Inverse Problem of Finding the Coefficient of the Lowest Term in Two-Dimensional Heat Equation with Ionkin-Type Boundary Condition
We consider an inverse problem of determining the time-dependent lowest order coefficient of two-dimensional (2D) heat equation with Ionkin boundary and total energy integral overdetermination condition. The global well-posedness of the problem is obtained by generalized Fourier method combined with...
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| Vydáno v: | Computational mathematics and mathematical physics Ročník 59; číslo 5; s. 791 - 808 |
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Pleiades Publishing
01.05.2019
Springer Nature B.V |
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| Abstract | We consider an inverse problem of determining the time-dependent lowest order coefficient of two-dimensional (2D) heat equation with Ionkin boundary and total energy integral overdetermination condition. The global well-posedness of the problem is obtained by generalized Fourier method combined with the unique solvability of the second kind Volterra integral equation. For obtaining a numerical solution of the inverse problem, we propose the discretization method from a new combination. On the one hand, it is known the traditional method of uniform finite difference combined with numerical integration on a uniform grid (trapezoidal and Simpson’s), on the other hand, we give the method of non-uniform finite difference is combined by a numerical integration on a non-uniform grid (with Gauss–Lobatto nodes). Numerical examples illustrate how to implement the method. |
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| AbstractList | We consider an inverse problem of determining the time-dependent lowest order coefficient of two-dimensional (2D) heat equation with Ionkin boundary and total energy integral overdetermination condition. The global well-posedness of the problem is obtained by generalized Fourier method combined with the unique solvability of the second kind Volterra integral equation. For obtaining a numerical solution of the inverse problem, we propose the discretization method from a new combination. On the one hand, it is known the traditional method of uniform finite difference combined with numerical integration on a uniform grid (trapezoidal and Simpson’s), on the other hand, we give the method of non-uniform finite difference is combined by a numerical integration on a non-uniform grid (with Gauss–Lobatto nodes). Numerical examples illustrate how to implement the method. |
| Author | Erkovan, S. Ismailov, M. I. |
| Author_xml | – sequence: 1 givenname: M. I. surname: Ismailov fullname: Ismailov, M. I. email: mismailov@gtu.edu.tr organization: Gebze Technical University, Department of Mathematics, Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan – sequence: 2 givenname: S. surname: Erkovan fullname: Erkovan, S. email: serkovan@gtu.edu.tr organization: Gebze Technical University, Department of Mathematics |
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| Cites_doi | 10.1515/1569394042248193 10.1016/j.cpc.2017.01.002 10.1137/1.9780898717570 10.1134/S0001434613070201 10.1134/S1064562414020161 10.1080/00207160108805077 10.1023/A:1012570031242 10.1007/978-1-4899-0030-2 10.1134/S0012266115050043 10.1098/rspa.2004.1430 10.1134/S0965542515070052 10.1088/0266-5611/10/5/009 10.1002/mma.1396 10.1515/jiip.2002.10.6.611 10.1017/S0334270000006962 10.1023/A:1011961822115 10.1515/9783110205794 10.1088/0266-5611/4/1/006 10.1007/BF02754498 10.1093/imamat/hxv005 10.1515/156939403770888246 10.1134/S0001434615030062 10.1134/S1064562406040272 10.1088/0266-5611/4/3/005 10.1088/0266-5611/10/2/004 10.1007/BF00971406 |
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| Keywords | generalized Fourier method non-uniform finite difference method Ionkin-type boundary condition numerical integration 2D heat equation uniform finite difference method Volterra integral equation |
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Nauk SSSR1969185739740247271 KozhanovA. I.Parabolic equations with an unknown absorption coefficientDokl. Math.20067457357610.1134/S10645624060402721152.35112 PrilepkoA. I.OrlovskyD. G.VasinI. A.Methods for Solving Inverse Problems in Mathematical Physics2000New YorkMarcel Dekker0947.35173 CannonJ. R.LinY.Determination of parameter p(t) in Hölder classes for some semilinear parabolic equationsInverse Probl.1988459560610.1088/0266-5611/4/3/0050688.35104 BedinL.BazanF. S. V.On the 2D bioheat equation with convective boundary conditions and its numerical realization via a highly accurate approachAppl. Math. Comput.201423642243631977391334.65165 PyatkovS. G.“Solvability of some inverse problems for parabolic equations,” J. InvIll-Posed Problems200412397412208538210.1515/15693940422481931081.35148 Il’inV. A.On the absolute and uniform convergence of the expansions in eigen- and associated functions of a non-self-adjoint elliptic operatorDokl. Akad. Nauk SSSR19842741922730157 KozhanovA. 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V.Simultaneous determination of two coefficients of a parabolic equation in the case of nonlocal and integral conditionsUkr. Math. J.200153674684185454910.1023/A:10125700312420991.35102 KerimovN. B.IsmailovM. I.Direct and inverse problems for the heat equation with a dynamic-type boundary conditionIMA J. Appl. Math.20158015191533340398910.1093/imamat/hxv0051327.35369 IonkinN. I.Solution of a boundary-value problem in heat conduction with a non-classical boundary conditionDiffer. Equations197713294304603291 AbramowitzM.StegunA.Handbook of Mathematical Functions1965Mineola, NYDover0171.38503 KostinA. B.Inverse problem of finding the coefficient of u in a parabolic equation on the basis of a nonlocal observation conditionDiffer. Equations201551605619337483610.1134/S00122661150500431332.35395 ChoulliM.An inverse problem for a semilinear parabolic equationInverse Probl.19941011231132129636310.1088/0266-5611/10/5/0090807.35154 KozhanovA. I.“An inverse problem with an unknown coefficient and right-hand side for a parabolic equation II,” J. InvIll-Posed Probl.200311505522201867510.1515/1569394037708882461142.35626 SamarskiiA. A.VabishchevichP. N.Numerical Methods for Solving Inverse Problems of Mathematical Physics2007BerlinDe Gruyter10.1515/97831102057941136.65105 CannonJ. R.LinY.Determination of a parameter p(t) in some quasi-linear parabolic differential equationsInverse Probl.19884354510.1088/0266-5611/4/1/0060697.35162 KozhanovA. I.Nonlinear loaded equations and inverse problemsComput. Math. Math. Phys. Fiz.20044465767820692121114.35148 N. I. Ionkin (1168_CR29) 1977; 13 A. I. Kozhanov (1168_CR13) 2006; 74 A. V. Bicadze (1168_CR19) 1969; 185 M. Abramowitz (1168_CR37) 1965 P. N. Vabishchevich (1168_CR26) 2014; 89 M. I. Ivanchov (1168_CR32) 2001; 53 A. I. Kozhanov (1168_CR14) 2002; 10 A. I. Kozhanov (1168_CR16) 2004; 44 J. R. Cannon (1168_CR31) 1991; 33 V. Isakov (1168_CR1) 1998 V. A. Il’in (1168_CR36) 1984; 274 N. B. Kerimov (1168_CR38) 2015; 80 F. S. V. Bazan (1168_CR27) 2017; 214 A. Tarantola (1168_CR22) 1987 M. Choulli (1168_CR9) 1994; 10 J. R. Cannon (1168_CR5) 1988; 4 A. I. Kozhanov (1168_CR12) 2003; 11 V. L. Kamynin (1168_CR8) 2015; 97 V. L. Kamynin (1168_CR10) 2013; 94 C. R. Vogel (1168_CR23) 2002 1168_CR33 A. I. Prilepko (1168_CR17) 1993; 34 A. A. Samarskii (1168_CR21) 2007 cr-split#-1168_CR2.2 M. Dehghan (1168_CR25) 2000; 77 A. I. Prilepko (1168_CR3) 2000 A. B. Kostin (1168_CR11) 2015; 51 J. R. Cannon (1168_CR7) 1994; 10 T. I. Bukharova (1168_CR4) 2015; 55 M. I. Ismailov (1168_CR35) 2011; 34 N. I. Ionkin (1168_CR30) 2000; 36 J. R. Cannon (1168_CR6) 1988; 4 O. M. Alifanov (1168_CR20) 2011 L. Bedin (1168_CR28) 2014; 236 A. V. Gulin (1168_CR34) 2001; 37 D. S. Daoud (1168_CR24) 2005; 166 A. I. Prilepko (1168_CR15) 1987; 23 cr-split#-1168_CR2.1 S. G. Pyatkov (1168_CR18) 2004; 12 1168_CR39 |
| References_xml | – reference: BukharovaT. I.KamyninV. L.Inverse problem of determining the absorption coefficient in the multidimensional heat equation with unlimited minor coefficientsComput. Math. Math. Phys.20155511641176337263810.1134/S09655425150700521323.35208 – reference: IonkinN. I.MorozovaV. A.The two-dimensional heat equation with nonlocal boundary conditionsDiffer. Equations200036982987181959110.1007/BF027544980979.35059 – reference: BazanF. S. V.BedinL.BorgesL. S.Space-dependent perfusion coefficient estimation in a 2D bioheat transfer problemComput. Phys. Commun.20172141830361411910.1016/j.cpc.2017.01.0021380.65297 – reference: DaoudD. S.SubasiD.A splitting up algorithm for the determination of the control parameter in multi-dimensional parabolic problemAppl. Math. Comput.200516658459521504911078.65082 – reference: BedinL.BazanF. S. V.On the 2D bioheat equation with convective boundary conditions and its numerical realization via a highly accurate approachAppl. Math. Comput.201423642243631977391334.65165 – reference: VogelC. R.Computational Methods for Inverse Problems2002Philadelphia, PASoc. Ind. Appl. Math10.1137/1.97808987175701008.65103 – reference: IsakovV.Inverse Problems for Partial Differential Equations1998BerlinSpringer10.1007/978-1-4899-0030-20908.35134 – reference: KostinA. B.Inverse problem of finding the coefficient of u in a parabolic equation on the basis of a nonlocal observation conditionDiffer. Equations201551605619337483610.1134/S00122661150500431332.35395 – reference: DehghanM.Numerical methods for two-dimensional parabolic inverse problem with energy overspecificationInt. J. Comput. Math.200077441455189899810.1080/002071601088050770986.65085 – reference: M. M. Lavrent’ev, V. G. Romanov, and S. P. Shishatskii, Ill-Posed Problems of Mathematical Physics and Analysis (Nauka, Moscow, 1980; Am. Math. Soc., Providence, R.I., 1986). – reference: IonkinN. I.Solution of a boundary-value problem in heat conduction with a non-classical boundary conditionDiffer. Equations197713294304603291 – reference: KozhanovA. I.Parabolic equations with an unknown absorption coefficientDokl. Math.20067457357610.1134/S10645624060402721152.35112 – reference: BicadzeA. V.SamarskiiA. A.Some elementary generalizations of linear elliptic boundary value problemsDokl. Akad. Nauk SSSR1969185739740247271 – reference: SamarskiiA. A.VabishchevichP. N.Numerical Methods for Solving Inverse Problems of Mathematical Physics2007BerlinDe Gruyter10.1515/97831102057941136.65105 – reference: CannonJ. R.LinY.Determination of parameter p(t) in Hölder classes for some semilinear parabolic equationsInverse Probl.1988459560610.1088/0266-5611/4/3/0050688.35104 – reference: KamyninV. L.The inverse problem of the simultaneous determination of the right-hand side and the lowest coefficient in parabolic equations with many space variablesMath. Notes201597349361337052510.1134/S00014346150300621325.35281 – reference: Il’inV. A.On the absolute and uniform convergence of the expansions in eigen- and associated functions of a non-self-adjoint elliptic operatorDokl. Akad. Nauk SSSR19842741922730157 – reference: KozhanovA. I.“An inverse problem with an unknown coefficient and right-hand side for a parabolic equation II,” J. InvIll-Posed Probl.200311505522201867510.1515/1569394037708882461142.35626 – reference: KozhanovA. I.Nonlinear loaded equations and inverse problemsComput. Math. Math. Phys. Fiz.20044465767820692121114.35148 – reference: AlifanovO. M.Inverse Heat Transfer Problems2011 – reference: M. K. Bowen and R. Smith, “Derivative formulas and errors for non-uniformly spaced points,” Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 461 (2059), 1975–1997 (2005). – reference: GulinA. V.IonkinN. I.MorozovaV. A.Stability of a nonlocal two-dimensional finite-difference problemDiffer. 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| Snippet | We consider an inverse problem of determining the time-dependent lowest order coefficient of two-dimensional (2D) heat equation with Ionkin boundary and total... |
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| SubjectTerms | Boundary conditions Computational Mathematics and Numerical Analysis Economic models Finite difference method Inverse problems Mathematical analysis Mathematics Mathematics and Statistics Numerical integration Thermodynamics Time dependence Volterra integral equations Well posed problems |
| Title | Inverse Problem of Finding the Coefficient of the Lowest Term in Two-Dimensional Heat Equation with Ionkin-Type Boundary Condition |
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