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Mat. Sb., 2013, Volume 204, Number 10, Pages 3–46 (Mi msb8104)  

This article is cited in 27 scientific papers (total in 27 papers)

The inverse problem of recovering the source in a parabolic equation under a condition of nonlocal observation

A. B. Kostin

National Engineering Physics Institute "MEPhI", Moscow

Abstract: We study the inverse problem for a parabolic equation of recovering the source, that is, the right-hand side $F(x,t)=h(x,t)f(x)$, where the function $f(x)$ is unknown. To find $f(x)$, along with the initial and boundary conditions, we also introduce an additional condition of nonlocal observation of the form $\displaystyle\int_{0}^{T}u(x,t) d\mu(t)=\chi(x)$. We prove the Fredholm property for the problem stated in this way, and obtain sufficient conditions for the existence and uniqueness of a solution. These conditions are of the form of readily verifiable inequalities and put no restrictions on the value of $T>0$ or the diameter of the domain $\Omega$ under consideration. The proof uses a priori estimates and the qualitative properties of solutions of initial-boundary value problems for parabolic equations.
Bibliography: 40 titles.

Keywords: inverse problems, parabolic equations, nonlocal overdetermination.

DOI: https://doi.org/10.4213/sm8104

Full text: PDF file (888 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2013, 204:10, 1391–1434

Bibliographic databases:

UDC: 517.95
MSC: Primary 35R30; Secondary 34A55
Received: 16.01.2012 and 17.06.2013

Citation: A. B. Kostin, “The inverse problem of recovering the source in a parabolic equation under a condition of nonlocal observation”, Mat. Sb., 204:10 (2013), 3–46; Sb. Math., 204:10 (2013), 1391–1434

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. Ashyralyev, D. Agirseven, “On source identification problem for a delay parabolic equation”, Nonlinear Anal. Model. Control, 19:3 (2014), 335–349  crossref  mathscinet  zmath  isi
    2. A. B. Kostin, “Recovery of the coefficient of $u_t$ in the heat equation from a condition of nonlocal observation in time”, Comput. Math. Math. Phys., 55:1 (2015), 85–100  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    3. T. K. Yuldashev, “Obratnaya zadacha dlya nelineinogo integro-differentsialnogo uravneniya Fredgolma chetvertogo poryadka s vyrozhdennym yadrom”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 19:4 (2015), 736–749  mathnet  crossref  zmath  elib
    4. A. B. Kostin, “Inverse problem of finding the coefficient of $u$ in a parabolic equation on the basis of a nonlocal observation condition”, Differ. Equ., 51:5 (2015), 605–619  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
    5. T. K. Yuldashev, “Inverse problem for a nonlinear Benney–Luke type integro-differential equations with degenerate kernel”, Russian Math. (Iz. VUZ), 60:9 (2016), 53–60  mathnet  crossref  isi
    6. T. K. Yuldashev, “Obratnaya zadacha dlya integro-differentsialnogo uravneniya Fredgolma tretego poryadka s vyrozhdennym yadrom”, Vladikavk. matem. zhurn., 18:2 (2016), 76–85  mathnet
    7. T. K. Yuldashev, “Obratnaya zadacha dlya obyknovennogo integro-differentsialnogo uravneniya s vyrozhdennym yadrom i nelokalnymi integralnymi usloviyami”, Vestnik TvGU. Seriya: Prikladnaya matematika, 2016, no. 3, 19–33  mathnet  crossref  elib
    8. A. B. Kostin, “Inverse problem with nonlocal observation of finding the coefficient multiplying $u_t$ in the parabolic equation”, Differ. Equ., 52:2 (2016), 220–239  crossref  mathscinet  zmath  isi  scopus
    9. A. U. Sazaklioglu, A. Ashyralyev, A. S. Erdogan, “Existence and uniqueness results for an inverse problem for semilinear parabolic equations”, Filomat, 31:4 (2017), 1057–1064  crossref  mathscinet  isi  scopus
    10. T. K. Yuldashev, “Determination of the coefficient and boundary regime in boundary value problem for integro-differential equation with degenerate kernel”, Lobachevskii J. Math., 38:3 (2017), 547–553  crossref  mathscinet  zmath  isi  scopus
    11. P. N. Vabishchevich, “Iterative computational identification of a space-wise dependent source in parabolic equation”, Inverse Probl. Sci. Eng., 25:8 (2017), 1168–1190  crossref  mathscinet  zmath  isi  scopus
    12. A. Ashyralyev, A. U. Sazaklioglu, “Investigation of a time-dependent source identification inverse problem with integral overdetermination”, Numer. Funct. Anal. Optim., 38:10 (2017), 1276–1294  crossref  mathscinet  zmath  isi  scopus
    13. A. Ashyralyev, F. Emharab, “Source identification problems for hyperbolic differential and difference equations”, International Conference Functional Analysis in Interdisciplinary Applications (FAIA 2017), AIP Conf. Proc., 1880, eds. T. Kalmenov, M. Sadybekov, Amer. Inst. Phys., Melville, NY, 2017, UNSP 040001  crossref  isi  scopus
    14. A. I. Prilepko, A. B. Kostin, V. V. Solovev, “Obratnye zadachi nakhozhdeniya istochnika i koeffitsientov dlya ellipticheskikh i parabolicheskikh uravnenii v prostranstvakh Geldera i Soboleva”, Sib. zhurn. chist. i prikl. matem., 17:3 (2017), 67–85  mathnet  crossref
    15. A. I. Prilepko, V. L. Kamynin, A. B. Kostin, “Inverse source problem for parabolic equation with the condition of integral observation in time”, J. Inverse Ill-Posed Probl., 26:4 (2018), 523–539  crossref  mathscinet  zmath  isi  scopus
    16. A. Ashyralyev, F. Emharab, “Identification hyperbolic problems with nonlocal conditions”, International Conference on Analysis and Applied Mathematics (ICAAM 2018), AIP Conf. Proc., 1997, eds. A. Ashyralyev, A. Lukashov, M. Sadybekov, Amer. Inst. Phys., 2018, UNSP 020031-1  crossref  isi  scopus
    17. A. Ashyralyev, M. Urun, “Source identification problems for Schrödinger differential and difference equations”, International Conference on Analysis and Applied Mathematics (ICAAM 2018), AIP Conf. Proc., 1997, eds. A. Ashyralyev, A. Lukashov, M. Sadybekov, Amer. Inst. Phys., 2018, UNSP 020014-1  crossref  isi  scopus
    18. A. Ashyralyev, F. Emharab, “Identification hyperbolic problems with the Neumann boundary condition”, Bull. Karaganda Univ-Math., 91:3 (2018), 89–98  crossref  isi
    19. V. V. Solov'ev, “On determining sources with compact supports in a bounded plane domain for the heat equation”, Comput. Math. Math. Phys., 58:5 (2018), 750–760  mathnet  crossref  crossref  isi  elib
    20. A. B. Rahimov, “An approach to solution to class of coefficient-inverse nonlocal problems for a parabolic equation”, Proceedings of the 6th International Conference on Control and Optimization With Industrial Applications, Vol I, eds. A. Fikret, B. Tamer, Baku State Univ, Inst Applied Mathematics, 2018, 309–311  isi
    21. S. N. Sidorov, “Obratnye zadachi dlya uravneniya smeshannogo parabolo-giperbolicheskogo tipa s vyrozhdayuscheisya parabolicheskoi chastyu”, Sib. elektron. matem. izv., 16 (2019), 144–157  mathnet  crossref
    22. S. N. Sidorov, “Inverse problems for a degenerate mixed parabolic-hyperbolic equation on finding time-depending factors in right hand sides”, Ufa Math. J., 11:1 (2019), 75–89  mathnet  crossref  isi
    23. Ashyralyev A., Emharab F., “Source Identification Problems For Hyperbolic Differential and Difference Equations”, J. Inverse Ill-Posed Probl., 27:3 (2019), 301–315  crossref  mathscinet  zmath  isi
    24. T. K. Yuldashev, “Opredelenie koeffitsienta v nelokalnoi zadache dlya integro-differentsialnogo uravneniya tipa Bussineska s vyrozhdennym yadrom”, Vladikavk. matem. zhurn., 21:2 (2019), 67–84  mathnet  crossref
    25. Soloviev V.V., Tkachenko D.S., Vii International Conference Problems of Mathematical Physics and Mathematical Modelling, Journal of Physics Conference Series, 1205, IOP Publishing Ltd, 2019  crossref  isi
    26. Ashyralyev A., Urun M., “Time-Dependent Source Identification Problem For the Schrodinger Equation With Nonlocal Boundary Conditions”, AIP Conference Proceedings, 2183, eds. Cakalli H., Kocinac L., Harte R., Cavalcanti V., Ashyralyev A., Sakalli I., Canak I., Gurtug O., Cavalcanti M., Turkoglu D., Tez M., Kandemir H., Uyaver S., Akay K., Gul I., Yilmazturk T., Akyel T., Ucgun F., Sahin H., Amer Inst Physics, 2019, UNSP 070016  crossref  isi
    27. Ashyralyev A., Al-Hammouri A., “A Numerical Algorithm For a Source Identification Problem For the Elliptic-Telegraph Equation”, AIP Conference Proceedings, 2183, eds. Cakalli H., Kocinac L., Harte R., Cavalcanti V., Ashyralyev A., Sakalli I., Canak I., Gurtug O., Cavalcanti M., Turkoglu D., Tez M., Kandemir H., Uyaver S., Akay K., Gul I., Yilmazturk T., Akyel T., Ucgun F., Sahin H., Amer Inst Physics, 2019, UNSP 070008  crossref  isi
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