On the uniqueness of kernel determination in the integro-differential equation of parabolic type

We study the problem of determining the kernel of the integral term in the one-dimensional integro-differential equation of heat conduction from the known solution of the Cauchy problem for this equation. First, the original problem is replaced by the equivalent problem where an additional condition contains the unknown kernel without integral. We study the question of the uniqueness of the determining of the kernel. Next, assuming that there are two solutions $ k_1 (x, t) $ and $ k_2 (x, t), $ integro-differential equations, Cauchy and additional conditions for the difference of solutions of the Cauchy problem corresponding to the functions $ k_1 (x, t), $ $ k_2 (x, t)$ are obtained. Further research is being conducted for the difference $k_1 (x, t) - k_2 (x, t) $ of solutions of the problem and using the techniques of integral equations estimates it is shown that if the unknown kernel $ k (x, t) $ can be represented as $ k_j (x, t) = \sum_ {i = 0} ^ N a_i (x) b_i (t)$, $ j = 1, 2, $ then $ k_1 (x, t ) \equiv k_2 (x, t). $ Thus, the theorem on the uniqueness of the solution of the problem is proved.

inverse problem, parabolic equation, Cauchy problem, integral equation, uniqueness