Uniqueness for multidimensional kernel determination problems from a
parabolic integro-differential equation

- Durdimurod Durdiev,
- Javlon Nuriddinov

## Abstract

We study two problems of determining the kernel of the integral terms in
a parabolic integro-differential equation. In the first problem the
kernel depends on time t and x = (x_{1}, …,
x_{n}) spatial variables in the multidimensional
integro-differential equation of heat conduction. In the second problem
the kernel it is determined from one dimensional integro-differential
heat equation with a time-variable coefficient of thermal conductivity.
In both cases it is supposed that the initial condition for this
equation depends on a parameter y = (y_{1}, …,
y_{n}) and the additional condition is given with respect
to a solution of direct problem on the hyperplanes x = y. It is shown
that if the unknown kernel has the form k(x, t)
=∑_{i=o}^{N}a_{i}(x)b_{i}(t),
then it can be uniquely determined.