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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]:
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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2014, Issue 1(34), Pages 56–65 (Mi vsgtu1299)  

This article is cited in 23 scientific papers (total in 23 papers)

Differential Equations

Inverse Problem for a Fredholm Third Order Partial Integro-differential Equation

T. K. Yuldashev

M. F. Reshetnev Siberian State Aerospace University, Krasnoyarsk, 660014, Russian Federation

Abstract: The solvability of various problems for partial differential equations of the third order is researched in many papers. But, partial Fredholm integro-differential equations of the third order are studied comparatively less. Integro-differential equations have traits in their one-valued solvability. The questions of solvability of linear inverse problems for partial differential equations are studied by many authors. We consider a nonlinear inverse problem, where the restore function appears in the equation nonlinearly and with delay. This equation with respect to the restore function is Fredholm implicit functional integral equation. The one- valued solvability of the nonlinear inverse problem for a partial Fredholm integro-differential equation of the third order is studied. First, the method of degenerate kernel designed for Fredholm integral equations is modified to the case of partial Fredholm integro-differential equations of the third order. The nonlinear Volterra integral equation of the first kind is obtained while solving the nonlinear inverse problem with respect to the restore function. This equation by the special non-classical integral transformation is reduced to a nonlinear Volterra integral equation of the second kind. Since the restore function, which entered into the integro-differential equation, is nonlinear and has delay time, we need an additional initial value condition with respect to restore function. This initial value condition ensures the uniqueness of solution of a nonlinear Volterra integral equation of the first kind and determines the value of the unknown restore function at the initial set. Further the method of successive approximations is used, combined with the method of contracting mapping.

Keywords: nonlinear inverse problem, partial differential equation of the third order, implicit functional-integral equation, integral transformation, method of successive approximations

DOI: https://doi.org/10.14498/vsgtu1299

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Bibliographic databases:

Document Type: Article
UDC: 517.968.7
MSC: Primary 35R30; Secondary 35K70, 35M12
Original article submitted 28/XII/2013
revision submitted – 24/II/2014

Citation: T. K. Yuldashev, “Inverse Problem for a Fredholm Third Order Partial Integro-differential Equation”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 1(34) (2014), 56–65

Citation in format AMSBIB
\Bibitem{Yul14}
\by T.~K.~Yuldashev
\paper Inverse Problem for a Fredholm Third Order Partial Integro-differential Equation
\jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]
\yr 2014
\vol 1(34)
\pages 56--65
\mathnet{http://mi.mathnet.ru/vsgtu1299}
\crossref{https://doi.org/10.14498/vsgtu1299}
\zmath{https://zbmath.org/?q=an:06968825}
\elib{http://elibrary.ru/item.asp?id=22813960}


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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. T. K. Yuldashev, “Obratnaya zadacha dlya odnogo nelineinogo uravneniya v chastnykh proizvodnykh vosmogo poryadka”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 19:1 (2015), 136–154  mathnet  crossref  zmath  elib
    2. Yuldashev T. K., “Upravlenie v nelineinoi obratnoi zadache dlya odnoi sistemy s differentsialnym uravneniem psevdoparabolicheskogo tipa”, Vestnik Voronezhskogo gosudarstvennogo universiteta. Seriya: Sistemnyi analiz i informatsionnye tekhnologii, 2015, no. 1, 23–31  elib
    3. T. K. Yuldashev, “On Fredholm partial integro-differential equation of the third order”, Russian Math. (Iz. VUZ), 59:9 (2015), 62–66  mathnet  crossref
    4. T. K. Yuldashev, “Ob odnoi obratnoi zadache dlya lineinogo integro-differentsialnogo uravneniya Fredgolma v chastnykh proizvodnykh chetvertogo poryadka”, Vestnik Voronezhskogo gosudarstvennogo universiteta. Seriya: Fizika. Matematika, 2015, no. 2, 180–189  zmath  elib
    5. 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
    6. T. K. Yuldashev, O. V. Solodova, “Ob odnom integro-differentsialnom uravnenii tipa Volterra s nelineinoi pravoi chastyu”, Reshetnevskie chteniya, 2:19 (2015), 136–138  elib
    7. T. K. Yuldashev, K. Kh. Shabadikov, “Obratnaya zadacha dlya psevdoparabolicheskogo integro-differentsialnogo uravneniya Fredgolma s vyrozhdennym yadrom”, Reshetnevskie chteniya, 2:19 (2015), 138–140  elib
    8. A. O. Bulov, T. K. Yuldashev, “Nelokalnaya obratnaya zadacha dlya parabolicheskogo integro-differentsialnogo uravneniya Fredgolma s vyrozhdennym yadrom”, Aktualnye problemy aviatsii i kosmonavtiki, 1:11 (2015), 299–301  elib
    9. A. G. Loskutova, T. K. Yuldashev, “Smeshannaya zadacha dlya nelineinogo integro-differentsialnogo uravneniya giperbolicheskogo tipa s vyrozhdennym yadrom”, Aktualnye problemy aviatsii i kosmonavtiki, 1:11 (2015), 343–346  elib
    10. 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
    11. T. K. Yuldashev, “Nelineinoe integro-differentsialnoe uravnenie psevdoparabolicheskogo tipa s nelokalnym integralnym usloviem”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2016, no. 1(32), 11–23  mathnet  crossref
    12. T. K. Yuldashev, “Smeshannoe differentsialnoe uravnenie tipa Bussineska”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2016, no. 2(33), 13–26  mathnet  crossref
    13. T. K. Yuldashev, “Nonlocal problem for a mixed type differential equation in rectangular domain”, Uch. zapiski EGU, ser. Fizika i Matematika, 2016, no. 3, 70–78  mathnet
    14. 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
    15. K. B. Matanova, B. K. Temirov, “Obratnaya zadacha ob istochnike dlya differentsialnogo uravneniya tretego poryadka s chastnymi proizvodnymi”, Estestvennye i matematicheskie nauki v sovremennom mire, 2016, no. 10 (45), 45–59  elib
    16. T. K. Yuldashev, “Nelineinoe integro-differentsialnoe uravnenie psevdoparabolicheskogo tipa s nelokalnym integralnym usloviem”, Vestnik VolGU. Seriya 1. Matematika. Fizika, 2016, no. 1 (32), 11–23  elib
    17. T. K. Yuldashev , K. Kh. Shabadikov, “Kvazilineinoe integro-differentsialnoe uravnenie psevdoparabolicheskogo tipa s vyrozhdennym yadrom i integralnym usloviem”, Zhurnal SVMO, 18:4 (2016), 76–88  mathnet  elib
    18. T. K. Yuldashev, “Obyknovennoe integro-differentsialnoe uravnenie s vyrozhdennym yadrom i integralnym usloviem”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 20:4 (2016), 644–655  mathnet  crossref  zmath  elib
    19. T. K. Yuldashev, “Mixed problem for pseudoparabolic integro-differential equation with degenerate kernel”, Diff. Equat., 53:1 (2017), 99–108  crossref  crossref  isi  elib  elib  scopus
    20. T. K. Yuldashev, “Determination of the coefficient and boundary regime in boundary value problem for integro-differential equation with degenerate kernel”, Lobachevskii journal of mathematics, 38:3 (2017), 547–553  crossref  scopus
    21. T. K. Yuldashev, “Nelokalnaya kraevaya zadacha dlya neodnorodnogo psevdoparabolicheskogo integro-differentsialnogo uravneniya s vyrozhdennym yadrom”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2017, no. 1(38), 42–54  mathnet  crossref
    22. T. K. Yuldashev, “Nonlocal Mixed-Value Problem for a Boussinesq-Type Integrodifferential Equation with Degenerate Kernel”, Ukrainian Mathematical Journal, 68:8 (2017), 1278–1296  crossref  isi  scopus
    23. S. K. Zaripov, “Postroenie analoga teoremy Fredgolma dlya odnogo klassa modelnykh integro-differentsialnykh uravnenii pervogo poryadka s logarifmicheskoi osobennostyu v yadre”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 21:2 (2017), 236–248  mathnet  crossref  zmath  elib
  • Вестник Самарского государственного технического университета. Серия: Физико-математические науки
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