Geodesic
Inverse Problem for an Integrodifferential Equation of the Hyperbolic Type protect in a Rectangular Domain
Matematičeskie zametki, Tome 114 (2023) no. 2, pp. 244-259

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The inverse problem of determining the solution and the kernel of the integral term in an inhomogeneous two-dimensional integrodifferential wave equation in a rectangular domain is considered. First, the uniqueness of the solution of the direct problem is established using the completeness of the eigenfunction system of the corresponding homogeneous Dirichlet problem for the two-dimensional Laplace operator, and the existence of a solution of the direct problem is proved. Using additional information about the solution of the direct problem, we obtain a Volterra integral equation of the second kind for the kernel of the integral term. The existence and uniqueness of a solution of this equation is proved by the contraction mapping method in the space of continuous functions with a weighted norm.
Keywords: integrodifferential equation, integral kernel, Fourier method, eigenfunctions, eigenvalues, Banach theorem. 
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     author = {D. K. Durdiev and J. Sh. Safarov and J. Sh. Safarov},
     title = {Inverse {Problem} for an {Integrodifferential} {Equation} of the {Hyperbolic} {Type} protect in a {Rectangular} {Domain}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {244--259},
     publisher = {mathdoc},
     volume = {114},
     number = {2},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2023_114_2_a6/}
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D. K. Durdiev; J. Sh. Safarov; J. Sh. Safarov. Inverse Problem for an Integrodifferential Equation of the Hyperbolic Type protect in a Rectangular Domain. Matematičeskie zametki, Tome 114 (2023) no. 2, pp. 244-259. http://geodesic.mathdoc.fr/item/MZM_2023_114_2_a6/