An inverse problem for a one-dimensional time-fractional diffusion problem

Bangti Jin; William Rundell

doi:10.1088/0266-5611/28/7/075010

An inverse problem for a one-dimensional time-fractional diffusion problem

Bangti Jin, William Rundell

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Abstract

We study an inverse problem of recovering a spatially varying potential term in a one-dimensional time-fractional diffusion equation from the flux measurements taken at a single fixed time corresponding to a given set of input sources. The unique identifiability of the potential is shown for two cases, i.e. the flux at one end and the net flux, provided that the set of input sources forms a complete basis in L 2(0, 1). An algorithm of the quasi-Newton type is proposed for the efficient and accurate reconstruction of the coefficient from finite data, and the injectivity of the Jacobian is discussed. Numerical results for both exact and noisy data are presented. © 2012 IOP Publishing Ltd.

Original language	English (US)
Pages (from-to)	075010
Journal	Inverse Problems
Volume	28
Issue number	7
DOIs	https://doi.org/10.1088/0266-5611/28/7/075010
State	Published - Jun 26 2012
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUS-C1-016-04
Acknowledgements: This work is supported by award no. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST), and NSF award DMS-0715060.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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10.1088/0266-5611/28/7/075010

Cite this

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title = "An inverse problem for a one-dimensional time-fractional diffusion problem",

abstract = "We study an inverse problem of recovering a spatially varying potential term in a one-dimensional time-fractional diffusion equation from the flux measurements taken at a single fixed time corresponding to a given set of input sources. The unique identifiability of the potential is shown for two cases, i.e. the flux at one end and the net flux, provided that the set of input sources forms a complete basis in L 2(0, 1). An algorithm of the quasi-Newton type is proposed for the efficient and accurate reconstruction of the coefficient from finite data, and the injectivity of the Jacobian is discussed. Numerical results for both exact and noisy data are presented. {\textcopyright} 2012 IOP Publishing Ltd.",

author = "Bangti Jin and William Rundell",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUS-C1-016-04 Acknowledgements: This work is supported by award no. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST), and NSF award DMS-0715060. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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doi = "10.1088/0266-5611/28/7/075010",

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AU - Jin, Bangti

AU - Rundell, William

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUS-C1-016-04 Acknowledgements: This work is supported by award no. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST), and NSF award DMS-0715060. This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2012/6/26

Y1 - 2012/6/26

N2 - We study an inverse problem of recovering a spatially varying potential term in a one-dimensional time-fractional diffusion equation from the flux measurements taken at a single fixed time corresponding to a given set of input sources. The unique identifiability of the potential is shown for two cases, i.e. the flux at one end and the net flux, provided that the set of input sources forms a complete basis in L 2(0, 1). An algorithm of the quasi-Newton type is proposed for the efficient and accurate reconstruction of the coefficient from finite data, and the injectivity of the Jacobian is discussed. Numerical results for both exact and noisy data are presented. © 2012 IOP Publishing Ltd.

AB - We study an inverse problem of recovering a spatially varying potential term in a one-dimensional time-fractional diffusion equation from the flux measurements taken at a single fixed time corresponding to a given set of input sources. The unique identifiability of the potential is shown for two cases, i.e. the flux at one end and the net flux, provided that the set of input sources forms a complete basis in L 2(0, 1). An algorithm of the quasi-Newton type is proposed for the efficient and accurate reconstruction of the coefficient from finite data, and the injectivity of the Jacobian is discussed. Numerical results for both exact and noisy data are presented. © 2012 IOP Publishing Ltd.

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UR - https://iopscience.iop.org/article/10.1088/0266-5611/28/7/075010

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DO - 10.1088/0266-5611/28/7/075010

M3 - Article

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VL - 28

SP - 075010

JO - Inverse Problems

JF - Inverse Problems

IS - 7

ER -

An inverse problem for a one-dimensional time-fractional diffusion problem

Abstract

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