The problem of determining the memory of an environment with weak horizontal heterogeneity

D. K. Durdiev   ; J. Sh. Safarov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 32 (2022) no. 3, pp. 383-402

Résumé

The problem of determining the convolutional kernel $k(t,x)$, $t>0$, $x \in {\mathbb{R}}$, included in a hyperbolic integro-differential equation of the second order, is investigated in a domain bounded by a variable $z$ and having weakly horizontal heterogeneity. It is assumed that this kernel weakly depends on the variable $x$ and decomposes into a power series by degrees of a small parameter $\varepsilon$. A method for finding the first two coefficients $k_{0}(t)$, $k_{1}(t)$ of this expansion is constructed according to the given first two moments in the variable $x$ of the solution of the direct problem at $z=0$.

Détail

Keywords: integro-differential equation, inverse problem, the Dirac delta function, the kernel of the integral, the norm.

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