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