAM L. de Duran, S.L. Kalla and H.M. Srivastava: Fractional calculus and the sums of certain families of infinite series, Jour. Ismail Ali and S.L. Kalla: An application of fractional calculus to the solution of a general class of differintegral equations. Boyadjiev, H.J. Dobner and S.L. Kalla: On a fractional integro-differential equation of Volterra-type. www.sci.kuniv.edu.kw /~kalla/recentpub.html   (924 words)

 Open Directory - Science: Math: Calculus   (Site not responding. Last check: 2007-10-21) Calculus and Physics Practice Exams - Practice exams for applied calculus and physics in PDF and HTML formats. Calculus Solutions - Reference site with a vast amount of information and example problems which are alphabetically listed by topic. The University of Minnesota Calculus Initiative - Offers calculus application examples for the mathematical properties of a rainbow, the fundamental theorem of calculus, methods of maximizing structural beams in a building, and modeling population growth. dmoz.org /Science/Math/Calculus   (916 words)

 Seminar on Scaling We use fractional integrals and derivatives to obtain representations of fractional Brownian motion. A network calculus is developed for processes whose burstiness is stochastically bounded by general decreasing functions. This new calculus is expected to be of special interest for the efficient implementation of network services providing statistical guarantees. math.bu.edu /INDIVIDUAL/murad/colloq-99-00.html   (1997 words)

 Wen Chen The fractional time derivative is a very effective means to describe anomalous attenuation of arbitrary frequency dependency. In contrast, the time fractional derivative is better suited to describe the anomalous diffusions of memory media. Besides the above fractional calculus modelings, we also developed a time-domain integer-order PDE strategy to model the broadband pulse signal propagation, which uses the standard damped wave equation and frequency decomposion. folk.uio.no /wenc/html/anomaloussummary.htm   (529 words)

 [No title]   (Site not responding. Last check: 2007-10-21) APPLICATIONS OF FRACTIONAL CALCULUS IN PHYSICS edited by R Hilfer (Universität Mainz & Universität Stuttgart, Germany) Fractional calculus is a collection of relatively little-known mathematical results concerning generalizations of differentiation and integration to noninteger orders. This situation is beginning to change, and there are now a growing number of research areas in physics which employ fractional calculus. This volume provides an introduction to fractional calculus for physicists, and collects easily accessible review articles surveying those areas of physics in which applications of fractional calculus have recently become prominent. www.worldscibooks.com /physics/3779.txt   (148 words)

 [No title] There are 165 math papers containing "fractional derivative" in their titles, classified under many fields of analysis, as well as under the fields of mechanics. There are a dozen or so books on fractional calculus (Nishimoto has a 5-volume work!), and even a Journal of Fractional Calculus. i attended a conference on fractional calculus in the 70's, when i was an undergrad. www.math.niu.edu /~rusin/known-math/98/fracder1   (670 words)

 Sandia National Laboratories - Computational Electromagnetics in Geophysics   (Site not responding. Last check: 2007-10-21) A growing body of literature on diffusion processes within spatially hierarchical, or multiscale, materials has been directed at a re-examination of the fractional order calculus as a tool for capturing the behavior of transport phenomena observed in the materials' natural analogs, such as rocks. Because low-frequency electromagnetic induction is also a diffusion process (in contrast to the wave-like behavior at higher frequencies), we have been investigating the utility of fractional calculus in modeling and interpretation of electromagnetic geophysical data. To better understand the effects of a frational derivative/itegral operator on simple functions, the case for the unit function, f=1, is presented here (based on figure 4.1.1 in "The Fractional Calculus" by K. Oldham and J. Spanier, Academic Press, 1970). www.sandia.gov /comp-em-geop/gallery/graphic2.htm   (127 words)

 [No title] This is the topic of Fractional Order Derivatives (or Integrals), aka Fractional Calculus, and was studied by a bunch of mathematicians in early 1900s and late 1800s, including Riemann and a whole bunch of big names my brain's too dusty and rusty to recall. Even sticking with the gamma, it turns out that fractional order derivatives have a kind of looseness - the operator used in the electrochemistry paper, and seen elsewhere, is not the one you and I are playing with. I think the title was "The fractional Calculus", and I remember it had the style of Academic Press. www.math.niu.edu /~rusin/known-math/99/frac_der   (1335 words)

 New Page 1 An application of fractional calculus to the solution of a general class of differential equations, Appl. S.L. Kalla and Ismail Ali: Fractional Calculus and It’s applications to differential equations, New Frontiers In Algebras, groups and geometries, Hadronic Press, pp. Fractional calculus and its application to differential equations, Proceedings of International Workshop, Institute for Basic Research, Monteroduni, Italy (Aug. 1995). www.sci.kuniv.edu.kw /~taqi/6.htm   (575 words)

 Wen Chen The research described below is a part of the ongoing project "mathematical and numerical modelings of medical ultrasound wave propagation", for which I am the project manager (other team memebers: Aicha Bounaim, Xing Cai, Sverre Holm, Aslak Tveito, Åsmund Ødegård). the introduction of the concept of the positive fractional time derivative and accordingly the presentation of the modified Szabo wave equations, where the hyper-singularity of the original Szabo wave equation models for anomalously attenuative media is eased and the integer-order initial condition is naturally included; the establishment of explicit links between fractional calculus equation models, 1/f power spectrum, Hurst exponent, fractals, fractional Brownian motion and Levy stable process, all of which reflect the memory dynamics and/or fractal (topology/molecular structure) microstructures of complex systems. heim.ifi.uio.no /~wenc/html/anomalous.htm   (415 words)

 Invited Lectures Abroad   (Site not responding. Last check: 2007-10-21) Fractional Calculus and Its Appl-s (100 years of Nihon Univ.); A generalized fractional calculus dealing with H-functions (in common with S.L. Kalla); U.K., England, Keele; 1992; Keele University; Applications of the G-functions and fractional calculus to the hyper-Bessel equations; Canary Islands, Las Palmas; 1993; Universidad de Las Palmas; A cycle of 7 invited lectures on the subject of the G-functions and generalized fractional calculus; (Doctoral Course, 2 credits). www.math.bas.bg /~complan/virginia/invited.html   (224 words)

 Mathematical Modeling of Complex Phenomena   (Site not responding. Last check: 2007-10-21) Research is being conducted on the development and application of mathematical models of phenomena that are scaling (e.g., phase transitions) and cannot be described by systems of ordinary differential equations. The mathematical techniques include renormalization group theory, fractional calculus, and stochastic fluctuations. A new aspect of this investigation is that the evolution of complex phenomena is described by fractional differential stochastic equations, whose solutions have scaling properties. www4.nationalacademies.org /pga/rap.nsf/44ac59cd53fc460885256a220069c796/5336385c6bc9196c85257076004b5457?OpenDocument   (111 words)

 [No title] The foundation of the derivative of fractional Wick-Ito Calculus is based upon Malliavin calculus for BM. Apparently, the fractional Black-Scholes PDE scales the concavity of the option price with respect to the stock process by a function dependent on both t and H. The Greeks with there analogies in the classical BM setting is given as the following. An introduction to white noise theory and Malliavin calculus for fractional Brownian motion. www.stat.purdue.edu /~zdaye/Projects/fBM.doc   (3454 words)

 Initialization, Conceptualization, and Application in the Generalized Fractional Calculus   (Site not responding. Last check: 2007-10-21) A modified set of definitions for the fractional calculus is provided which formally include the effects of initialization. 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. gltrs.grc.nasa.gov /cgi-bin/GLTRS/browse.pl?1998/TP-1998-208415.html   (278 words)

 Fractal Cauculus Project   (Site not responding. Last check: 2007-10-21) The Fractional Calculus Project is an interdisciplinary collaboration of mathematicians, statisticians, physicists and hydrologists to develop the theory and practical application of fractals, fractional derivatives, and heavy tailed stochastic processes. Fractional-order partial differential equations (PDEs) are used by physicists and hydrologists to model anomalous diffusion and Hamiltonian chaos. Fractional PDEs address shortcomings with previous methods in geophysics, but a number of important problems remain open. unr.edu /homepage/mcubed/FRG.html   (370 words)

 Jahrbuch-CD der MPG 2003 - Fractional Calculus via Functio This paper demonstrates the power of the functional-calculus definition of linear fractional (pseudo-)differential operators via generalised Fourier transforms. The suggested method via residue calculus separates an impulse response automatically into an exponentially damped (possibly oscillatory) part and a ''slow' relaxation. If an impulse response is stable it becomes automatically causal, otherwise one has to add a homogeneous solution to get causality. www.mpg.de /forschungsergebnisse/wissVeroeffentlichungen/archivListenJahrbuch/2002/06/publZIM28.html   (199 words)

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