On accurate product integration rules for linear fractional differential equations

Journal of Computational and Applied Mathematics

Volumn 235, Issue 5, 2011, Pages 1085-1097

  Garrappa, Roberto ^a   Popolizio, Marina ^a  

^a UNIVERSITY OF BARI   (Italy)

Author keywords

Accuracy; Contour integral; Fractional differential equation; Linear problem; Mittag Leffler function; Product integration

Indexed keywords

ACCURACY; CONTOUR INTEGRALS; FRACTIONAL DIFFERENTIAL EQUATIONS; LINEAR PROBLEMS; MITTAG-LEFFLER FUNCTIONS; PRODUCT INTEGRATION;

COMPUTATIONAL COMPLEXITY; DIFFERENTIAL EQUATIONS; ERROR ANALYSIS; FUNCTIONS; INTEGRAL EQUATIONS;

DIFFERENTIATION (CALCULUS);

EID: 77957868556     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2010.07.008     Document Type: Article

References (36)

1. I. Podlubny Fractional Differential Equations Mathematics in Science and Engineering vol. 198 1999 Academic Press Inc. San Diego, CA
2. A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo Theory and Applications of Fractional Differential Equations North-Holland Mathematics Studies vol. 204 2006 Elsevier Science B.V. Amsterdam
3. A. Young Approximate product-integration Proc. Roy. Soc. London Ser. A. 224 1954 552 561
4. K. Diethelm, N.J. Ford, and A.D. Freed Detailed error analysis for a fractional Adams method Numer. Algorithms 36 1 2004 31 52
5. J. Dixon On the order of the error in discretization methods for weakly singular second kind Volterra integral equations with nonsmooth solutions BIT 25 4 1985 624 634
6. K. Diethelm, and A.D. Freed The Frac PECE subroutine for the numerical solution of differential equations of fractional order S. Heinzel, T. Plesser, Forschung und Wissenschaftliches Rechnen 1998 1999 Gessellschaft fr Wissenschaftliche Datenverarbeitung Gttingen 57 71
7. R. Garrappa On linear stability of predictorcorrector algorithms for fractional differential equations Internat. J. Comput. Math. 87 10 2010 2281 2290
8. R. Garrappa A comparison of some explicit methods for fractional differential equations AIP 1048 2008 217 220
9. R.F. Cameron, and S. McKee Product integration methods for second-kind Abel integral equations J. Comput. Appl. Math. 11 1 1984 1 10
10. C. Lubich RungeKutta theory for Volterra and Abel integral equations of the second kind Math. Comp. 41 163 1983 87 102
11. R. Gorenflo, and F. Mainardi Fractional calculus: integral and differential equations of fractional order Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996) CISM Courses and Lectures vol. 378 1997 Springer Vienna 223 276 Reprinted with revisions in: arxiv:0805.3823 http://arxiv.org/abs/0805.3823
12. C.F. Lorenzo, T.T. Hartley, Generalized functions for the fractional calculus, Tech. Rep. TP-1999-209424, NASA 1999.
13. H. Berland, B. Skaflestad, and W.M. Wright EXPINTA MATLAB package for exponential integrators ACM Trans. Math. Softw. 33 1 2007 4
14. J.N. Lyness, and B.W. Ninham Numerical quadrature and asymptotic expansions Math Comp. 21 1967 162 178
15. I.S. Gradshteyn, and I.M. Ryzhik Table of Integrals, Series, and Products 1965 Academic Press New York Fourth edition prepared by Ju. V. Geronimus and M. Ju. Cetlin. Translated from the Russian by Scripta Technica, Inc. Translation edited by Alan Jeffrey
16. F. de Hoog, and R. Weiss Asymptotic expansions for product integration Math. Comp. 27 1973 295 306
17. O.R. Espinosa, and V.H. Moll On some integrals involving the Hurwitz zeta function The Ramanujan J. 6 2 2002 159 188
18. R. Gorenflo, J. Loutchko, and Y. Luchko Computation of the MittagLeffler function and its derivatives Fract. Calc. Appl. Anal. 5 4 2002 491 518
19. R. Hilfer, and H.J. Seybold Computation of the generalized MittagLeffler function and its inverse in the complex plane Integral Transforms Spec. Funct. 17 9 2006 637 652
20. I. Podlubny, M. Kacenak, The Matlab mlf code, http://www.mathworks.com/ matlabcentral/fileexchange/8738 (20012009).
21. H. Seybold, and R. Hilfer Numerical algorithm for calculating the generalized MittagLeffler function SIAM J. Numer. Anal. 47 1 2008/09 69 88
22. L.N. Trefethen, J.A.C. Weideman, and T. Schmelzer Talbot quadratures and rational approximations BIT 46 3 2006 653 670
23. P. Petrushev, and V. Popov Rational Approximation of Real Function 1987 Cambridge University Press Cambridge
24. W.J. Cody, G. Meinardus, and R.S. Varga Chebyshev rational approximations to e - x in [ 0, + ∞ ) and applications to heat-conduction problems J. Approx. Theory 2 1 1969 50 65
25. L.N. Trefethen, and M.H. Gutknecht The CarathodoryFejr method for real rational approximation SIAM J. Numer. Anal. 20 2 1983 420 436
26. L.N. Trefethen Rational Chebyshev approximation on the unit disk Numer. Math. 37 2 1981 297 320
27. L.N. Trefethen Circularity of the error curve and sharpness of the CF method in complex Chebyshev approximation SIAM J. Numer. Anal. 20 6 1983 1258 1263
28. A.P. Magnus Asymptotics and super asymptotics for best rational approximation error norms to the exponential function (the "1/9" problem) by the CarathodoryFejr method Nonlinear Numer. methods Ration. Approx. II 296 1994 173 185
29. R. Garrappa, M. Popolizio, On the use of matrix functions for fractional partial differential equations, Tech. Report n. 44, Dipartimento di Matematica, Univ. di Bari, 2008.
30. I. Moret, P. Novati, On the convergence of Krylov subspace methods for MittagLeffler functions. Available at: www.math.unipd.it/~novati/dati/ finalpreprints/MLK.pdf.
31. T. Schmelzer, and L.N. Trefethen Computing the gamma function using contour integrals and rational approximations SIAM J. Numer. Anal. 45 2 2007 558 571 (electronic)
32. E. Hairer, C. Lubich, and M. Schlichte Fast numerical solution of nonlinear Volterra convolution equations SIAM J. Sci. Statist. Comput. 6 3 1985 532 541
33. C. Li, and C. Tao On the fractional Adams method Comput. Math. Appl. 58 8 2009 1573 1588
34. K. Diethelm, and Y. Luchko Numerical solution of linear multi-term initial value problems of fractional order J. Comput. Anal. Appl. 6 3 2004 243 263
35. C. Lubich Convolution quadrature and discretized operational calculus. I Numer. Math. 52 2 1988 129 145
36. Y. Luchko, and R. Gorenflo An operational method for solving fractional differential equations with the Caputo derivatives Acta Math. Vietnam. 24 2 1999 207 233