Quasi Solution of a Nonlinear Inverse Parabolic Problem
In this paper, we study the existence of a quasi solution to nonlinear inverse parabolic problem related to ℵ ( u ) : ≡ u t - ∇ · ( F ( x , ∇ u ) ) where the function F is unknown. We consider a methodology, involving minimization of a least squares cost functional, to identify the unknown function...
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| Published in | Bulletin of the Iranian Mathematical Society Vol. 45; no. 1; pp. 1 - 12 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Singapore
Springer Singapore
07.02.2019
|
| Subjects | |
| Online Access | Get full text |
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| Abstract | In this paper, we study the existence of a quasi solution to nonlinear inverse parabolic problem related to
ℵ
(
u
)
:
≡
u
t
-
∇
·
(
F
(
x
,
∇
u
)
)
where the function
F
is unknown. We consider a methodology, involving minimization of a least squares cost functional, to identify the unknown function
F
. At the first step of the methodology, we give a stability result corresponding to connectivity of
F
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 last step, we conclude that the nonlinear inverse parabolic problem has at least one quasi solution in that class of functions. |
|---|---|
| AbstractList | In this paper, we study the existence of a quasi solution to nonlinear inverse parabolic problem related to
ℵ
(
u
)
:
≡
u
t
-
∇
·
(
F
(
x
,
∇
u
)
)
where the function
F
is unknown. We consider a methodology, involving minimization of a least squares cost functional, to identify the unknown function
F
. At the first step of the methodology, we give a stability result corresponding to connectivity of
F
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 last step, we conclude that the nonlinear inverse parabolic problem has at least one quasi solution in that class of functions. |
| Author | Zakeri, Ali Nikazad, Touraj Shayegan, Amir Hossein Salehi |
| Author_xml | – sequence: 1 givenname: Amir Hossein Salehi surname: Shayegan fullname: Shayegan, Amir Hossein Salehi organization: Faculty of Mathematics, K. N. Toosi University of Technology – sequence: 2 givenname: Ali surname: Zakeri fullname: Zakeri, Ali organization: Faculty of Mathematics, K. N. Toosi University of Technology – sequence: 3 givenname: Touraj surname: Nikazad fullname: Nikazad, Touraj email: tnikazad@iust.ac.ir organization: School of Mathematics, Iran University of Science and Technology |
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| Cites_doi | 10.1016/j.aml.2007.06.007 10.1088/0266-5611/13/5/011 10.1007/s10114-008-6384-0 10.1016/S0898-1221(01)00220-6 10.1016/j.apm.2013.06.018 |
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| Copyright | Iranian Mathematical Society 2018 |
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| DOI | 10.1007/s41980-018-0115-9 |
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| Keywords | Primary 35K55 Quasi solution Time-dependent problems Nonlinear inverse parabolic problem 49J20 Secondary 35R30 |
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| References | Ladyzhenskaia, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and quasi-linear equations of parabolic type, vol. 23. American Mathematical Society, Providence (1988) FaragóIKarátsonJThe gradient-finite element method for elliptic problemsComput. Math. Appl.2001428–910431053185122410.1016/S0898-1221(01)00220-60987.65121 RektorysKThe method of discretization in time and partial differential equations1982DordrechtD. Reidel Publishing Company0522.65059 Faragó, I., Karátson, J.: Numerical solution of nonlinear elliptic problems via preconditioning operators: Theory and applications, vol. 11. Nova Science Publishers, Inc., New York (2002) WilanskyATopology for analysis1970BostonGinn0229.54001 HasanovAInverse coefficient problems for monotone potential operatorsInverse Probl.199713512651278147436810.1088/0266-5611/13/5/0110883.35128 Di Nardo, R.: Nonlinear elliptic and parabolic equations with measure data. PhD thesis, Università degli Studi di Napoli Federico II (2009) OuYHHasanovALiuZHInverse coefficient problems for nonlinear parabolic differential equationsActa Math. Sin. (Engl. Ser.)2008241016171624245304710.1007/s10114-008-6384-01157.35402 ZakeriASalehi ShayeganAHGradient WEB-spline finite element method for solving two-dimensional quasilinear elliptic problemsAppl. Math. Model2014382775783314165910.1016/j.apm.2013.06.01806968229 HasanovALiuZHAn inverse coefficient problem for a nonlinear parabolic variational inequalityAppl. Math. Lett.2008216563570241237910.1016/j.aml.2007.06.0071143.35385 A Wilansky (115_CR9) 1970 A Hasanov (115_CR4) 1997; 13 YH Ou (115_CR7) 2008; 24 K Rektorys (115_CR8) 1982 I Faragó (115_CR2) 2001; 42 115_CR6 A Hasanov (115_CR5) 2008; 21 115_CR1 115_CR3 A Zakeri (115_CR10) 2014; 38 |
| References_xml | – reference: WilanskyATopology for analysis1970BostonGinn0229.54001 – reference: Ladyzhenskaia, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and quasi-linear equations of parabolic type, vol. 23. American Mathematical Society, Providence (1988) – reference: OuYHHasanovALiuZHInverse coefficient problems for nonlinear parabolic differential equationsActa Math. Sin. (Engl. Ser.)2008241016171624245304710.1007/s10114-008-6384-01157.35402 – reference: ZakeriASalehi ShayeganAHGradient WEB-spline finite element method for solving two-dimensional quasilinear elliptic problemsAppl. Math. Model2014382775783314165910.1016/j.apm.2013.06.01806968229 – reference: HasanovALiuZHAn inverse coefficient problem for a nonlinear parabolic variational inequalityAppl. Math. Lett.2008216563570241237910.1016/j.aml.2007.06.0071143.35385 – reference: Di Nardo, R.: Nonlinear elliptic and parabolic equations with measure data. PhD thesis, Università degli Studi di Napoli Federico II (2009) – reference: HasanovAInverse coefficient problems for monotone potential operatorsInverse Probl.199713512651278147436810.1088/0266-5611/13/5/0110883.35128 – reference: FaragóIKarátsonJThe gradient-finite element method for elliptic problemsComput. Math. Appl.2001428–910431053185122410.1016/S0898-1221(01)00220-60987.65121 – reference: Faragó, I., Karátson, J.: Numerical solution of nonlinear elliptic problems via preconditioning operators: Theory and applications, vol. 11. Nova Science Publishers, Inc., New York (2002) – reference: RektorysKThe method of discretization in time and partial differential equations1982DordrechtD. Reidel Publishing Company0522.65059 – ident: 115_CR3 – volume: 21 start-page: 563 issue: 6 year: 2008 ident: 115_CR5 publication-title: Appl. Math. Lett. doi: 10.1016/j.aml.2007.06.007 – volume-title: Topology for analysis year: 1970 ident: 115_CR9 – volume: 13 start-page: 1265 issue: 5 year: 1997 ident: 115_CR4 publication-title: Inverse Probl. doi: 10.1088/0266-5611/13/5/011 – ident: 115_CR6 – volume: 24 start-page: 1617 issue: 10 year: 2008 ident: 115_CR7 publication-title: Acta Math. Sin. (Engl. Ser.) doi: 10.1007/s10114-008-6384-0 – volume: 42 start-page: 1043 issue: 8–9 year: 2001 ident: 115_CR2 publication-title: Comput. Math. Appl. doi: 10.1016/S0898-1221(01)00220-6 – ident: 115_CR1 – volume: 38 start-page: 775 issue: 2 year: 2014 ident: 115_CR10 publication-title: Appl. Math. Model doi: 10.1016/j.apm.2013.06.018 – volume-title: The method of discretization in time and partial differential equations year: 1982 ident: 115_CR8 |
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| Snippet | In this paper, we study the existence of a quasi solution to nonlinear inverse parabolic problem related to
ℵ
(
u
)
:
≡
u
t
-
∇
·
(
F
(
x
,
∇
u
)
)
where the... |
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| Title | Quasi Solution of a Nonlinear Inverse Parabolic Problem |
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