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Some identification problems for integro-differential operator equations

Alfredo Lorenzi·Alexander Ramm
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Abstract

We consider, in a Hilbert space $H$, the convolution integro-differential equation $u''(t)-h*Au(t)=f(t)$, $0\le t\le T$, $h*v(t)=\int_0^t h(t-s)v(s) ds$, where $A$ is a linear closed densely defined (possibly selfadjoint and/or positive definite) operator in $H$. Under suitable assumptions on the data we solve the inverse problem consisting of finding the kernel $h$ from the extra data (measured data) of the type $g(t):=(u(t),ϕ)$, where $ϕ$ is some eigenvector of $A^*$. An inverse problem for the first-order equation $u'(t)-l*Au(t)=f(t)$, $0\le t\le T$, is also studied when $A$ enjoys the same properties as in the previous case.

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Alfredo Lorenzi, Alexander Ramm. 2000-11-17. Some identification problems for integro-differential operator equations. https://arxiv.org/abs/math/0011132

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