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A piecewise memory principle for fractional derivatives

  • Chunye Gong EMAIL logo , Weimin Bao and Jie Liu
Published/Copyright: August 8, 2017
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 20 Issue 4

Abstract

In the numerical approximation of fractional order derivatives, the crucial point is to balance the computing complexity and the computing accuracy. We proposed a piecewise memory principle for fractional derivatives, in which the past history is divided into several segments instead of discarded. The piecewise approximation is performed on each segment. Error estimation of piecewise memory principle is analyzed also. Numerical examples show that the contradiction of computing accuracy and complexity is effectively relaxed and the piecewise memory principle is superior to the existing short, variable and equal-weight memory principles. The impacts of the memory length, step size and segment size are also discussed.

MSC 2010: Primary 26A33; Secondary 33E12; 34A08; 34K07; 35R11
Key Words and Phrases: fractional calculus; fractional derivative; piecewise memory principle; short memory principle; variable memory principle; equal-weight memory principle

Acknowledgements

This research work is supported in part by the National Key Research and Development Program of China under Grant no. 2017YFB0202100, 2016YFB0200400, National Natural Science Foundation of China (NSFC) under Grant no. 61402039, Major research plan of NSFC under Grants no. 91430218, 91530324, China Postdoctoral Science Foundation (CPSF) under Grant no. 2014M562570, Special Financial Grant from CPSF under Grant no. 2015T81127, Science and Technology on reactor system design technology laboratory under Grant no. SQ-KFKT-02-2016-04.

We would like to thank the anonymous reviewers for their helpful comments as well.

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Received: 2015-7-15
Revised: 2017-6-25
Published Online: 2017-8-8
Published in Print: 2017-8-28

© 2017 Diogenes Co., Sofia

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  3. FCAA related news, events and books (FCAA–volume 20–4–2017)
  4. Research Paper
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https://doi.org/10.1515/fca-2017-0052
Keywords for this article
fractional calculus; fractional derivative; piecewise memory principle; short memory principle; variable memory principle; equal-weight memory principle

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–volume 20–4–2017)
  4. Research Paper
  5. Perturbation methods for nonlocal Kirchhoff–type problems
  6. Research Paper
  7. On infinite order differential operators in fractional viscoelasticity
  8. Research Paper
  9. Fundamental solution of the multi-dimensional time fractional telegraph equation
  10. Research Paper
  11. Robustness and convergence of fractional systems and their applications to adaptive schemes
  12. Research Paper
  13. Stability analysis of linear distributed order fractional systems with distributed delays
  14. Research Paper
  15. Fractional sobolev spaces and functions of bounded variation of one variable
  16. Research Paper
  17. Approximate controllability for fractional differential equations of sobolev type via properties on resolvent operators
  18. Research Paper
  19. On moduli of smoothness and averaged differences of fractional order
  20. Research Paper
  21. A piecewise memory principle for fractional derivatives
  22. Research Paper
  23. A computational approach for the solution of a class of variable-order fractional integro-differential equations with weakly singular kernels
  24. Shopt Paper
  25. Analytic approximate solutions for a class of variable order fractional differential equations using the polynomial least squares method
  26. Corrigendum
  27. Corrigendum: Fractional integral on martingale hardy spaces with variable exponents
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