A real regularised fractional derivative

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• 作者：Manuel D. Ortigueira (1) mdo@fct.unl.pt
  Richard L. Magin (2) rmagin@uic.edu
  Juan J. Trujillo (3) jtrujill@ullmat.es
  M. Pilar Velasco (4) mvcebrian@mat.ucm.es
• 关键词：Fractional derivative – Regularised – Grünwald–Letnikov – Riemann–Liouville – Caputo
• 刊名：Signal, Image and Video Processing
• 出版年：2012
• 出版时间：September 2012
• 年：2012
• 卷：6
• 期：3
• 页码：351-358
• 全文大小：195.8 KB
• 参考文献：1. Chaudhry M.A., Zubair S.M.: On a Class of Incomplete Gamma Functions with Applications. Chapman Hall/CRC, Boca Raton (2002)
  2. Gel’fand I.M., Shilov G.E.: Generalised Functions. Academic Press, New York (1964)
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  4. Kilbas A.A., Srivastava H.M., Trujillo J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
  5. Magin R., Ortigueira M.D., Podlubny I., Trujillo J.: On the fractional signals and systems. Sig. Process. 91, 350–371 (2011)
  6. Ortigueira M.D., Coito F.J.V.: From differences to differintegrations. Fract. Calc. Appl. Anal. 7(4), 459–471 (2004)
  7. Ortigueira M.D., Tenreiro-Machado J.A., Sáda Costa J.: Which differintegration?. IEE Proc. Vis. Image Sig. Process. 152(6), 846–850 (2005)
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  11. Ortigueira M.D.: Fractional Calculus for Scientists and Engineers. Springer, New York (2011)
  12. Podlubny I.: Fractional Differential Equations. Academic Press, San Diego (1999)
  13. Samko S.G., Kilbas A.A., Marichev O.I.: Fractional Integrals and Derivatives—Theory and Applications. Gordon and Breach Science Publishers, New York (1987)
• 作者单位：1. UNINOVA and Department of Electrical Engineering, Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa, Campus da FCT da UNL, Quinta da Torre, 2829-516 Monte da Caparica, Portugal2. Bioengineering Department, University of Illinois at Chicago, 851 South Morgan, Room 212, Chicago, IL 60607-7052, USA3. Departamento de Análisis Matemático, University of La Laguna, 38271 La Laguna, Tenerife, Spain4. Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
• ISSN：1863-1711

文摘

A real regularised integral formulation of the fractional derivative is obtained from the generalised Grünwald–Letnikov derivative without using the Cauchy derivative. This new approach is based on the properties of the Mellin transform. The usual Riemann–Liouville and Caputo derivatives are expressed in a similar way emphasising their regularising capabilities. Some examples involving the Heaviside unit step function are presented in the last section of the paper.