Uniqueness of the Kernel Determination Problem in a Integro-Differential Parabolic Equation with Variable Coefficients
Abstract
We investigate the inverse problem of determining the time and space dependent kernel of the integral term in the $$n$$ -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 in which an additional condition contains the unknown kernel without integral. We study the question of the uniqueness of the determining of this kernel. Next, assuming that there are two solutions $${{k}_{1}}(x,t)$$ and $${{k}_{2}}(x,t)$$ of the stated problem, an equation is formed for the difference of this solution. 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 $$k(x,t) = \sum\limits_{i = 0}^N {{a}_{i}}(x){{b}_{i}}(t)$$ , then $${{k}_{1}}(x,t) \equiv {{k}_{2}}(x,t)$$ . Thus, the theorem on the uniqueness of the solution of the problem is proved.