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An adaptive memory method for accurate and efficient computation of the Caputo fractional derivative

  • Daegeun Yoon and Donghyun You EMAIL logo
Published/Copyright: October 28, 2021
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 24 Issue 5

Abstract

A fractional derivative is a temporally nonlocal operation which is computationally intensive due to inclusion of the accumulated contribution of function values at past times. In order to lessen the computational load while maintaining the accuracy of the fractional derivative, a novel numerical method for the Caputo fractional derivative is proposed. The present adaptive memory method significantly reduces the requirement for computational memory for storing function values at past time points and also significantly improves the accuracy by calculating convolution weights to function values at past time points which can be non-uniformly distributed in time. The superior accuracy of the present method to the accuracy of the previously reported methods is identified by deriving numerical errors analytically. The sub-diffusion process of a time-fractional diffusion equation is simulated to demonstrate the accuracy as well as the computational efficiency of the present method.

Key Words and Phrases: Caputo fractional derivative; fractional differential equations; adaptive memory method
MSC 2010: Primary 26A33; Secondary 34A08; 35R11; 74S40

Appendix A. B(m, α) ⩽ c(α)m−α

For 0 < α < 1, the function B(m, α) in Eq. (4.23) is expressed as follows:

B(m,α)=∑k=0m−1(2m−k)1−α{2(2m−k−1)+α}−(2m−k−1)1−α{2(2m−k)−α}. (A.1)

By letting p = 2m − k, Eq. (A.1) is rewritten as follows:

B(m,α)=∑p=m+12mp1−α{2(p−1)+α}−(p−1)1−α(2p−α). (A.2)

Using the Taylor series expansion, (p − 1)1−α is expanded as follows:

(p−1)1−α=p1−α−(1−α)p−α−12(1−α)αp−α−1−16(1−α)α(α+1)p−α−2−124(1−α)α(α+1)(α+2)p−α−3−O(p−α−4). (A.3)

By substituting Eq. (A.3) into Eq. (A.2), Eq. (A.2) is recast in terms of p as follows:

B(m,α)=16α(1−α)(2−α)∑p=m+12mp−α−1+O(p−α−2). (A.4)

For the summation in Eq. (A.4) we have the following inequality:

∑p=m+12m(2m)−α−1<∑p=m+12mp−α−1<∑p=m+12mm−α−1,(2−α−1)m−α<∑p=m+12mp−α−1<m−α. (A.5)

Therefore, there exists a real constant c(α) such that B(m, α) ⩽ c(α)m−α.

Acknowledgements

This research was supported by the Basic Science Research Program of the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2015R1A2A1A15056086 and NRF-2019K1A3A1A74107685).

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Received: 2020-03-01
Revised: 2021-07-25
Published Online: 2021-10-28
Published in Print: 2021-10-26

© 2021 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books
  4. Survey Paper
  5. Towards a unified theory of fractional and nonlocal vector calculus
  6. Research Paper
  7. An adaptive memory method for accurate and efficient computation of the Caputo fractional derivative
  8. Analysis of solutions of some multi-term fractional Bessel equations
  9. Existence of solutions for the semilinear abstract Cauchy problem of fractional order
  10. Summability of formal solutions for a family of generalized moment integro-differential equations
  11. Analysis and fast approximation of a steady-state spatially-dependent distributed-order space-fractional diffusion equation
  12. Green’s function for the fractional KdV equation on the periodic domain via Mittag–Leffler function
  13. First order plus fractional diffusive delay modeling: Interconnected discrete systems
  14. On a solution of a fractional hyper-Bessel differential equation by means of a multi-index special function
  15. On the decomposition of solutions: From fractional diffusion to fractional Laplacian
  16. Output error MISO system identification using fractional models
  17. Short Paper
  18. Identification of system with distributed-order derivatives
  19. Short note
  20. On the Green function of the killed fractional Laplacian on the periodic domain
https://doi.org/10.1515/fca-2021-0058
Keywords for this article
Caputo fractional derivative; fractional differential equations; adaptive memory method

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books
  4. Survey Paper
  5. Towards a unified theory of fractional and nonlocal vector calculus
  6. Research Paper
  7. An adaptive memory method for accurate and efficient computation of the Caputo fractional derivative
  8. Analysis of solutions of some multi-term fractional Bessel equations
  9. Existence of solutions for the semilinear abstract Cauchy problem of fractional order
  10. Summability of formal solutions for a family of generalized moment integro-differential equations
  11. Analysis and fast approximation of a steady-state spatially-dependent distributed-order space-fractional diffusion equation
  12. Green’s function for the fractional KdV equation on the periodic domain via Mittag–Leffler function
  13. First order plus fractional diffusive delay modeling: Interconnected discrete systems
  14. On a solution of a fractional hyper-Bessel differential equation by means of a multi-index special function
  15. On the decomposition of solutions: From fractional diffusion to fractional Laplacian
  16. Output error MISO system identification using fractional models
  17. Short Paper
  18. Identification of system with distributed-order derivatives
  19. Short note
  20. On the Green function of the killed fractional Laplacian on the periodic domain
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