Applications of Mathematics

, Volume 62, Issue 1, pp 15–36

# Convergence of the matrix transformation method for the finite difference approximation of fractional order diffusion problems

Article

## Abstract

Numerical solution of fractional order diffusion problems with homogeneous Dirichlet boundary conditions is investigated on a square domain. An appropriate extension is applied to have a well-posed problem on R2 and the solution on the square is regarded as a localization. For the numerical approximation a finite difference method is applied combined with the matrix transformation method. Here the discrete fractional Laplacian is approximated with a matrix power instead of computing the complicated approximations of fractional order derivatives. The spatial convergence of this method is proved and demonstrated by some numerical experiments.

## Keywords

fractional diffusion problem finite differences matrix transformation method

## MSC 2010

35R11 65M06 65M12

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## References

1. [1]
N. Abatangelo, L. Dupaigne: Nonhomogeneous boundary conditions for the spectral fractional Laplacian. To appear in Ann. Inst. Henri Poincaré, Anal. Non. Linéaire (2016).Google Scholar
2. [2]
A. Bátkai, P. Csomós, B. Farkas: Semigroups for Numerical Analysis. Internet-Seminar Manuscript, 2012, https://isem-mathematik.uibk.ac.at.Google Scholar
3. [3]
D. A. Benson, S. W. Wheatcraft, M. M. Meerschaert: Application of a fractional advection-dispersion equation. Water Resour. Res. 36 (2000), 1403–1412.
4. [4]
C. Canuto, A. Quarteroni: Spectral and pseudo-spectral methods for parabolic problems with non periodic boundary conditions. Calcolo 18 (1981), 197–217.
5. [5]
E. Di Nezza, G. Palatucci, E. Valdinoci: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012), 521–573.
6. [6]
Q. Du, M. Gunzburger, R. B. Lehoucq, K. Zhou: Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 54 (2012), 667–696.
7. [7]
Q. Du, M. Gunzburger, R. B. Lehoucq, K. Zhou: A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws. Math. Models Methods Appl. Sci. 23 (2013), 493–540.
8. [8]
A. M. Edwards, R. A. Phillips, N. W. Watkins, M. P. Freeman, E. J. Murphy, V. Afanasyev, S. V. Buldyrev, M. G. E. da Luz, E. P. Raposo, H. E. Stanley, G. M. Viswanathan: Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer. Nature 449 (2007), 1044–1048.
9. [9]
K.-J. Engel, R. Nagel: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics 194, Springer, Berlin, 2000.
10. [10]
L. B. Feng, P. Zhuang, F. Liu, I. Turner: Stability and convergence of a new finite volume method for a two-sided space-fractional diffusion equation. Appl. Math. Comput. 257 (2015), 52–65.
11. [11]
I. S. Gradshteyn, I. M. Ryzhik: Table of Integrals, Series, and Products. Academic Press, San Diego, 2000.
12. [12]
M. Ilic, F. Liu, I. Turner, V. Anh: Numerical approximation of a fractional-in-space diffusion equation. I. Fract. Calc. Appl. Anal. 8 (2005), 323–341.
13. [13]
M. Ilic, F. Liu, I. Turner, V. Anh: Numerical approximation of a fractional-in-space diffusion equation. II. With nonhomogeneous boundary conditions. Fract. Calc. Appl. Anal. 9 (2006), 333–349.
14. [14]
M. Ilić, I. W. Turner, D. P. Simpson: A restarted Lanczos approximation to functions of a symmetric matrix. IMA J. Numer. Anal. 30 (2010), 1044–1061.
15. [15]
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204, Elsevier, Amsterdam, 2006.
16. [16]
C. Li, Z. Zhao, Y. Chen: Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Comput. Math. Appl. 62 (2011), 855–875.
17. [17]
F. Liu, P. Zhuang, V. Anh, I. Turner: A fractional-order implicit difference approximation for the space-time fractional diffusion equation. ANZIAM J. 47 (2005), C48–C68.
18. [18]
F. Mainardi, Y. Luchko, G. Pagnini: The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4 (2001), 153–192.
19. [19]
M. M. Meerschaert, C. Tadjeran: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172 (2004), 65–77.
20. [20]
T. Michelitsch, G. Maugin, A. Nowakowski, F. Nicolleau, M. Rahman: The fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion. Fract. Calc. Appl. Anal. 16 (2013), 827–859.
21. [21]
R. H. Nochetto, E. Otárola, A. J. Salgado: A PDE approach to fractional diffusion in general domains: a priori error analysis. Found. Comput. Math. 15 (2015), 733–791.
22. [22]
J. E. Pasciak: Spectral and pseudo spectral methods for advection equations. Math. Comput. 35 (1980), 1081–1092.
23. [23]
I. Podlubny: Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications. Mathematics in Science and Engineering 198, Academic Press, San Diego, 1999.
24. [24]
S. G. Samko, A. A. Kilbas, O. I. Marichev: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York, 1993.
25. [25]
B. J. Szekeres, F. Izsák: A finite difference method for fractional diffusion equations with Neumann boundary conditions. Open Math. (electronic only) 13 (2015), 581–600.
26. [26]
B. J. Szekeres, F. Izsák: Finite element approximation of fractional order elliptic boundary value problems. J. Comput. Appl. Math. 292 (2016), 553–561.
27. [27]
C. Tadjeran, M. M. Meerschaert, H.-P. Scheffler: A second-order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213 (2006), 205–213.
28. [28]
W. Tian, H. Zhou, W. Deng: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84 (2015), 1703–1727.
29. [29]
Q. Yang, I. Turner, T. Moroney, F. Liu: A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction-diffusion equations. Appl. Math. Model. 38 (2014), 3755–3762.
30. [30]
H. Zhou, W. Tian, W. Deng: Quasi-compact finite difference schemes for space fractional diffusion equations. J. Sci. Comput. 56 (2013), 45–66.