Geodesic
An inverse problem of determining the kernel in an integro-differential equation of vibrations of a bounded string
Matematičeskie zametki SVFU, Tome 29 (2022), pp. 21-36.

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We consider an integro-differential equation of hyperbolic type in the domain $D={(x, t) : 0 x l, t > 0}$ bounded in the variable $x$. The direct problem is investigated rst. For the direct problem, the inverse problem of determining the kernel of the integral term of the integro-differential equation is studied on the basis of the available additional information about the solution of the direct problem for $x=0$. Differentiating the obtained integral equation for $u(x, t)$ three times with respect to $t$ and using some additional condition, we reduce the solution of the inverse problem to solving a system of integral equations for unknown functions. The contraction mapping principle is applied to this system in the space of continuous functions with weighted norms. A theorem on the global unique solvability is proved. An estimate for the conditional stability of the solution to the inverse problem is also obtained.
Keywords: integro-differential equation, inverse problem, kernel of integral, Banach theorem. 
@article{SVFU_2022_29_a2,
     author = {J. Sh. Safarov},
     title = {An inverse problem of determining the kernel in an integro-differential equation of vibrations of a bounded string},
     journal = {Matemati\v{c}eskie zametki SVFU},
     pages = {21--36},
     publisher = {mathdoc},
     volume = {29},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVFU_2022_29_a2/}
}
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J. Sh. Safarov. An inverse problem of determining the kernel in an integro-differential equation of vibrations of a bounded string. Matematičeskie zametki SVFU, Tome 29 (2022), pp. 21-36. http://geodesic.mathdoc.fr/item/SVFU_2022_29_a2/