An introduction to fractional calculus | Advanced Electromagnetics

PDF

Published: Feb 19, 2020

DOI: https://doi.org/10.7716/aem.v9i1.1192

Keywords:

fractional calculus fractional quadrature linear response numerical methods

A. Persechino

Brazilian Center for Research in Physics

Abstract

The aim of this work is to introduce the main concepts of Fractional Calculus, followed by one of its application to classical electrodynamics, illustrating how non-locality can be interpreted naturally in a fractional scenario. In particular, a result relating fractional dynamics to high frequency dielectric response is used as motivation. In addition to the theoretical discussion, a comprehensive review of two numerical procedures for fractional integration is carried out, allowing one immediately to build numerical models applied to high frequency electromagnetics and correlated fields.

Downloads

Download data is not yet available.

How to Cite

Persechino, A. (2020). An introduction to fractional calculus: Numerical methods and application to HF dielectric response. Advanced Electromagnetics, 9(1), 19–30. https://doi.org/10.7716/aem.v9i1.1192

Issue

Vol. 9 No. 1 (2020)

Section

Review Articles (only if soclicited by an Editor)

Authors who publish with this journal agree to the following terms:

Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).

References

Ross, B. Fractional Calculus and its Applications. (Lecture Notes in Mathematics) 1ed. Berlin: Springer, 1975. 381 p.

Hermann, R. Fractional Calculus: an introduction for physicists. 1ed. Singapore: World Scientific Publishing, 2014. 479 p.

Tarasov, V. E. Review of Some Promising Fractional Physical Models. International Journal of Modern Physics B. v. 27. n. 9 1-38. 2013.

Oughstun, K. E. Electromagnetic and Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media. (Springer Series in Optical Sciences) 1ed. New York: Springer, 2006. 456 p.

Hilfer, R. Applications of fractional calculus in physics. 1ed. Singapore: World Scientific Publishing, 2000. 463 p.

Jonscher, A. K. The 'universal' dielectric response. Nature. v. 267. n. 673-679. 1977.

Jonscher, A. K. Dielectric relaxation in solids. Journal of Physics D: Applied Physics. v. 32. n. R57. 1999.

Tarasov, V. E. Universal electromagnetic waves in dielectric. Journal of Physics: condensed matter. v. 20. n. 17 175223. 2008.

Tarasov, V. E. Fractional dynamics: Applications of fractional calculus to dynamics of particles, fields and media. (Nonlinear Physical Sciences Series) 1ed. Beijing: Higher Education Press, 2010. 504 p.

Tarasov, V. E. Fractional integro-differential equations for electromagnetic waves in dielectric media. Theoretical and Mathematical Physics. v. 158. n. 3 355-359. 2009.

Samko, S.G.; Kilbas, A. A.; Marichev, O. I. Fractional Integrals and Derivatives: theory and applications. 1ed. Singapore: Gordon and Breach Science Publishers, 1993. 976 p.

Oldham, K. B.; Spanier, J. The Fractional Calculus:Theory and Applications of Differentiation and Integration to Arbitrary Order. 1ed. New York: Dover Publications, 2006. 256 p.

Greiner, W.. Classical Electrodynamics. (Classical Theoretical Physics) 1ed. New York: Springer, 1998. 555 p.

Dahlquist, G.; Björck, Å. Numerical Methods. (Dover Books on Mathematics) 1ed. New York: Dover Publications, 2003. 573 p.

Ralston, A,; Rabinowitz, P. A First Course in Numerical Analysis. (Dover Books on Mathematics) 2ed. New York: Dover Publications, 2001. p.

Golub, G. H.; Ortega, J. M. Scientific Computing and Differential Equations: An Introduction to Numerical Methods. 1ed. San Diego: Academic Press, 1991. 337 p.

Lubich, C. Discretized Fractional Calculus. SIAM Journal on Mathematical Analysis. v. 17. n. 3 704-719. 1986.

Matthys, J. A-Stable linear multistep methods for Volterra Integro-Differential Equations. Numerische Mathematik. v. 27. n. 1 85-94. 1976.

Wolkenfelt, P. H. M. Reducible quadrature methods for Volterra integral equations of the first kind. BIT Numerical Mathematics. v. 21. n. 2 232-241. 1981.

Lubich, C. Convolution quadrature and discretized operational calculus. I. Numerische Mathematik. v. 52. n. 2 . 1988.

Lubich, C. Convolution quadrature and discretized operational calculus. II. Numerische Mathematik. v. 52. n. 4 . 1988.

Li, C.; Zeng, F. Numerical Methods for Fractional Calculus. (Chapman & Hall/CRC Numerical Analysis and Scientific Computing Series) 1ed. Abingdon: Chapman and Hall/CRC, 2015. 281 p.

de Oliveira, E. C.; Machado, J. A. T. A review of definitions for fractional derivatives and integrals. Mathematical Problems in Engineering. v. 2014. n. 2014 . 2014.

Tarasov, V. E. No violation of the Leibniz rule. No fractional derivative. Communications in Nonlinear Science and Numerical Simulation. v. 18. n. 11 2945 - 2948. 2013.

Ortigueira, M. D.; Machado, J. A. T. . Journal of Computational Physics. v. 293. n. 4-13. 2015.

Whittaker, K. A.; Keaveney, J.; Hughes, I. G.; Adams, C. S. Hilbert transform: Applications to atomic spectra. Physical Review A. v. 91. n. 032513. 2015.

Garrappa, R.; Mainardi, F.; Maione, G. Models of dielectric relaxation based on completely monotone functions. Fractional Calculus and Applied Analysis. v. 19. n. 5 1105-1160. 2016.

Zhao, X.; Peng, G.; Jiang, X.; Liu, W.; Zhan, Z.; Meng, W.; Wang, Y.; Song, T.; Li, J.,; Feng, H. Investigation of relaxation process in poly(vinylidene fluoride-hexafluoropropylene) using dielectric relaxation spectroscopy. Journal of Materials Science: Materials in Electronics. v. 27. n. 1 718-723. 2016.

Gao, Y.; Chen, Z.; Tu, W.; Li, X.; Tian, Y.; Liu, R.; Wang, L. Anomaly in dielectric relaxation dispersion of glass-forming alkoxy alcohols. The Journal of Chemical Physics. v. 142. n. 21 214505. 2015.

Coughlan, J. J.; Hill, A. T.; Logemann, H. The Z-transform and linear multistep stability. IMA Journal of Numerical Analysis. v. 27. n. 1 45-73. 2007.

Galeone, L.; Garrappa, R. On Multistep Methods for Differential Equations of Fractional Order. Mediterranean Journal of Mathematics. v. 3. n. 3 565-580. 2006.

Lubich, Ch. IMA Journal of Numerical Analysis. IMA Journal of Numerical Analysis. v. 3. n. 4 439-465. 1983.

Lovoie, J.; Olser, T.; Tremblay, R. Fractional Derivatives and Special Functions. SIAM Review. v. 18. n. 2 240-268. 1976.

Scilab Enterprises. Scilab: Free and Open Source software for numerical computation (Linux, Version 6.0.1).2012. Available in http://www.scilab.org.

Li, C.; Chen, A.; Ye, J. Numerical approaches to fractional calculus and fractional ordinary differential equation. Journal of Computational Physics. v. 230. n. 9 3352-3368. 2011.

Aberge, R.; Moisan, L. Fast and accurate evaluation of a generalized incomplete gamma function. 2016. Available in https://hal.archives-ouvertes.fr/hal-01329669.