Identification of the diffusion coefficient in a time fractional diffusion equation
In this paper, we study the existence and uniqueness of a quasi solution to a time fractional diffusion equation related to , where the function is unknown. We consider a methodology, involving minimization of a least squares cost functional, to identify the unknown function . At the first step of t...
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| Published in | Journal of inverse and ill-posed problems Vol. 28; no. 2; pp. 299 - 306 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin
De Gruyter
01.04.2020
Walter de Gruyter GmbH |
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| Abstract | In this paper, we study the existence and uniqueness of a quasi solution to a time fractional diffusion equation related to
, where the function
is unknown.
We consider a methodology, involving minimization of a least squares cost functional, to identify the unknown function
.
At the first step of the methodology, we give a stability result corresponding to connectivity of
and
which leads to the continuity
of the cost functional.
We next construct an appropriate class of admissible functions and show that a solution of the minimization problem exists for the continuous cost functional.
At the end, convexity of the introduced cost functional and subsequently the uniqueness theorem of the quasi solution are given. |
|---|---|
| AbstractList | In this paper, we study the existence and uniqueness of a quasi solution to a time fractional diffusion equation related to
, where the function
is unknown.
We consider a methodology, involving minimization of a least squares cost functional, to identify the unknown function
.
At the first step of the methodology, we give a stability result corresponding to connectivity of
and
which leads to the continuity
of the cost functional.
We next construct an appropriate class of admissible functions and show that a solution of the minimization problem exists for the continuous cost functional.
At the end, convexity of the introduced cost functional and subsequently the uniqueness theorem of the quasi solution are given. In this paper, we study the existence and uniqueness of a quasi solution to a time fractional diffusion equation related to D t α C u - ∇ ⋅ ( k ( x ) ∇ u ) = f {{}^{C}D_{t}^{\alpha}u-\nabla\cdot(k(x)\nabla u)=f} , where the function k = k ( x ) {k=k(x)} is unknown. We consider a methodology, involving minimization of a least squares cost functional, to identify the unknown function k . At the first step of the methodology, we give a stability result corresponding to connectivity of k and u which leads to the continuity of the cost functional. We next construct an appropriate class of admissible functions and show that a solution of the minimization problem exists for the continuous cost functional. At the end, convexity of the introduced cost functional and subsequently the uniqueness theorem of the quasi solution are given. In this paper, we study the existence and uniqueness of a quasi solution to a time fractional diffusion equation related to [Image omitted], where the function [Image omitted] is unknown. We consider a methodology, involving minimization of a least squares cost functional, to identify the unknown function k. At the first step of the methodology, we give a stability result corresponding to connectivity of k and u which leads to the continuity of the cost functional. We next construct an appropriate class of admissible functions and show that a solution of the minimization problem exists for the continuous cost functional. At the end, convexity of the introduced cost functional and subsequently the uniqueness theorem of the quasi solution are given. |
| Author | Zakeri, Ali Bodaghi, Soheila Salehi Shayegan, Amir Hossein Heshmati, M. |
| Author_xml | – sequence: 1 givenname: Amir Hossein surname: Salehi Shayegan fullname: Salehi Shayegan, Amir Hossein email: ah.salehi@mail.kntu.ac.ir organization: Mathematics Department, Faculty of Basic Science, Khatam-ol-Anbia (PBU) University, Tehran, Iran – sequence: 2 givenname: Ali surname: Zakeri fullname: Zakeri, Ali email: azakeri@kntu.ac.ir organization: Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran – sequence: 3 givenname: Soheila surname: Bodaghi fullname: Bodaghi, Soheila email: sbodaghi@mail.kntu.ac.ir organization: Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran – sequence: 4 givenname: M. surname: Heshmati fullname: Heshmati, M. email: mmheshmati.kau@gmail.com organization: Mathematics Department, Faculty of Basic Science, Khatam-ol-Anbia (PBU) University, Tehran, Iran |
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| Cites_doi | 10.1016/0020-7225(87)90098-X 10.1137/S0036139997331628 10.1080/17415977.2017.1384826 10.1007/978-3-319-62797-7 10.2478/s13540-011-0028-2 10.1515/156939406778247615 10.1515/jiip-2018-0042 10.4208/cicp.020709.221209a |
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| References | Hanke, M.; Scherzer, O. (j_jiip-2018-0109_ref_003) 1999; 59 Ford, N.; Xiao, J.; Yan, Y. (j_jiip-2018-0109_ref_002) 2011; 14 Salehi Shayegan, A. H.; Zakeri, A. (j_jiip-2018-0109_ref_007) 2018; 26 Hasanov, A.; Duchateau, P.; Pektas, B. (j_jiip-2018-0109_ref_004) 2006; 14 Li, X.; Xu, C. (j_jiip-2018-0109_ref_006) 2010; 8 Cannon, J. R.; DuChateau, P. (j_jiip-2018-0109_ref_001) 1987; 25 Salehi Shayegan, A. H.; Zakeri, A. (j_jiip-2018-0109_ref_008) 2019; 27 2023040100055454152_j_jiip-2018-0109_ref_003_w2aab3b7d796b1b6b1ab2b1b3Aa 2023040100055454152_j_jiip-2018-0109_ref_010_w2aab3b7d796b1b6b1ab2b1c10Aa 2023040100055454152_j_jiip-2018-0109_ref_002_w2aab3b7d796b1b6b1ab2b1b2Aa 2023040100055454152_j_jiip-2018-0109_ref_008_w2aab3b7d796b1b6b1ab2b1b8Aa 2023040100055454152_j_jiip-2018-0109_ref_001_w2aab3b7d796b1b6b1ab2b1b1Aa 2023040100055454152_j_jiip-2018-0109_ref_009_w2aab3b7d796b1b6b1ab2b1b9Aa 2023040100055454152_j_jiip-2018-0109_ref_006_w2aab3b7d796b1b6b1ab2b1b6Aa 2023040100055454152_j_jiip-2018-0109_ref_007_w2aab3b7d796b1b6b1ab2b1b7Aa 2023040100055454152_j_jiip-2018-0109_ref_005_w2aab3b7d796b1b6b1ab2b1b5Aa 2023040100055454152_j_jiip-2018-0109_ref_004_w2aab3b7d796b1b6b1ab2b1b4Aa |
| References_xml | – volume: 14 start-page: 454 issue: 3 year: 2011 end-page: 474 ident: j_jiip-2018-0109_ref_002 article-title: A finite element method for time fractional partial differential equations publication-title: Fract. Calc. Appl. Anal. – volume: 8 start-page: 1016 issue: 5 year: 2010 end-page: 1051 ident: j_jiip-2018-0109_ref_006 article-title: Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation publication-title: Commun. Comput. Phys. – volume: 59 start-page: 1012 year: 1999 end-page: 1027 ident: j_jiip-2018-0109_ref_003 article-title: Error analysis of an equation error method for the identification of the diffusion coefficient in a quasilinear parabolic differential equation publication-title: SIAM J. Appl. Math. – volume: 27 start-page: 795 issue: 6 year: 2019 end-page: 814 ident: j_jiip-2018-0109_ref_008 article-title: Quasi solution of a backward space fractional diffusion equation publication-title: J. Inverse Ill-Posed Probl. – volume: 26 start-page: 1130 issue: 8 year: 2018 end-page: 1154 ident: j_jiip-2018-0109_ref_007 article-title: A numerical method for determining a quasi solution of a backward time-fractional diffusion equation publication-title: Inverse Probl. Sci. Eng. – volume: 25 start-page: 1067 year: 1987 end-page: 1078 ident: j_jiip-2018-0109_ref_001 article-title: Design of an experiment for the determination of a coefficient in a nonlinear diffusion equation publication-title: Int. J. Eng. Sci. – volume: 14 start-page: 435 year: 2006 end-page: 463 ident: j_jiip-2018-0109_ref_004 article-title: An adjoint problem approach and coarse-fine mesh method for identification of the diffusion coefficient in a linear parabolic equation publication-title: J. Inverse Ill-Posed Probl. – ident: 2023040100055454152_j_jiip-2018-0109_ref_001_w2aab3b7d796b1b6b1ab2b1b1Aa doi: 10.1016/0020-7225(87)90098-X – ident: 2023040100055454152_j_jiip-2018-0109_ref_003_w2aab3b7d796b1b6b1ab2b1b3Aa doi: 10.1137/S0036139997331628 – ident: 2023040100055454152_j_jiip-2018-0109_ref_007_w2aab3b7d796b1b6b1ab2b1b7Aa doi: 10.1080/17415977.2017.1384826 – ident: 2023040100055454152_j_jiip-2018-0109_ref_005_w2aab3b7d796b1b6b1ab2b1b5Aa doi: 10.1007/978-3-319-62797-7 – ident: 2023040100055454152_j_jiip-2018-0109_ref_010_w2aab3b7d796b1b6b1ab2b1c10Aa – ident: 2023040100055454152_j_jiip-2018-0109_ref_002_w2aab3b7d796b1b6b1ab2b1b2Aa doi: 10.2478/s13540-011-0028-2 – ident: 2023040100055454152_j_jiip-2018-0109_ref_004_w2aab3b7d796b1b6b1ab2b1b4Aa doi: 10.1515/156939406778247615 – ident: 2023040100055454152_j_jiip-2018-0109_ref_008_w2aab3b7d796b1b6b1ab2b1b8Aa doi: 10.1515/jiip-2018-0042 – ident: 2023040100055454152_j_jiip-2018-0109_ref_006_w2aab3b7d796b1b6b1ab2b1b6Aa doi: 10.4208/cicp.020709.221209a – ident: 2023040100055454152_j_jiip-2018-0109_ref_009_w2aab3b7d796b1b6b1ab2b1b9Aa |
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| Snippet | In this paper, we study the existence and uniqueness of a quasi solution to a time fractional diffusion equation related to
, where the function
is unknown.
We... In this paper, we study the existence and uniqueness of a quasi solution to a time fractional diffusion equation related to D t α C u - ∇ ⋅ ( k ( x ) ∇ ... In this paper, we study the existence and uniqueness of a quasi solution to a time fractional diffusion equation related to [Image omitted], where the function... |
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| SubjectTerms | 35R30 Continuity (mathematics) Convexity Diffusion Diffusion coefficient Functionals Inverse coefficient problem Optimization quasi solution time fractional diffusion equation Uniqueness theorems |
| Title | Identification of the diffusion coefficient in a time fractional diffusion equation |
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