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

M. L. Morgado; M. Rebelo

doi:10.1063/1.4965348

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

M. L. Morgado, M. Rebelo

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

3 Citations (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 language	English
Title of host publication	Numerical Computations
Subtitle of host publication	Theory and Algorithms, NUMTA 2016: Proceedings of the 2nd International Conference "Numerical Computations: Theory and Algorithms"
Publisher	AIP - American Institute of Physics
Volume	1776
ISBN (Electronic)	9780735414389
DOIs	https://doi.org/10.1063/1.4965348
Publication status	Published - 20 Oct 2016
Event	2nd International Conference on Numerical Computations: Theory and Algorithms, NUMTA 2016 - Pizzo Calabro, Italy Duration: 19 Jun 2016 → 25 Jun 2016

Conference

Conference	2nd International Conference on Numerical Computations: Theory and Algorithms, NUMTA 2016
Country/Territory	Italy
City	Pizzo Calabro
Period	19/06/16 → 25/06/16

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10.1063/1.4965348Licence: Unspecified

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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). Article 070002 AIP - American Institute of Physics. https://doi.org/10.1063/1.4965348

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title = "Introducing graded meshes in the numerical approximation of distributed-order diffusion equations",

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{\c c}{\~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{\c c} {\~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, AIP - American Institute of Physics, 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 AIP - American Institute of Physics, 2016. 070002.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AU - Rebelo, M.

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N2 - 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).

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

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