Обратные задачи восстановления параметров в дифференциальном уравнении с кратными характеристиками
DOI:
https://doi.org/10.26577/JMMCS.2022.v113.i1.01Аннотация
Обратные задачи - задача нахождения причин известных или заданных следствий. Они возникают, когда характеристики интересующего нас объекта недоступны для непосредственного наблюдения. Это, например, восстановление характеристик источников поля в соответствии с их заданными значениями в некоторых точках, восстановление или интерпретация исходного сигнала из известного выходного сигнала и т.д. В данной работе исследуется разрешимость нахождения решения дифференциального уравнения обратных задач. Работа посвящена исследованию разрешимости в пространствах Соболева нелинейных обратных коэффициентных задач для дифференциальных уравнений третьего порядка с множественными характеристиками. В этой статье, наряду с поиском решения того или иного дифференциального уравнения, также требуется найти один или несколько коэффициентов самого уравнения, чтобы мы назвали их обратными коэффициентными задачами. Отличительной особенностью задач, изучаемых в данной работе, является то, что неизвестный коэффициент является числовым параметром, а не функцией определенных независимых переменных.
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[4] Kozhanov, A. I. Composite Type Equations and Inverse Problems. Utrecht: VSP, 1999.
[5] Lorenzi, A. Introduction to Identification Problems via Functional Analysis. Utrecht: VSP, 2001.
[6] Anikonov, Yu.E. Inverse Problems for Kinetic and Other Evolution Equation. Utrecht: VSP, 2001.
[7] Danilaev, P.G. Coefficient Inverse Problems for Parabolic Type Equations and Their Applications. Utrecht: VSP, 2001.
[8] Belov, Yu.Ya. Inverse Problems for Partial Differntial Equations. Utrecht: VSP, 2002.
[9] Lavrentiev, M.M. Inverse Problems of Mathematical Physics. Utrecht: VSP, 2003.
[10] Megrabov, A.G. Forward and Inverse Problems for Hyperbolic, Elliptic and Mixed Type Equations. Utrecht: VSP, 2003.
[11] Ivanchov, M. Inverse Problems for Equations of Parabolic Type. Lviv: WNTL Publishers, 2003.
[12] Romanov V. G. Stability in inverse problems. Moscow: Nauchny mir, 2005.(In Russian)
[13] Isakov, V. Inverse Problem for Partial Differential Equations. Springer Verlag, 2009.
[14] Kabanikhin S. I. Inverse and ill-posed problems. Novosibirsk: Siberian Book Publishing House. 2009. (In Russian)
[15] Hasanov, H.A., Romanov, V.G. Introduction to Inverse Problems for Differential Equations; Springer: New York, NY, USA, 2017.
[16] Vatulyan A. O. Coefficient inverse problems of mechanics. Moscow: Fizmatlit, 2019.(In Russian)
[17] Kozhanov A. I. On the definition of sources in the linearized Kortweg-de Vries equation. Math. notes of YSU. 2002. Vol. 9, vol. 2. P. 74-82.(In Russian)
[18] Kozhanov A. I. On solvability of some nonlinear inverse problems for a single equation with multiple characteristics. Vestnik NSU, series: mathematics, mechanics, Informatics. 2003. Vol. 3, issue. 3. P. 34-51.(In Russian)
[19] Kozhanov A. I. Inverse problem for equations with multiple characteristics: the case of the unknown coefficient is timedependent. The reports of the Adyghe (Circassian) international Academy of Sciences. 2005. Vol. 8, No. 1. P. 38-49.(In Russian)
[20] Lorenzi, A. Recovering Two Constans in a Linear Parabolic Equation. Inverse Problems in Applied Sciences. J. Phys. Conf. Ser. 2007, 73, 012014.
[21] Lyubanova, A.S. Identification of a Constant Coefficient in an Elliptic Equation. Appl. Anal. 2008, 87, 1121–1128.
[22] Lorenzi, A.; Mola, G. Identification of a Real Constant in Linear Evolution Equation in a Hilbert Spaces. Inverse Probl. Imaging. 2011, 5, 695–714.
[23] Mola, G. Identification of the Diffusion Coefficient in Linear Evolution Equation in a Hilbert Spaces. J. Abstr. Differ. Appl. 2011, 2, 18–28.
[24] Lorenzi, A.; Mola, G. Recovering the Reaction and the Diffusion Coefficients in a Linear Parabolic Equations. Inverse Probl. 2012, 28, 075006.
[25] Mola, G.; Okazawa, N.; Pru?ss, J.; Yokota, T. Semigroup–Theoretic Approach to Identification of Linear Diffusion Coefficient. Discret. Contin. Dyn. Syst. S. 2016, 9, 777–790.
[26] Kozhanov, A.I.; Safiullova, R.R. Determination of Parameter in Telegraph Equation. Ufa Math. J. 2017, 9, 62–75.
[27] Kozhanov, A.I. Inverse Problems of Finding of Absorption Parameter in the Diffusion Equation. Math. Notes. 2019, 106, 378–389.
[28] Kozhanov, A.I. The Heat Transfer Equation with an Unknown Heat Capacity Coefficient. J. Appl. Ind. Math. 2020, 14, 104–114. 15
[29] Kozhanov, A. I. Hyperbolic Equations with Unknown Coefficients Symmetry. 2020. V. 12. Issue 9. 1539.
[30] V. S. Vladimirov, equations of mathematical physics. M.: Nauka, 1974.
[31] Courant, R. Partial Differential Equations. New York–London, 1962.
[32] Sobolev, S.L. Some Applications of Functional Analysis in Mathematical Physics, Providence: Amer. Math. Soc., 1991.
[33] Ladyzhenskaya O.A., Uraltseva N.N. Linear and Quasilinear Elliptic Equations, New York and London: Academic Press, 1987.
[34] Triebel, H. Interpolation Theory, Function Spaces, Differential Operators. North–Holland Publ., Amsterdam, 1978.
[35] Dzhenaliev M. T. To the theory of linear boundary value problems for loaded differential equations. Almaty. Institute of theoretical and applied mathematics, 1995.(In Russian)
[36] Nakhushev a.m. Loaded equations and their application. Moscow: Nauka. 2012 (In Russian)
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Опубликован
2022-03-31
Выпуск
Раздел
Математика
