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Fractional Calculus and Applied Analysis

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Volume 16, Issue 1


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
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/ Agnieszka Malinowska / Delfim Torres
  • Center for Research and Development in Mathematics and Applications Department of Mathematics, University of Aveiro, 3810-193, Aveiro, Portugal
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Published Online: 2012-12-27 | DOI: https://doi.org/10.2478/s13540-013-0005-z


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.

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

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