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Linked References

1. F. Lorenzo and T.T. Hartley, “Initialization, conceptualization, and application in the generalized fractional calculus,” NASA Center for Aerospace Information, Hanover, Md, USA, 1998.
2. R. Metzler and J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach,” Physics Reports, vol. 339, no. 1, pp. 1–77, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
3. R. Metzler and J. Klafter, “The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics,” Journal of Physics A, vol. 37, no. 31, pp. R161–R208, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
4. D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, “Application of a fractional advectiondispersion equation,” Water Resources Research, vol. 36, no. 6, pp. 1403–1412, 2000.
5. D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, “The fractional-order governing equation of levy motion,” Water Resources Research, vol. 36, no. 6, pp. 1413–1423, 2000.
6. M. M. Meerschaert, J. Mortensen, and S. W. Wheatcraft, “Fractional vector calculus for fractional advection-dispersion,” Physica A, vol. 367, pp. 181–190, 2006. View at Publisher · View at Google Scholar
7. B. Friedman, Principles and Techniques of Applied Mathematics, John Wiley & Sons, New York, NY, USA, 1966.
8. M. Ilić, F. Liu, I. W. Turner, and V. Anh, “Numerical approximation of a fractional-in-space diffusion equation—II-with nonhomogeneous boundary conditions,” Fractional Calculus & Applied Analysis, vol. 9, no. 4, pp. 333–349, 2006. View at Zentralblatt MATH · View at MathSciNet
9. B. J. West and V. Seshadri, “Linear systems with Lévy fluctuations,” Physica A, vol. 113, no. 1-2, pp. 203–216, 1982. View at Publisher · View at Google Scholar · View at MathSciNet
10. R. Gorenflo and F. Mainardi, “Random walk models for space-fractional diffusion processes,” Fractional Calculus & Applied Analysis, vol. 1, no. 2, pp. 167–191, 1998. View at Zentralblatt MATH · View at MathSciNet
11. R. Gorenflo and F. Mainardi, “Approximation of Lévy-Feller diffusion by random walk,” Journal for Analysis and Its Applications, vol. 18, no. 2, pp. 231–246, 1999. View at Zentralblatt MATH · View at MathSciNet
12. I. P. Gavrilyuk, W. Hackbusch, and B. N. Khoromskij, “Data-sparse approximation to the operator-valued functions of elliptic operator,” Mathematics of Computation, vol. 73, no. 247, pp. 1297–1324, 2004. View at Zentralblatt MATH · View at MathSciNet
13. W. Hackbusch and B. N. Khoromskij, “Low-rank Kronecker-product approximation to multidimensional nonlocal operators—part I. Separable approximation of multi-variate functions,” Computing, vol. 76, no. 3, pp. 177–202, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
14. W. Hackbusch and B. N. Khoromskij, “Low-rank Kronecker-product approximation to multidimensional nonlocal operators—part II. HKT representation of certain operators,” Computing, vol. 76, no. 3, pp. 203–225, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
15. M. Ilić, F. Liu, I. W. Turner, and V. Anh, “Numerical approximation of a fractional-in-space diffusion equation—I,” Fractional Calculus & Applied Analysis, vol. 8, no. 3, pp. 323–341, 2005. View at Zentralblatt MATH · View at MathSciNet
16. G. Gripenberg and I. Norros, “On the prediction of fractional Brownian motion,” Journal of Applied Probability, vol. 33, no. 2, pp. 400–410, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
17. Y. Hu, “Prediction and translation of fractional Brownian motions,” in Stochastics in Finite and Infinite Dimensions, T. Hida, R. L. Karandikar, H. Kunita, B. S. Rajput, S. Watanabe, and J. Xiong, Eds., Trends in Mathematics, pp. 153–171, Birkhäuser, Boston, Mass, USA, 2001. View at MathSciNet
18. G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences, The Johns Hopkins University Press, Baltimore, Md, USA, 3rd edition, 1996. View at Zentralblatt MATH · View at MathSciNet
19. Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, Pa, USA, 2nd edition, 2003. View at Zentralblatt MATH · View at MathSciNet
20. H. A. van der Vorst, Iterative Krylov Methods for Large Linear Systems, vol. 13 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, UK, 2003. View at Zentralblatt MATH · View at MathSciNet
21. N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, Pa, USA, 2008. View at MathSciNet
22. V. Druskin and L. Knizhnerman, “Krylov subspace approximation of eigenpairs and matrix functions in exact and computer arithmetic,” Numerical Linear Algebra with Applications, vol. 2, no. 3, pp. 205–217, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
23. M. Hochbruck and C. Lubich, “On Krylov subspace approximations to the matrix exponential operator,” SIAM Journal on Numerical Analysis, vol. 34, no. 5, pp. 1911–1925, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
24. M. Eiermann and O. G. Ernst, “A restarted Krylov subspace method for the evaluation of matrix functions,” SIAM Journal on Numerical Analysis, vol. 44, no. 6, pp. 2481–2504, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
25. L. Lopez and V. Simoncini, “Analysis of projection methods for rational function approximation to the matrix exponential,” SIAM Journal on Numerical Analysis, vol. 44, no. 2, pp. 613–635, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
26. J. van den Eshof, A. Frommer, T. Lippert, K. Schilling, and H. A. van der Vorst, “Numerical methods for the QCD overlap operator—I. Sign-function and error bounds,” Computer Physics Communications, vol. 146, pp. 203–224, 2002.
27. M. Ilić and I. W. Turner, “Approximating functions of a large sparse positive definite matrix using a spectral splitting method,” The ANZIAM Journal, vol. 46(E), pp. C472–C487, 2005. View at Zentralblatt MATH · View at MathSciNet
28. M. Ilić, I. W. Turner, and A. N. Pettitt, “Bayesian computations and efficient algorithms for computing functions of large, sparse matrices,” The ANZIAM Journal, vol. 45(E), pp. C504–C518, 2004. View at MathSciNet
29. R. B. Lehoucq and D. C. Sorensen, “Deflation techniques for an implicitly restarted Arnoldi iteration,” SIAM Journal on Matrix Analysis and Applications, vol. 17, no. 4, pp. 789–821, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
30. A. Stathopoulos, Y. Saad, and K. Wu, “Dynamic thick restarting of the Davidson, and the implicitly restarted Arnoldi methods,” SIAM Journal on Scientific Computing, vol. 19, no. 1, pp. 227–245, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
31. M. Ilić and I. W. Turner, “Krylov subspaces and the analytic grade,” Numerical Linear Algebra with Applications, vol. 12, no. 1, pp. 55–76, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
32. R. B. Morgan, “A restarted GMRES method augmented with eigenvectors,” SIAM Journal on Matrix Analysis and Applications, vol. 16, no. 4, pp. 1154–1171, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
33. J. Baglama, D. Calvetti, G. H. Golub, and L. Reichel, “Adaptively preconditioned GMRES algorithms,” SIAM Journal on Scientific Computing, vol. 20, no. 1, pp. 243–269, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
34. J. Erhel, K. Burrage, and B. Pohl, “Restarted GMRES preconditioned by deflation,” Journal of Computational and Applied Mathematics, vol. 69, no. 2, pp. 303–318, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
35. R. V. Churchill, Complex Variables and Applications, McGraw-Hill, New York, NY, USA, 2nd edition, 1960. View at Zentralblatt MATH · View at MathSciNet