Popolizio, Marina:
Numerical Approximation of Matrix Functions for Fractional Differential Equations
Bollettino dell'Unione Matematica Italiana Serie 9 6 (2013), fasc. n.3, p. 793-815, (English)
pdf (382 Kb), djvu (224 Kb). | MR 3202856
Sunto
In questo lavoro si presentano delle connessioni rilevanti tra le funzioni di matrice e la soluzione di equazioni differenziali di ordine frazionario. Questo nesso è stato notato solo recentemente ed ora riscuote notevole interesse. Si presenta qui una rassegna dei fondamenti del calcolo frazionario e della teoria dell'approssimazione di funzioni di matrice; si mostrano inoltre i contributi che, insieme ad i miei coautori, abbiamo recentemente elaborato su questo argomento [13, 14, 15, 16, 32].
Referenze Bibliografiche
[1]
BENZI M. and
BERTACCINI D.,
Block Preconditioning of Real-Valued Iterative Algorithms for Complex Linear Systems,
IMA Journal of Numerical Analysis,
28, 598-618 (
2008). |
fulltext (doi) |
MR 2433214 |
Zbl 1145.65022[2]
BERTACCINI D.,
Efficient preconditioning for sequences of parametric complex symmetric linear systems,
Electronic Transactions on Numerical Analysis,
18, 49-64 (
2004). |
fulltext EuDML |
MR 2083294 |
Zbl 1066.65048[3] BERTACCINI D. and POPOLIZIO M., Adaptive updating techniques for the approximation of functions of large matrices, preprint (2012).
[4]
CODY W. J.,
MEINARDUS G. and
VARGA R. S.,
Chebyshev rational approximations to $e^{-x}$ in $[0, +\infty)$ and applications to heat-conduction problems,
J. Approximation Theory,
2, 50-65 (
1969). |
fulltext (doi) |
MR 245224 |
Zbl 0187.11602[5]
DEL BUONO N.,
LOPEZ L. and
PELUSO R.,
Computation of the exponential of large sparse skew-symmetric matrices,
SIAM J. Sci. Comput.,
27 (1), 278-293 (
2005). |
fulltext (doi) |
MR 2201184 |
Zbl 1108.65037[6]
DEL BUONO N.,
LOPEZ L. and
POLITI T.,
Computation of functions of Hamiltonian and skew-symmetric matrices,
Math. Comput. Simulation,
79 (4), 1284-1297 (
2008). |
fulltext (doi) |
MR 2487801 |
Zbl 1162.65338[9]
GARRAPPA R.,
On linear stability of predictor-corrector algorithms for fractional differential equations,
International Journal of Computer Mathematics,
87 (10), 2281-2290 (
2010). |
fulltext (doi) |
MR 2680147 |
Zbl 1206.65197[10]
GARRAPPA R.,
On some generalizations of the implicit Euler method for discontinuous fractional differential equations,
Mathematics and Computers in Simulation, ,
95 (
2014), 213-228. |
fulltext (doi) |
MR 3127766[11]
GARRAPPA R.,
Stability-preserving high-order methods for multiterm fractional differential equations,
International Journal of Bifurcation and Chaos,
22 (4) (
2012), 1-13. |
fulltext (doi) |
MR 2926049 |
Zbl 1258.34011[12]
GARRAPPA R.,
A family of Adams exponential integrators for fractional linear systems,
Computers and Mathematics with Applications, (
2013), in print, doi: http://dx.doi.org/10.1016/j.camwa.2013.01.022 |
fulltext (doi) |
MR 3089380 |
Zbl 1350.65078[13]
GARRAPPA R. and
POPOLIZIO M.,
On the use of matrix functions for fractional partial differential equations,
Math. Comput. Simulation,
81 (5), (
2011), 1045-1056. |
fulltext (doi) |
MR 2769818 |
Zbl 1210.65162[14]
GARRAPPA R. and
POPOLIZIO M.,
Generalized exponential time differencing methods for fractional order problems,
Comput. Math. Appl.,
62(3) (
2011), 876-890 |
fulltext (doi) |
MR 2824677 |
Zbl 1228.65235[15]
GARRAPPA R. and
POPOLIZIO M.,
On accurate product integration rules for linear fractional differential equations,
J. Comput. Appl. Math.,
235 (5) (
2011), 1085-1097. |
fulltext (doi) |
MR 2728050 |
Zbl 1206.65176[16]
GARRAPPA R. and
POPOLIZIO M.,
Evaluation of generalized Mittag-Leffler functions on the real line,
Adv. Comput. Math.,
39 (1) (
2013), 205-225. |
fulltext (doi) |
MR 3068601 |
Zbl 1272.33020[17]
GOLUB G.H. and
VAN LOAN C.F.,
Matrix Computations,
Johns Hopkins Studies in the Mathematical Sciences (
1996). |
MR 1417720[18]
GORENFLO R. and
MAINARDI F.,
Some recent advances in theory and simulation of fractional diffusion processes.,
J. Comput. Appl. Math.,
229 (2) (
2009), 400-415. |
fulltext (doi) |
MR 2527894 |
Zbl 1166.45004[20]
HOCHBRUCK M. and
LUBICH C.,
On Krylov subspace approximations to the matrix exponential operator,
SIAM Journal on Numerical Analysis,
34 (
1987), 1911-1925. |
fulltext (doi) |
MR 1472203 |
Zbl 0888.65032[22]
KILBAS A. A.,
SRIVASTAVA H. M. and
TRUJILLO J. J.,
Theory and applications of fractional differential equations, vol.
204 of
North-Holland Mathematics Studies,
Elsevier Science B.V., Amsterdam (
2006). |
MR 2218073 |
Zbl 1092.45003[24]
LOPEZ L. and
SIMONCINI V.,
Analysis of projection methods for rational function approximation to the matrix exponential,
SIAM J. Numer. Anal.,
44 (2) (
2006), 613-635. |
fulltext (doi) |
MR 2218962 |
Zbl 1158.65031[25]
LOPEZ L. and
SIMONCINI V.,
Preserving geometric properties of the exponential matrix by block Krylov subspace methods,
BIT,
46 (4) (
2006), 813-830. |
fulltext (doi) |
MR 2285209 |
Zbl 1107.65039[26]
MAGNUS A. P.,
Asymptotics and super asymptotics for best rational approximation error norms to the exponential function (the ``1=9'' problem) by the Carathéodory-Fejér method,
Nonlinear numerical methods and rational approximation,
296 (II) (
1994), 173-185. |
MR 1307197[28]
MOLER C. and
VAN LOAN C.,
Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later,
SIAM Review,
45 (1) (
2003), 3-49. |
fulltext (doi) |
MR 1981253 |
Zbl 1030.65029[29]
MORET I.,
Rational Lanczos approximations to the matrix square root and related functions,
Numerical Linear Algebra with Applications,
16 (
2009), 431-445. |
fulltext (doi) |
MR 2522957 |
Zbl 1224.65124[31]
MORET I. and
NOVATI P.,
On the convergence of Krylov subspace methods for matrix Mittag-Leffler functions,
SIAM Journal on Numerical Analysis,
49 (
2011), 2144-2164 |
fulltext (doi) |
MR 2861713 |
Zbl 1244.65065[32]
MORET I. and
POPOLIZIO M.,
The restarted shift-and-invert Krylov method for matrix functions,
Numerical Linear Algebra with Applications,
21 (
2014), 68-80. |
fulltext (doi) |
MR 3150610 |
Zbl 1324.65079[33]
PETRUSHEV P.P. and
POPOV V.A.,
Rational approximation of real function,
Cambridge University Press, Cambridge (
1987). |
MR 940242 |
Zbl 0644.41010[34]
PODLUBNY I.,
Fractional differential equations,
Mathematics in Science and Engineering,
Academic Press Inc., San Diego, CA (
1999). |
MR 1658022 |
Zbl 0924.34008[35] PODLUBNY I. and KACENAK M., Matlab implementation of the Mittag-Leffler function, available online: http://www.mathworks.com (2005).
[36]
POLITI T. and
POPOLIZIO M.,
Schur Decomposition Methods for the Computation of Rational Matrix Functions,
Computational science-ICCS 2006. Part IV,
Springer, 3994, 708-715 (
2006). |
Zbl 1157.65344[37]
POPOLIZIO M.,
Tecniche di accelerazione per approssimare l'esponenziale di matrice,
La Matematica nella Società e nella Cultura, Rivista dell'Unione Matematica Italiana, Serie I, Vol.
II, Agosto (
2009), 275-278. |
fulltext bdim |
fulltext EuDML |
MR 3558968[38]
POPOLIZIO M. and
SIMONCINI V.,
Acceleration Techniques for Approximating the Matrix Exponential Operator,
SIAM J. Matrix Analysis and Appl.,
30 (
2008), 657-683. |
fulltext (doi) |
MR 2421464 |
Zbl 1168.65021[39]
SAAD Y.,
Analysis of some Krylov subspace approximations to the matrix exponential operator,
SIAM Journal on Numerical Analysis,
29 (
1992), 209-228. |
fulltext (doi) |
MR 1149094 |
Zbl 0749.65030[41]
TREFETHEN L. N.,
Circularity of the error curve and sharpness of the CF method in complex Chebyshev approximation,
SIAM J. Numer. Anal.,
20 (6) (
1983), 1258-1263. |
fulltext (doi) |
MR 723844 |
Zbl 0551.41042[43]
VAN DEN ESHOF J. and
HOCHBRUCK M.,
Preconditioning Lanczos approximations to the matrix exponential,
SIAM Journal on Scientific Computing,
27 (
2006), 1438-1457. |
fulltext (doi) |
MR 2199756 |
Zbl 1105.65051[44]
WEIDEMAN J. A. C. and
TREFETHEN L. N.,
Parabolic and hyperbolic contours for computing the Bromwich integral,
Mathematics of Computation,
78 (
2007), 1341-1358. |
fulltext (doi) |
MR 2299777 |
Zbl 1113.65119
Con il contributo del Ministero per i Beni e le Attività Culturali