Startseite An adaptive memory method for accurate and efficient computation of the Caputo fractional derivative
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

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

  • Daegeun Yoon und Donghyun You EMAIL logo
Veröffentlicht/Copyright: 28. Oktober 2021
Veröffentlichen auch Sie bei De Gruyter Brill
Informationen für Autor*innen Erkunden Sie dieses Fachgebiet
Fractional Calculus and Applied Analysis
Aus der Zeitschrift Fractional Calculus and Applied Analysis Band 24 Heft 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).

References

[1] D. Craiem, R.L. Magin, Fractional order models of viscoelasticity as an alternative in the analysis of red blood cell (RBC) membrane mechanics. Physical Biology 7, No 1 (2010), # 13001.10.1088/1478-3975/7/1/013001Suche in Google Scholar PubMed PubMed Central

[2] K. Diethelm, A.D. Freed, An efficient algorithm for the evaluation of convolution integrals. Computers and Mathematics with Applications 51, No 1 (2006), 51–72.10.1016/j.camwa.2005.07.010Suche in Google Scholar

[3] N.J. Ford, A.C. Simpson, The numerical solution of fractional differential equations: Speed versus accuracy. Numerical Algorithms 26, No 4 (2001), 333–346.10.1023/A:1016601312158Suche in Google Scholar

[4] T.A.M. Langlands, B.I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation. Journal of Computational Physics 205, No 2 (2005), 719–736.10.1016/j.jcp.2004.11.025Suche in Google Scholar

[5] C. Li, F. Zeng, Numerical Methods for Fractional Calculus. CRC Press, Boca Raton (2015).10.1201/b18503Suche in Google Scholar

[6] C. Li, M. Cai, Theory and Numerical Approximations of Fractional Integrals and Derivatives. SIAM, Philadelphia, PA (2019).10.1137/1.9781611975888Suche in Google Scholar

[7] M. Lopez-Fernandez, C. Lubich, A. Schadle, Adaptive, fast, and oblivious convolution in evolution equations with memory. SIAM Journal on Scientific Computing 30, No 2 (2008), 1015–1037.10.1137/060674168Suche in Google Scholar

[8] A. Loverro, Fractional Calculus: History, Definitions and Applications for Engineer. University of Notre Dame, Notre Dame (2004).Suche in Google Scholar

[9] C. Lubich, Convolution quadrature and discretized operational calculus. I. Numerische Mathematik 52, No 2 (1988), 129–145.10.1007/BF01398686Suche in Google Scholar

[10] C. Lubich, A. Schadle, Fast convolution for nonreflecting boundary conditions. SIAM Journal on Scientific Computing 24, No 1 (2002), 161–182.10.1137/S1064827501388741Suche in Google Scholar

[11] C.L. MacDonald, N. Bhattacharya, B.P. Sprouse, G.A. Silva, Efficient computation of the Grünwald-Letnikov fractional diffusion derivative using adaptive time step memory. Journal of Computational Physics 297, No 15 (2015), 221–236.10.1016/j.jcp.2015.04.048Suche in Google Scholar

[12] K. Oldham, J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Elsevier Science, New York (2006).Suche in Google Scholar

[13] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1999).Suche in Google Scholar

[14] N. Sebaa, Z.E.A. Fellah, W. Lauriks, C. Depollier, Application of fractional calculus to ultrasonic wave propagation in human cancellous bone. Signal Processing 86, No 10 (2006), 2668–2677.10.1016/j.sigpro.2006.02.015Suche in Google Scholar

[15] C. Selhuber-Unkel, P. Yde, K. Berg-Sorensen, L.B. Oddershede, Variety in intracellular diffusion during the cell cycle. Physical Biology 6, No 2 (2009), 025015.10.1088/1478-3975/6/2/025015Suche in Google Scholar PubMed

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

© 2021 Diogenes Co., Sofia

Artikel in diesem Heft

  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
Schlagwörter für diesen Artikel
Caputo fractional derivative; fractional differential equations; adaptive memory method

Artikel in diesem Heft

  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
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 16.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/fca-2021-0058/html?lang=de
Button zum nach oben scrollen