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References (48)
-
F. Stenger (1973)
Integration Formulae Based on the Trapezoidal FormulaIma Journal of Applied Mathematics, 12
-
(1975)
Proceedings to the international conference on fractional calculus and its applications -
M. Ortigueira (2008)
On the fractional central differences and derivatives -
K. Oldham, J. Spanier (1974)
The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order -
D. Widder (1955)
Review: Tables of integral transforms. Vol. IIBulletin of the American Mathematical Society, 61
-
J. Miller (1966)
Numerical AnalysisNature, 212
-
L. Debnath (2012)
Tables of Integral Transforms -
(2015)
The MPFR library for multiple-precision floating-point computations with correct rounding -
M. Mori (1990)
Developments in the Double Exponential Formulas for Numerical Integration -
(2015)
gmp: GMP is a free library for arbitrary precision arithmetic (version 6.0.0a) -
W. Press, S. Teukolsky, W. Vetterling, B. Flannery (2007)
Numerical recipes: the art of scientific computing, 3rd Edition -
J. Waldvogel (2010)
Towards a General Error Theory of the Trapezoidal Rule -
J. Machado (2015)
Numerical calculation of the left and right fractional derivativesJ. Comput. Phys., 293
-
D. Murio (2006)
On the stable numerical evaluation of caputo fractional derivativesComput. Math. Appl., 51
-
T. Gerber (2016)
Handbook Of Mathematical Functions -
(1965)
Metody Numeryczne -
G. Milovanović, D. Tošić, Miloljub Albijanic (2012)
Numerical integration of analytic functions, 1479
-
H. Takahasi, M. Mori (1973)
Quadrature formulas obtained by variable transformationNumerische Mathematik, 21
-
(2006)
Zarys rachunku różniczkowego i całkowego ułamkowych rzȩdów -
K. Diethelm, N. Ford, A. Freed, Yury Luchko (2005)
Algorithms for the fractional calculus: A selection of numerical methodsComputer Methods in Applied Mechanics and Engineering, 194
-
D. Brzezinski, P. Ostalczyk (2015)
Accuracy Assessment of Fractional Order Derivatives and Integrals Numerical ComputationsJournal of Applied Nonlinear Dynamics, 4
-
W. Press, S. Teukolsky (1990)
Numerical recipesComputers in Physics, 4
-
W. Marsden (2012)
I and J -
D. Valério, J. Trujillo, M. Rivero, J. Machado, D. Baleanu, D. Baleanu (2013)
Fractional calculus: A survey of useful formulasThe European Physical Journal Special Topics, 222
-
P. Kythe, M. Schaferkotter (2004)
Handbook of Computational Methods for Integration -
J. Kiusalaas (2005)
Numerical methods in engineering with Python -
A. Folkesson (2007)
Numerical methods for engineers -
S. Samko, A. Kilbas, O. Marichev (1993)
Fractional Integrals and Derivatives: Theory and Applications -
(2005)
Which differintegration? -
Abd Elgadir, Ahmed Mohammed (2009)
Integral transforms and their applications -
Problems of Numerical Calculation of Fractional Order Derivatives and Integrals -
K. Miller, Bertram Ross (1993)
An Introduction to the Fractional Calculus and Fractional Differential Equations -
P. Ostalczyk (2015)
Discrete Fractional Calculus: Applications In Control And Image Processing -
K. Diethelm, G. Walz (1997)
Numerical solution of fractional order differential equations by extrapolationNumerical Algorithms, 16
-
H. Sugiura, T. Hasegawa (2008)
Quadrature rule for Abel's equations : uniformly approximating fractional derivatives uniformly approximating fractional derivatives (High Performance Algorithms for Computational Science and Their Applications), 1614
-
Z. Odibat (2006)
Approximations of fractional integrals and Caputo fractional derivativesAppl. Math. Comput., 178
-
R. Herrmann (2014)
Fractional Calculus: an Introduction for Physicists (2nd Edition) -
S. Koonin (1986)
Computational physicsPhysics Education, 21
-
Nicholas Hale, Alex Townsend (2013)
Fast and Accurate Computation of Gauss-Legendre and Gauss-Jacobi Quadrature Nodes and WeightsSIAM J. Sci. Comput., 35
-
(1962)
Leibniz. Letter from hannover, germany, to g -
S. Haber (1977)
The Tanh Rule for Numerical IntegrationSIAM Journal on Numerical Analysis, 14
-
(1967)
Priblizhennoe wychislenie integralov, 2e izd. Nauka, -
F. Mainardi (2018)
Fractional CalculusFractional Calculus
-
(1962)
Letter from hannover, germany, to g. f. a. l'hôpital, september 30. 1695 -
C. Goodrich, A. Peterson (2016)
Discrete Fractional Calculus -
H. Takahasi, M. Mori (1973)
Double Exponential Formulas for Numerical IntegrationPublications of The Research Institute for Mathematical Sciences, 9
-
T. Kaczorek, K. Rogowski (2015)
Fractional Differential Equations -
(1963)
Teoretikocislwyje metody w priblizennom analize
- Publisher
- de Gruyter
- Copyright
- © 2016 Dariusz W. Brzeziński, published by Sciendo
- eISSN
- 2444-8656
- DOI
- 10.21042/AMNS.2016.1.00003
- Publisher site
- See Article on Publisher Site
Abstract
AbstractIn the paper there are presented and evaluated for effectiveness three methods of accuracy increase of fractional order derivatives and integrals computations for application with the Riemann-Liouville/Caputo formulas. They are based on the ideas of either transforming difficult integrand in the formulas to high-accuracy computations requirements of a applied method of numerical integration or adapting a numerical method of integration to handle with high-accuracy a difficult feature in the integrand. Additional accuracy gain is obtained by incorporating increased precision into computations. The actual accuracy improvement by applying presented methods is compared with the capabilities of wide range of available methods of integration.
Journal
Applied Mathematics and Nonlinear Sciences – de Gruyter
Published: Jan 1, 2016