Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia


IMPACT FACTOR 2018: 3.514
5-year IMPACT FACTOR: 3.524

CiteScore 2018: 3.44

SCImago Journal Rank (SJR) 2018: 1.891
Source Normalized Impact per Paper (SNIP) 2018: 1.808

Mathematical Citation Quotient (MCQ) 2017: 0.98

Online
ISSN
1314-2224
See all formats and pricing
More options …
Volume 16, Issue 1

Issues

Green’s theorem for generalized fractional derivatives

Tatiana Odzijewicz
  • Center for Research and Development in Mathematics and Applications Department of Mathematics, University of Aveiro, 3810-193, Aveiro, Portugal
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Agnieszka Malinowska / Delfim Torres
  • Center for Research and Development in Mathematics and Applications Department of Mathematics, University of Aveiro, 3810-193, Aveiro, Portugal
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2012-12-27 | DOI: https://doi.org/10.2478/s13540-013-0005-z

Abstract

We study three types of generalized partial fractional order operators. An extension of Green’s theorem, by considering partial fractional derivatives with more general kernels, is proved. New results are obtained, even in the particular case when the generalized operators are reduced to the standard partial fractional derivatives and fractional integrals in the sense of Riemann-Liouville or Caputo.

MSC: Primary 26B20; Secondary 35R11

Keywords: fractional calculus; generalized operators; Green’s theorem

  • [1] O.P. Agrawal, Generalized variational problems and Euler-Lagrange equations. Comput. Math. Appl. 59, No 5 (2010), 1852–1864. http://dx.doi.org/10.1016/j.camwa.2009.08.029CrossrefGoogle Scholar

  • [2] O.P. Agrawal, Some generalized fractional calculus operators and their applications in integral equations. Fract. Calc. Appl. Anal. 15, No 4 (2012), 700–711; DOI:10.2478/s13540-012-0047-7; at http://link.springer.com/article/10.2478/s13540-012-0047-7. CrossrefGoogle Scholar

  • [3] R. Almeida, A.B. Malinowska, D.F.M. Torres, A fractional calculus of variations for multiple integrals with application to vibrating string, J. Math. Phys. 51 (2010), 033503, 12 pp. http://dx.doi.org/10.1063/1.3319559Web of ScienceCrossrefGoogle Scholar

  • [4] D.L. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Second Edition. Springer (1998). Google Scholar

  • [5] J. Cresson, Fractional embedding of differential operators and Lagrangian systems. J. Math. Phys. 48 (2007), 033504, 34 pp. http://dx.doi.org/10.1063/1.2483292CrossrefGoogle Scholar

  • [6] G.G. Emch, C. Liu, The Logic and Thermostatical Physics. Springer-Verlag, New York (2002). http://dx.doi.org/10.1007/978-3-662-04886-3CrossrefGoogle Scholar

  • [7] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam (2006). Google Scholar

  • [8] V. Kiryakova, Generalized Fractional Calculus and Applications. Pitman Research Notes in Mathematics Series, 301, Longman Sci. Tech., Harlow & J. Wiley, N. York (1994). Google Scholar

  • [9] M. Klimek, On Solutions of Linear Fractional Differential Equations of a Variational Type. The Publ. Office of Czenstochowa University of Technology, Czestochowa (2009). Google Scholar

  • [10] A.B. Malinowska, D.F.M. Torres, Introduction to the Fractional Calculus of Variations. Imperial College Press, London & World Scientific Publishing, Singapore (2012). Google Scholar

  • [11] T. Odzijewicz, A.B. Malinowska, D.F.M. Torres, Generalized fractional calculus with applications to the calculus of variations. Comput. Math. Appl. 64, No 10 (2012), 3351–3366. http://dx.doi.org/10.1016/j.camwa.2012.01.073Web of ScienceCrossrefGoogle Scholar

  • [12] T. Odzijewicz, A.B. Malinowska, D.F.M. Torres, Fractional calculus of variations in terms of a generalized fractional integral with applications to physics. Abstr. Appl. Anal. 2012 (2012), 871912, 24 pp. Google Scholar

  • [13] T. Odzijewicz, D.F.M. Torres, Fractional calculus of variations for double integrals. Balkan J. Geom. Appl. 16, No 2 (2011), 102–113. Google Scholar

  • [14] I. Podlubny, Fractional Differential Equations. Ser. Mathematics in Science and Engineering, 198, Academic Press, San Diego, CA (1999). Google Scholar

  • [15] S. Russenschuck, Field Computation for Accelerator Magnets: Analytical and Numerical Methods for Electromagnetic Design and Optimization. Wiley-VCH Verlag GmbH and Co. KGaA, Weinheim (2010). http://dx.doi.org/10.1002/9783527635467CrossrefGoogle Scholar

  • [16] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Gordon and Breach, Yverdon (1993); Transl. and updated from the 1987 Russian original. Google Scholar

  • [17] C.H. Sherman, J.L. Butler, Transducers and Arrays for Underwater Sound. Springer-Verlag (2007). Google Scholar

  • [18] V.E. Tarasov, Fractional vector calculus and fractional Maxwell’s equations. Ann. Physics 323, No 11 (2008), 2756–2778. http://dx.doi.org/10.1016/j.aop.2008.04.005CrossrefGoogle Scholar

About the article

Published Online: 2012-12-27

Published in Print: 2013-03-01


Citation Information: Fractional Calculus and Applied Analysis, Volume 16, Issue 1, Pages 64–75, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.2478/s13540-013-0005-z.

Export Citation

© 2013 Diogenes Co., Sofia. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Sabrina Roscani and Domingo Tarzia
Fractional Calculus and Applied Analysis, 2018, Volume 21, Number 4, Page 901
[2]
Michał Szymczyk, Marcin Nowak, and Wojciech Sumelka
Symmetry, 2018, Volume 10, Number 7, Page 282
[3]
Wojciech Sumelka
Mechanics Research Communications, 2017
[4]
Wojciech Sumelka and Marcin Nowak
Mechanics of Materials, 2017
[5]
T. Odzijewicz, A. B. Malinowska, and D. F. M. Torres
The European Physical Journal Special Topics, 2013, Volume 222, Number 8, Page 1813
[6]
Tomasz Blaszczyk and Mariusz Ciesielski
Applied Mathematics and Computation, 2015, Volume 257, Page 428
[7]
Om P. Agrawal and Yufeng Xu
Communications in Nonlinear Science and Numerical Simulation, 2015, Volume 23, Number 1-3, Page 129
[8]
Tatiana Odzijewicz
Discrete and Continuous Dynamical Systems - Series B, 2014, Volume 19, Number 8, Page 2617
[9]
Giorgio S. Taverna and Delfim F.M. Torres
Mathematical Methods in the Applied Sciences, 2015, Volume 38, Number 9, Page 1808
[10]
Tomasz Blaszczyk and Mariusz Ciesielski
Fractional Calculus and Applied Analysis, 2014, Volume 17, Number 2

Comments (0)

Please log in or register to comment.
Log in