Numerical Solution Methods for a Nonlinear Operator Equation Arising in an Inverse Coefficient Problem
We consider the inverse problem of determining two unknown coefficients in a linear system of partial differential equations using additional information about one of the solution components. The problem is reduced to a nonlinear operator equation for one of the unknown coefficients. The successive...
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| Published in: | Differential equations Vol. 57; no. 7; pp. 868 - 875 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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Moscow
Pleiades Publishing
01.07.2021
Springer |
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| ISSN: | 0012-2661, 1608-3083 |
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| Abstract | We consider the inverse problem of determining two unknown coefficients in a linear system of partial differential equations using additional information about one of the solution components. The problem is reduced to a nonlinear operator equation for one of the unknown coefficients. The successive approximation method and the Newton method are used to solve this operator equation numerically. Results of calculations illustrating the convergence of numerical methods for solving the inverse problem are presented. |
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| AbstractList | We consider the inverse problem of determining two unknown coefficients in a linear system of partial differential equations using additional information about one of the solution components. The problem is reduced to a nonlinear operator equation for one of the unknown coefficients. The successive approximation method and the Newton method are used to solve this operator equation numerically. Results of calculations illustrating the convergence of numerical methods for solving the inverse problem are presented. We consider the inverse problem of determining two unknown coefficients in a linearsystem of partial differential equations using additional information about one of the solutioncomponents. The problem is reduced to a nonlinear operator equation for one of the unknowncoefficients. The successive approximation method and the Newton method are used to solve thisoperator equation numerically. Results of calculations illustrating the convergence of numericalmethods for solving the inverse problem are presented. |
| Audience | Academic |
| Author | Gavrilov, S. V. Denisov, A. M. |
| Author_xml | – sequence: 1 givenname: S. V. surname: Gavrilov fullname: Gavrilov, S. V. email: gvrlserg@gmail.com organization: Lomonosov Moscow State University – sequence: 2 givenname: A. M. surname: Denisov fullname: Denisov, A. M. email: den@cs.msu.ru organization: Lomonosov Moscow State University |
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| References_xml | – reference: BaevA.V.GavrilovS.V.An iterative way of solving the inverse scattering problem for an acoustic system of equations in an absorptive layered nonhomogeneous mediumMosc. Univ. Comput. Math. Cybern.20184225562381376210.3103/S0278641918020024 – reference: DenisovA.M.Iterative method for solving an inverse problem for a hyperbolic equation with a small parameter multiplying the highest derivativeDiffer. Equations2019557940948399919810.1134/S0012266119070073 – reference: KabanikhinS.I.ScherzerO.ShichleninM.A.Iteration method for solving a two-dimensional inverse problem for hyperbolic equationJ. Inverse Ill-Posed Probl.2003111123197216710.1515/156939403322004955 – reference: Yan-BoMa.Newton method for estimation of the Robin coefficientJ. Nonlin. Sci. Appl.201585660669336131410.22436/jnsa.008.05.18 – reference: KantorovichL.V.AkilovG.P.Funktsional’nyi analiz (Functional Analysis)1977MoscowNauka – reference: DenisovA.M.LukshinA.V.Matematicheskie modeli neravnovesnoi dinamiki sorbtsii (Mathematical Models of Nonequilibrium Sorption Dynamics)1989MoscowIzd. Mosk. Gos. Univ. – reference: DenisovA.M.Existence and uniqueness of a solution of a system of nonlinear integral equationsDiffer. Equations202056911401147416524110.1134/S0012266120090049 – reference: BimuratovS.Sh.KabanikhinS.I.Solution of one-dimensional inverse problems of electrodynamics by the Newton–Kantorovich methodComput. Math. Math. Phys.199232121729174312059930786.65099 – reference: TikhonovA.N.SamarskiiA.A.Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics)1999MoscowIzd. Mosk. Gos. Univ. – reference: SamarskiiA.A.VabishchevichP.N.Chislennye metody resheniya obratnykh zadach matematicheskoi fiziki (Numerical Methods for Solving Inverse Problems of Mathematical Physics)2004MoscowEditorial URSS – reference: MonchL.A Newton method for solving inverse scattering problem for a sound-hard obstacleInverse Probl.1996123309324139154110.1088/0266-5611/12/3/010 – reference: DenisovA.M.Iterative method for solving an inverse coefficient problem for a hyperbolic equationDiffer. Equations2017537943949369162910.1134/S0012266117070084 – volume: 42 start-page: 55 issue: 2 year: 2018 ident: 2232_CR10 publication-title: Mosc. Univ. Comput. Math. Cybern. doi: 10.3103/S0278641918020024 – volume-title: Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics) year: 1999 ident: 2232_CR1 – volume-title: Chislennye metody resheniya obratnykh zadach matematicheskoi fiziki (Numerical Methods for Solving Inverse Problems of Mathematical Physics) year: 2004 ident: 2232_CR7 – volume: 55 start-page: 940 issue: 7 year: 2019 ident: 2232_CR11 publication-title: Differ. Equations doi: 10.1134/S0012266119070073 – volume: 32 start-page: 1729 issue: 12 year: 1992 ident: 2232_CR4 publication-title: Comput. Math. Math. Phys. – volume: 12 start-page: 309 issue: 3 year: 1996 ident: 2232_CR5 publication-title: Inverse Probl. doi: 10.1088/0266-5611/12/3/010 – volume: 11 start-page: 1 issue: 1 year: 2003 ident: 2232_CR6 publication-title: J. Inverse Ill-Posed Probl. doi: 10.1515/156939403322004955 – volume: 53 start-page: 943 issue: 7 year: 2017 ident: 2232_CR9 publication-title: Differ. Equations doi: 10.1134/S0012266117070084 – volume-title: Funktsional’nyi analiz (Functional Analysis) year: 1977 ident: 2232_CR12 – volume: 8 start-page: 660 issue: 5 year: 2015 ident: 2232_CR8 publication-title: J. Nonlin. Sci. Appl. doi: 10.22436/jnsa.008.05.18 – volume-title: Matematicheskie modeli neravnovesnoi dinamiki sorbtsii (Mathematical Models of Nonequilibrium Sorption Dynamics) year: 1989 ident: 2232_CR2 – volume: 56 start-page: 1140 issue: 9 year: 2020 ident: 2232_CR3 publication-title: Differ. Equations doi: 10.1134/S0012266120090049 |
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| SubjectTerms | Difference and Functional Equations Differential equations Mathematics Mathematics and Statistics Methods Numerical Methods Ordinary Differential Equations Partial Differential Equations |
| Title | Numerical Solution Methods for a Nonlinear Operator Equation Arising in an Inverse Coefficient Problem |
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