In this article, we deal with the efficient computation of the Wright function in the cases of interest for the expression of solutions of some fractional differential equations. The proposed algorithm is based on the inversion of the Laplace transform of a particular expression of the Wright function for which we discuss in detail the error analysis. We also present a code package that implements the algorithm proposed here in different programming languages. The analysis and implementation are accompanied by an extensive set of numerical experiments that validate both the theoretical estimates of the error and the applicability of the proposed method for representing the solutions of fractional differential equations.
Accepté le :
Publié le :
DOI : 10.1051/m2an/2022069
Keywords: Wright function, Laplace transform, trapezoidal rule, fractional PDEs
@article{M2AN_2022__56_6_2181_0,
author = {Aceto, Lidia and Durastante, Fabio},
title = {Efficient computation of the {Wright} function and its applications to fractional diffusion-wave equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2181--2196},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {6},
doi = {10.1051/m2an/2022069},
mrnumber = {4516169},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2022069/}
}
TY - JOUR AU - Aceto, Lidia AU - Durastante, Fabio TI - Efficient computation of the Wright function and its applications to fractional diffusion-wave equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2022 SP - 2181 EP - 2196 VL - 56 IS - 6 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2022069/ DO - 10.1051/m2an/2022069 LA - en ID - M2AN_2022__56_6_2181_0 ER -
%0 Journal Article %A Aceto, Lidia %A Durastante, Fabio %T Efficient computation of the Wright function and its applications to fractional diffusion-wave equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2022 %P 2181-2196 %V 56 %N 6 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2022069/ %R 10.1051/m2an/2022069 %G en %F M2AN_2022__56_6_2181_0
Aceto, Lidia; Durastante, Fabio. Efficient computation of the Wright function and its applications to fractional diffusion-wave equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 6, pp. 2181-2196. doi: 10.1051/m2an/2022069
[1] , On the Coefficients of Power Series Having Exponential Singularities. J. London Math. Soc. 8 (1933) 71–79. | MR | Zbl | DOI
[2] , The generalized Bessel function of order greater than one. Quart. J. Math. Oxford Ser. 11 (1940) 36–48. | MR | JFM | DOI
[3] , On the function of E. M. Wright. Publ. Inst. Math. (Beograd) (N.S.) 10 (1970) 113–124. | MR | Zbl
[4] , The Wright function and its applications. In Vol. 1 Basic Theory, De Gruyter (2019) 241–268. | MR | DOI
[5] and , The Wright Functions of the Second Kind in Mathematical Physics. Mathematics 8 (2020). | DOI
[6] , and , The M-Wright function in time-fractional diffusion processes: a tutorial survey. Int. J. Differ. Equ. 2010 (2010) 104505. | MR | Zbl
[7] , A tutorial on the basic special functions of Fractional Calculus. WSEAS Trans. Math. 19 (2020) 74–98. | DOI
[8] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.1.3 of 2021-09-15. , , , , , , , , and , eds.
[9] , Algorithms for evaluation of the Wright function for the real arguments’ values. Fract. Calc. Appl. Anal. 11 (2008) 57–75. | MR | Zbl
[10] , and , The Wright function and its numerical evaluation. Int. J. Pure Appl. Math 64 (2010) 567–575. | MR | Zbl
[11] and , An improved Talbot method for numerical Laplace transform inversion. Numer. Algorithms 68 (2015) 167–183. | MR | Zbl | DOI
[12] and , Evaluation of generalized Mittag-Leffler functions on the real line. Adv. Comput. Math. 39 (2013) 205–225. | MR | Zbl | DOI
[13] and , A contour method for time-fractional PDEs and an application to fractional viscoelastic beam equations. J. Comput. Phys. 454 (2022) 110995. | MR | DOI
[14] , Computing semigroups with error control. SIAM J. Numer. Anal. 60 (2022) 396–422. | MR | DOI
[15] and , Parabolic and hyperbolic contours for computing the Bromwich integral. Math. Comp. 76 (2007) 1341–1356. | MR | Zbl | DOI
[16] , Improved contour integral methods for parabolic PDEs. IMA J. Numer. Anal. 30 (2010) 334–350. | MR | Zbl | DOI
[17] and , The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000) 77. | MR | Zbl | DOI
[18] , The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9 (1996) 23–28. | MR | Zbl | DOI
[19] and , The Wright functions as solutions of the time-fractional diffusion equation. Appl. Math. Comput. 141 (2003) 51–62. | MR | Zbl
[20] and , Fractional diffusive waves in the Cauchy and Signalling problems, in Nonlocal and fractional operators, Edited by , and , In Vol. 26 of SEMA SIMAI Springer Ser., Edited by L. Formaggia and P. Pedregal, Springer, Cham (2021) 133–153. | MR | DOI
[21] , Fractional heat conduction in infinite one-dimensional composite medium. J. Therm. Stress. 36 (2013) 351–363. | DOI
[22] , Fundamental solutions to time-fractional heat conduction equations in two joint half-lines. Open Phys. 11 (2013) 1284–1294. | DOI
[23] , Arb: efficient arbitrary-precision midpoint-radius interval arithmetic. IEEE Trans. Comput. 66 (2017) 1281–1292. | MR | DOI
[24] , , and , Pitfalls in fast numerical solvers for fractional differential equations. J. Comput. Appl. Math. 186 (2006) 482–503. | MR | Zbl | DOI
[25] , Trapezoidal methods for fractional differential equations: theoretical and computational aspects. Math. Comput. Simulation 110 (2015) 96–112. | MR | DOI
Cité par Sources :
