Third-order differential equations with multiple characteristics: degeneration and unknown external influence

A. I. Kozhanov ; G. R. Ashurova

Čelâbinskij fiziko-matematičeskij žurnal, Tome 9 (2024) no. 4, pp. 585-595.

Résumé

Inverse problems of determination together with the solution of a degenerate differential equation with multiple characteristics of an unknown coefficient representing external influence (the free term) are studied. The nature of degeneration in the studied equation, as well as the type of the unknown coefficient, are determined by the time variable. Theorems of existence and uniqueness of regular solutions (solutions that possess all generalized derivatives according to S.L. Sobolev, which are involved in the equation) are proved for the studied problems.

Détail

Keywords: differential equation with multiple characteristics, degeneration, inverse problem, unknown external influence, regular solution, existence, uniqueness.

Bibliographie

[1] Romanov V.G., Inverse Problems of Mathematical Physics, BRILL, Utrecht, 1987

[2] Anikonov Yu. E., Bubnov B. A., Erokhin G. N., Inverse and Ill-Posed Sources Problems, VSP, Utrecht, 1997

[3] Kozhanov A. I., Composite Type Equations and Inverse Problems, VSP, Utrecht, 1999

[4] Prilepko A. I., Orlovsky D. G., Vasin I. A., Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York, 1999

[5] Anikonov Yu. E., Inverse Problems for Kinetic and Other Evolution Equation, VSP, Utrecht, 2001

[6] Belov Yu. Ya., Inverse Problems for Partial Differential Equations, VSP, Utrecht, 2002

[7] Ivanchov M., Inverse Problems for Equations of Parabolic Type, WNTI Publ., Lviv, 2003

[8] Lavrentiev M. M., Inverse Problems of Mathematical Physics, VSP, Utrecht, 2003

[9] Isakov V., An Inverse Problems for Partial Differential Equations, Springer, New York, 2006

[10] Kabanihin S.I., Inverse and ill-posed problems, Siberian book publishing house, Novosibirsk, 2009 (In Russ.)

[11] Lesnic D., Inverse Problems with Applications in Science and Engineering, Taylor and Frencis Group, New York, 2022

[12] Ablabekov B., Inverse problems for third-order differential equations, LAP Lambert Academic Publ., 2013 (In Russ.)

[13] Kozhanov A.I., “Inverse problems for differential equations with multiple characteristics: a case of unknown element depending on time”, Reports of Adyghe (Circassian) International Academy of Science, 8:1 (2005), 38–49 (In Russ.)

[14] Fedorov V. E., Ivanova N. D., “Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel”, Discrete and Continuous Dynamical Systems — S, 9:3 (2016), 687–696

[15] Fedorov V. E., Ivanova N. D., “Inverse problems for a class of linear Sobolev type equations with overdetermination on the kernel of operator at the derivative”, Journal of Inverse and Ill-posed Problems, 28:1 (2020), 53–61

[16] Kozhanov A. I., Abylkayrov U. U., Ashurova G. R., “Inverse problems of parameter recovery in differential equation with multiple characteristics”, KazNU Bulletin. Mathematics, Mechanics, Computer Science Series, 113:1 (2022), 3–16

[17] Faminskij A.V., Direct and inverse problems for quasilinear evolution equations of an odd order, Scientific workshop on differential and functional-differential equations. 14 May 2024. Moscow } (In Russ.) {\tt https://www.youtube.com/watch?v=dTKHDZGWfO4

[18] Egorov I.E., Fedorov V.E., Nonclassical mathematical physics equations of higher order, Computational Center of SB RAS, Novosibirsk, 1995 (In Russ.)

[19] Kozhanov A.I., Zikirov O.S., “Boundary problems for twice degenerate differential equations with multiple characteristics”, Mathematical Notes of NEFU, 25:4 (2018), 34–44

[20] Kozhanov A.I., Matsievskaya E.E., “Degenerate parabolic equations with variable direction of evolution”, Siberian Electronic Mathematical Reports, 16 (2019), 718–731 (In Russ.)

[21] Ladyzhenskaya O.A., Ural'tseva N.N., Linear and Quasi-linear Elliptic Equations, Academic Press, New York, 1968

[22] Triebel H., Interpolation Theory. Functional Spaces. Differential Operators, Veb Deutsher. Verl. Wiss., Berlin, 1978

[23] Sobolev S.L., Some Applications of Functional Analysis in Mathematical Physics, American Mathematical Soc., Providence, 2008

[24] Dzhenaliev M.T., To theory of linear boundary problems for loaded differential equations, Institute of Theoretical and Applied Mathematics, Almaty, 1995 (In Russ.)

[25] Nakhushev A.M., Loaded equations and their applications, Nauka, Moscow, 2012 (In Russ.)

[26] Ladyzhenskaya O.A., Solonnikov V.A., Ural'tseva N.N., Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, Providence, 1968

[27] Yakubov S.I., Linear differential operator equations and their applications, Elm, Baku, 1985 (In Russ.)

[28] Trenogin V.A., Functional analysis, Nauka, Moscow, 1980 (In Russ.)