In this paper we further develop Podlubny’s matrix approach to discretization of integrals and derivatives of arbitrary real order. Numerical integration and differentiation on a set of non-equidistant nodes is described and illustrated by several examples of numerical solution of fractional differential equations. In this paper, for the first time, we present a variable step length approach that we call “the method of large steps”, since it is applied in combination with the matrix approach for each “large step”. This new method is also illustrated by an example. The presented approach allows fractional order differentiation and integration of non-uniformly sampled signals, and opens the way to development of variable and adaptive step length techniques for fractional differential equations.
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ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
August 28–31, 2011
Washington, DC, USA
Conference Sponsors:
- Design Engineering Division and Computers and Information in Engineering Division
ISBN:
978-0-7918-5480-8
PROCEEDINGS PAPER
Discrete Fractional Calculus: Non-Equidistant Grids and Variable Step Length
Igor Podlubny,
Igor Podlubny
Technical University of Kosice, Kosice, Slovakia
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Tomas Skovranek,
Tomas Skovranek
Technical University of Kosice, Kosice, Slovakia
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Ivo Petras,
Ivo Petras
Technical University of Kosice, Kosice, Slovakia
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Viktor V. Verbickij,
Viktor V. Verbickij
Odessa National University, Odessa, Ukraine
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YangQuan Chen,
YangQuan Chen
Utah State University, Logan, UT
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Blas M. Vinagre Jara
Blas M. Vinagre Jara
University of Extremadura, Badajoz, Spain
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Igor Podlubny
Technical University of Kosice, Kosice, Slovakia
Tomas Skovranek
Technical University of Kosice, Kosice, Slovakia
Ivo Petras
Technical University of Kosice, Kosice, Slovakia
Viktor V. Verbickij
Odessa National University, Odessa, Ukraine
YangQuan Chen
Utah State University, Logan, UT
Blas M. Vinagre Jara
University of Extremadura, Badajoz, Spain
Paper No:
DETC2011-47623, pp. 211-217; 7 pages
Published Online:
June 12, 2012
Citation
Podlubny, I, Skovranek, T, Petras, I, Verbickij, VV, Chen, Y, & Vinagre Jara, BM. "Discrete Fractional Calculus: Non-Equidistant Grids and Variable Step Length." Proceedings of the ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Volume 3: 2011 ASME/IEEE International Conference on Mechatronic and Embedded Systems and Applications, Parts A and B. Washington, DC, USA. August 28–31, 2011. pp. 211-217. ASME. https://doi.org/10.1115/DETC2011-47623
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