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Inverse Problems for a Two-Dimensional Heat Equation with Unknown Right-Hand Side

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Abstract

Based on the solution of the first initial-boundary value problem for an inhomogeneous two-dimensional heat equation, we state and study inverse problems, whose right-hand sides contain unknown factors depending on spatial and time variables. Preliminarily, we explicitly construct a solution to the direct initial-boundary-value problem. We prove the uniqueness of the solution to direct and inverse problems, making use of the completeness property of the system of eigenfunctions of the corresponding homogeneous Dirichlet problem for the Laplace operator. We also establish existence theorems for solutions of inverse problems and construct solutions explicitly.

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REFERENCES

  1. Lavrent'ev, M.M., Vasil'ev, V.G., Romanov, V.G. Multidimensional Inverse Problems for Differential Equations (Nauka, Novosibirsk, 1969) [in Russian].

  2. Ivanov, V.K., Vasin, V.V., Tanana, V.P. Theory of Linear Ill-Posed Problems and its Applications (Nauka, Moscow, 1978) [in Russian].

  3. Lavrent'ev, M.M., Reznitskaya, K.G., Yakhno, V.G. One-Dimensional Inverse Problems of Mathematical Physics (Nauka, Novosibirsk, 1982) [in Russian].

  4. Romanov, V.G. Inverse Problems of Mathematical Physics (Nauka, Moscow, 1984) [in Russian].

  5. Romanov, V.G., Kabanikhin, S.I. Inverse Problems of Geoelectrics (Nauka, Moscow, 1991) [in Russian].

  6. Denisov, A.M. Introduction to the Theory of Inverse Problems (Mosk. Gos. Univ., Moscow, 1994) [in Russian].

  7. Prilepko, A.I., Orlovsky, D.G., Vasin, I.A. Methods for Solving Inverse Problems in Mathematical Physics (Marcel Dekker Inc., New York–Basel, 1999).

  8. Isakov, V. Inverse Problems for Partial Differential Equations (Springer, New York, 2006).

  9. Kabanikhin, S.I. Inverse and Ill-Posed Problems (Sib. Nauchn. Izd-vo, Novosibirsk, 2009) [in Russian].

  10. Prilepko, A.I., Solov'ev, V.V. “Solvability Theorems and the Rothe Method in Inverse Problems for an Equation of Parabolic Type. I”, Differents. Uravn. 23 (10), 1791–1799 (1987).

  11. Prilepko, A.I., Solov'ev, V.V. “Solvability Theorems and the Rothe Method in Inverse Problems for an Equation of Parabolic Type. II”, Differents. Uravn. 23 (11), 1971–1980 (1987).

  12. Solov'ev, V.V. “Determination of the Source and Coefficients in a Parabolic Equation in the Multidimensional Case”, Differents. Uravn. 31 (6), 1060–1069 (1995).

  13. Kostin, A.B. “The Inverse Problem of Recovering the Source in a Parabolic Equation under a Condition of Nonlocal Observation”, Matem. Sborn. 204 (10), 3–46 (2013).

  14. Orlovskii, D.G. “On a Problem of Determining the Parameter of an Evolution Equation”, Differents. Uravn. 26 (9), 1614–1621 (1990).

  15. Tikhonov, I.V., Eidel'man, Yu.S. “Uniqueness Criterion in an Inverse Problem for an Abstract Differential Equation with Nonstationary Inhomogeneous Term”, Matem. Zametki 77 (2), 273–290 (2005).

  16. Sabitov, K.B., Safin, E.M. “Inverse Problem for a Parabolic-Hyperbolic Equation in a Rectangular Domain”, Dokl. Ross. Akad. Nauk 429 (4), 451–454 (2009).

  17. Sabitov, K.B., Safin, E.M. “The Inverse Problem for a Mixed-Type Parabolic-Hyperbolic Equation in a Rectangular Domain”, Russian Mathematics 54, 48–54 (2010).

  18. Sabitov, K.B., Safin, E.M. “The Inverse Problem for an Equation of Mixed Parabolic-Hyperbolic Type”, Matem. Zametki 87 (6), 907–918 (2010).

  19. Sabitov, K.B. “Dirichlet Problem for Mixed-Type Equations in a Rectangular Domain”, Dokl. Ross. Akad. Nauk 413 (1), 23–26 (2007).

  20. Il'in, V.A., Sadovnichii, V.A., Sendov, Bl.Kh. Mathematical Analysis, Vol. 2 (Mosk. Gos. Univ., Moscow, 1987) [in Russian].

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Authors and Affiliations

  1. Institute for Strategic Studies of the Republic of Bashkortostan, 68 Odessa str., 453103, Sterlitamak, Russia

    K. B. Sabitov

  2. Sterlitamak branch of Bashkir State University, 49 Lenin Ave., 453103, Sterlitamak, Russia

    K. B. Sabitov & A. R. Zainullov

Authors
  1. K. B. Sabitov
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  2. A. R. Zainullov
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Correspondence to K. B. Sabitov or A. R. Zainullov.

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Russian Text © The Author(s), 2021, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2021, No. 3, pp. 83–97.

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Sabitov, K.B., Zainullov, A.R. Inverse Problems for a Two-Dimensional Heat Equation with Unknown Right-Hand Side. Russ Math. 65, 75–88 (2021). https://doi.org/10.3103/S1066369X21030087

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  • DOI: https://doi.org/10.3103/S1066369X21030087

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