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High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II)

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Published/Copyright: May 23, 2015
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 18 Issue 3

Abstract

In this paper, we first establish a high-order numerical algorithm for α-th (0 < α < 1) order Caputo derivative of a given function f(t), where the convergence rate is (4 − α)-th order. Then by using this new formula, an improved difference scheme with high order accuracy in time to solve Caputo-type fractional advection-diffusion equation with Dirichlet boundary conditions is constructed. Finally, numerical examples are carried out to confirm the efficiency of the constructed algorithm.

Keywords : Caputo derivative; advection-diffusion equation; difference scheme; stability; convergence

References

[1] A. A. Alikhanov, A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280 (2015), 424-438.Search in Google Scholar

[2] J. X. Cao, C. P. Li, Finite difference scheme for the time-space fractional diffusion equations. Centr. Eur. J. Phys. 11, No 10 (2013), 1440-1456.Search in Google Scholar

[3] C. Chen, F. Liu, I. Turner, V. Anh, A Fourier method for the fractional diffusion equation describing sub-diffusion. J. Comput. Phys. 227, No 2 (2007), 886-897.Search in Google Scholar

[4] J. X. Cao, C. P. Li, Y. Q. Chen, Compact differencemethod for solving the fractional reaction-subdiffusion equation with neumann boundary value condition. Int. J. Comput. Math. 92, No 1 (2015), 167-180.Search in Google Scholar

[5] G. Gao, Z. Sun, H. Zhang, A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259 (2014), 33-50.Search in Google Scholar

[6] B. Jin, R. Lazarov, Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal.; First Publ. online: Jan. 24, 2015; doi: 10.1093/imanum/dru063, 2015.10.1093/imanum/dru063Search in Google Scholar

[7] I. Karatay, N. Kale, S. R. Bayramoglu, A new difference scheme for time fractional heat equations based on the Crank-Nicholson method. Fract. Calc. Appl. Anal. 16, No 4 (2013), 892-910; DOI: 10.2478/s13540-013-0055-2; http://www.degruyter.com/view/j/fca.2013.16.issue-4/s13540-013-0055-2/s13540-013-0055-2.xml; http://link.springer.com/article/10.2478/s13540-013-0055-2.Search in Google Scholar

[8] C. P. Li, R. F. Wu, H. F. Ding, High-order approximation to Caputo derivative and Caputo-type advection-diffusion equation (I). Commun. Appl. Ind. Math. 7, No 1 (2015), In press.Search in Google Scholar

[9] C. Lubich, Discretized fractional calculus. SIAM J. Math. Anal. 17, No 3 (1986), 704-719.Search in Google Scholar

[10] C. P. Li, A. Chen, J. J. Ye, Numerical approaches to fractional calculus and fractional ordinary differential equation. J. Comput. Phys. 230, No 9 (2011), 3352-3368.Search in Google Scholar

[11] K. B. Oldham, J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, New York, 2006.Search in Google Scholar

[12] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego, 1999.Search in Google Scholar

[13] I. Podlubny, A. Chechkin, T. Skovranek, Y. Chen, B. M. Vinagre Jara, Matrix approach to discrete fractional calculus II: Partial fractional differential equations. J. Comput. Phys. 228, No 8 (2009), 3137-3153.Search in Google Scholar

[14] E. Sousa, Numerical approximations for fractional diffusion equations via splines. Comput. Math. Appl. 62, No 3 (2011), 938-944.Search in Google Scholar

[15] E. Sousa, How to approximate the fractional derivative of order 1 < α ≤ 2. Int. J. Bifurcation Chaos. 22, No 4 (2012).10.1142/S0218127412500757Search in Google Scholar

[16] F. H. Zeng, C. P. Li, F. W. Liu, I. Turner, The use of finite difference/ element approaches for solving the time-fractional subdiffusion equation. SIAM J. Sci. Comput. 35, No 6 (2013), A2976-A3000.10.1137/130910865Search in Google Scholar

[17] F. H. Zeng, C. P. Li, F. Liu, I. Turner, Numerical algorithms for timefractional subdiffusion equation with second-order accuracy. SIAM. J. Sci. Comput. 37, No 1 (2015), A55-A78. 10.1137/14096390XSearch in Google Scholar

Received: 2014-12-1
Published Online: 2015-5-23
Published in Print: 2015-6-1

© Diogenes Co., Sofia

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  15. High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II)
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https://doi.org/10.1515/fca-2015-0045
Keywords for this article
Caputo derivative; advection-diffusion equation; difference scheme; stability; convergence

Articles in the same Issue

  1. Frontmatter
  2. Fcaa Related News, Events And Books (Fcaa-Volume 18-3-2015)
  3. Decay solutions for a class of fractional differential variational inequalities
  4. A biomathematical view on the fractional dynamics of cellulose degradation
  5. The spreading property for a prey-predator reaction-diffusion system with fractional diffusion
  6. Fractional variation of Hölderian functions
  7. Periodic disturbance rejection for fractional-order dynamical systems
  8. Successive approximation: A survey on stable manifold of fractional differential systems
  9. When do fractional differential equations have solutions that are bounded by the Mittag--Leffler function ?
  10. On explicit stability conditions for a linear fractional difference system
  11. Fractional differential inclusions in the Almgren sense
  12. Time-optimal control of fractional-order linear systems
  13. Analytical solutions for the multi-term time-space fractional reaction-diffusion equations on an infinite domain
  14. Nonexistence results for a class of evolution equations in the Heisenberg group
  15. High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II)
  16. Dyadic nonlocal diffusions in metric measure spaces
  17. Fractional derivative anomalous diffusion equation modeling prime number distribution
  18. Time-fractional diffusion equation in the fractional Sobolev spaces
  19. Continuous time random walk models associated with distributed order diffusion equations
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