Inverse Problem for Incomplete Sobolev Type Equation of Higher Order
Автор(ы):
Alyona Aleksandrovna Zamyshlyaeva
Director of the Institute of Natural Scinces and Mathematics,
Doctor of Sciences (Phys&Math), Professor
South Ural State University (National Research University) in Chelyabinsk
zamyshliaevaaa@susu.ru
Aleksandr Valeryevich Lut
Post graduate student,
Assistant of the Department of Applied Mathematics and Programming
South Ural State University (National Research University) in Chelyabinsk
lutav@susu.ru
Аннотация:
The article is devoted to the study of the inverse problem for
a high-order Sobolev type equation. Mathematical models based
on such equations describe various problems of hydrodynamics
and elasticity theory. The main result of the article is to find
sufficient conditions for the existence and uniqueness of
a solution to the original problem. At first using the theory
of relatively bounded operators, the original problem is reduced
to the equivalent system of two problems, which are usually
called regular and singular. Thus, the solution to the original problem
is represented as the sum of the solutions of these two problems.
Further, the regular problem is reduced to a first-order equation.
Then, by the method of successive approximations, the required smoothness
for the function
Ключевые слова
- inverse problem
- method of successive approximations
- relative boundedness of operator
- Sobolev type equation
Ссылки:
- Zamyshlyaeva A. A. and Sviridyuk G. A. Nonclassical Equations of Mathematical Physics. Linear Sobolev Type Equations of Higher order. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2016; 8(4):5-16. (In English)
- Prilepko A. I., Orlovsky D. G. and Vasin I. A. Methods for Solving Inverse Problems in Mathematical Physics. New York: Marcel Dekker, 2000. 744 p. (In English)
- Zamyshlyaeva A. A. and Lut A. V. Inverse Problem for Sobolev Type Mathematical Models. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2019; 12(2):25-36. (In English)
- Banasiak J., Manakova N. A. and Sviridyuk G. A. Positive Solutions to Sobolev Type Equations with Relatively $p$-Sectorial Operators. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2020; 13(2):17-32. (In English)
- Zamyshlyaeva A. A., Manakova N. A. and Tsyplenkova O. N. Optimal Control in Linear Sobolev Type Mathematical Models. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2020; 13(1):5-27. (In English)
- Zamyshlyaeva A. A. and Bychkov E. V. Research of One Semilinear Mathematical Model of the Sobolev Type of Higher Order. Mathematical Methods in Technics and Technologies - MMTT, 2020; 8:3-6. (in Russian)
- Kozhanov A. I. and Namsaraeva G. V. Linear Inverse Problems for a Class of Equations of Sobolev Type. Chelyabinsk Physical and Mathematical Journal, 2018; 3(2):153-171. (in Russian)
- Kokurin M. Y. Completeness of asymmetric products of solutions of a second order elliptic equation and uniqueness of a solution to an inverse problem for a wave equation. Differential Equations, 2021; 57(2):255-264. (in Russian)
- Pyatkov S. G., Uvarova M. V. and Pronkina T. V. Inverse Problems for a Quasilinear Parabolic System with Integral Overdetermination Conditions. Mathematical Notes of NEFU, 2020; 27(4):43-59. (in Russian)
- Sulaimanov B. E., Myrzapayazova Z. K. and Toktogulova A. S. Inverse Problem for Differential Equations in Individual Derivatives. Bulletin of the Kyrgyz State Technical University named after Razzakov, 2020; 2(54):95-102. (in Kyrgyz)
- Umarov KH. G. The Cauchy Problem for The Nonlinear Long Longitudinal Waves Equation in a Viscoelastic Rod. Siberian Mathematical Journal, 2021; 62(1):198-209. (in Russian)
- Beshtokov M. KH. Numerical study of initial-boundary value problems for a Sobolev type equation with a time-fractional derivative. Computational Mathematics and Mathematical Physics, 2019; 59(2):185-202. (in Russian)
