Introducing graded meshes in the numerical approximation of distributed-order diffusion equations

M. L. Morgado, M. Rebelo

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

Abstract

In this paper we deal with the numerical approximation of initial-boundary value problems to the diffusion equation with distributed order in time. As it is widely known, the solutions of fractional differential equations may present a singularity at t = 0 and therefore in these cases, standard finite difference schemes usually suffer a convergence order reduction with respect to time discretization. In order to overcome this, here we propose a finite difference scheme with a graded time mesh, constructed in such a way that the time step-size is smaller near the potential singular point. Numerical results are presented and compared with those obtained with finite difference schemes with uniform meshes.Grant: The research of both authors was financed by Portuguese Funds through FCT Fundação para a Ciência e a Tecnologia, within, respectively, Project UID/MAT/00013/2013 (Centro de Matemática) and Project UID/MAT/00297/2013 (Centro de Matemática e Aplicaç ões).

Original languageEnglish
Title of host publicationNumerical Computations
Subtitle of host publicationTheory and Algorithms, NUMTA 2016: Proceedings of the 2nd International Conference "Numerical Computations: Theory and Algorithms"
PublisherAmerican Institute of Physics Inc.
Volume1776
ISBN (Electronic)9780735414389
DOIs
Publication statusPublished - 20 Oct 2016
Event2nd International Conference on Numerical Computations: Theory and Algorithms, NUMTA 2016 - Pizzo Calabro, Italy
Duration: 19 Jun 201625 Jun 2016

Conference

Conference2nd International Conference on Numerical Computations: Theory and Algorithms, NUMTA 2016
CountryItaly
CityPizzo Calabro
Period19/06/1625/06/16

Fingerprint

mesh
approximation
boundary value problems
differential equations

Cite this

Morgado, M. L., & Rebelo, M. (2016). Introducing graded meshes in the numerical approximation of distributed-order diffusion equations. In Numerical Computations: Theory and Algorithms, NUMTA 2016: Proceedings of the 2nd International Conference "Numerical Computations: Theory and Algorithms" (Vol. 1776). [070002] American Institute of Physics Inc.. https://doi.org/10.1063/1.4965348
Morgado, M. L. ; Rebelo, M. / Introducing graded meshes in the numerical approximation of distributed-order diffusion equations. Numerical Computations: Theory and Algorithms, NUMTA 2016: Proceedings of the 2nd International Conference "Numerical Computations: Theory and Algorithms". Vol. 1776 American Institute of Physics Inc., 2016.
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abstract = "In this paper we deal with the numerical approximation of initial-boundary value problems to the diffusion equation with distributed order in time. As it is widely known, the solutions of fractional differential equations may present a singularity at t = 0 and therefore in these cases, standard finite difference schemes usually suffer a convergence order reduction with respect to time discretization. In order to overcome this, here we propose a finite difference scheme with a graded time mesh, constructed in such a way that the time step-size is smaller near the potential singular point. Numerical results are presented and compared with those obtained with finite difference schemes with uniform meshes.Grant: The research of both authors was financed by Portuguese Funds through FCT Funda{\cc}{\~a}o para a Ci{\^e}ncia e a Tecnologia, within, respectively, Project UID/MAT/00013/2013 (Centro de Matem{\'a}tica) and Project UID/MAT/00297/2013 (Centro de Matem{\'a}tica e Aplica{\cc} {\~o}es).",
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Morgado, ML & Rebelo, M 2016, Introducing graded meshes in the numerical approximation of distributed-order diffusion equations. in Numerical Computations: Theory and Algorithms, NUMTA 2016: Proceedings of the 2nd International Conference "Numerical Computations: Theory and Algorithms". vol. 1776, 070002, American Institute of Physics Inc., 2nd International Conference on Numerical Computations: Theory and Algorithms, NUMTA 2016, Pizzo Calabro, Italy, 19/06/16. https://doi.org/10.1063/1.4965348

Introducing graded meshes in the numerical approximation of distributed-order diffusion equations. / Morgado, M. L.; Rebelo, M.

Numerical Computations: Theory and Algorithms, NUMTA 2016: Proceedings of the 2nd International Conference "Numerical Computations: Theory and Algorithms". Vol. 1776 American Institute of Physics Inc., 2016. 070002.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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AB - In this paper we deal with the numerical approximation of initial-boundary value problems to the diffusion equation with distributed order in time. As it is widely known, the solutions of fractional differential equations may present a singularity at t = 0 and therefore in these cases, standard finite difference schemes usually suffer a convergence order reduction with respect to time discretization. In order to overcome this, here we propose a finite difference scheme with a graded time mesh, constructed in such a way that the time step-size is smaller near the potential singular point. Numerical results are presented and compared with those obtained with finite difference schemes with uniform meshes.Grant: The research of both authors was financed by Portuguese Funds through FCT Fundação para a Ciência e a Tecnologia, within, respectively, Project UID/MAT/00013/2013 (Centro de Matemática) and Project UID/MAT/00297/2013 (Centro de Matemática e Aplicaç ões).

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Morgado ML, Rebelo M. Introducing graded meshes in the numerical approximation of distributed-order diffusion equations. In Numerical Computations: Theory and Algorithms, NUMTA 2016: Proceedings of the 2nd International Conference "Numerical Computations: Theory and Algorithms". Vol. 1776. American Institute of Physics Inc. 2016. 070002 https://doi.org/10.1063/1.4965348