
Inverse coefficient problem for the 2D wave equation with initial and nonlocal boundary conditions
D. K. Durdiev^{ab}, T. R. Suyarov^{ab} ^{a} Bukhara Branch of Romanovskiy Institute of Mathematics
of the Academy of Sciences of the Republic of Uzbekistan,
11 M. Ikbol St., Bukhara 200100, Uzbekistan
^{b} Bukhara State University, 11 M. Ikbol St., Bukhara 200100, Uzbekistan
Abstract:
In this paper, we consider direct and inverse problems for the twodimensional wave equation. The direct problem is an initial boundary value problem for this equation with nonlocal boundary conditions. In the inverse problem, it is required to find the timevariable coefficient at the lower term of the equation. The classical solution of the direct problem is presented in the form of a biorthogonal series in eigenvalues and associated functions, and the uniqueness and stability of this solution are proven. For solution to the inverse problem, theorems of existence in local, uniqueness in global, and an estimate of conditional stability are obtained. The problems of determining the righthand sides and variable coefficients at the lower terms from initial boundary value problems for secondorder linear partial differential equations with local boundary conditions have been studied by many authors. Since the nonlinearity is convolutional, the unique solvability theorems in them are proven in a global sense. In the works, the method of separation of variables is used to find the classical solution of the direct problem in the form of a biorthogonal series in terms of eigenfunctions and associated functions. The nonlocal integral condition is used as the overdetermination condition with respect to the solution of the direct problem. The direct problem reduces to equivalent integral equations of the Fourier method. To establish integral inequalities, the generalized GronwallBellman inequality is used. We obtain an a priori estimate of the solution in terms of an unknown coefficient, that are useful for studying the inverse problem.
Key words:
wave equation, nonlocal boundary conditions, inverse problem, Banach's theorem.
Received: 06.10.2023
Citation:
D. K. Durdiev, T. R. Suyarov, “Inverse coefficient problem for the 2D wave equation with initial and nonlocal boundary conditions”, Vladikavkaz. Mat. Zh., 26:2 (2024), 5–25
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https://www.mathnet.ru/eng/vmj906 https://www.mathnet.ru/eng/vmj/v26/i2/p5

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