Problems of Determining the Kernel of anIntegro-Differential Equation in a Bounded Domain
Abstract
The paper studies the inverse problem of the depending on a time variable kernel of the integral term of a multidimensional hyperbolic-type integro-differential equation. First, a direct problem is investigated, assuming that the kernel of the integral term is known. The Fourier method reduces this problem to solving a Volterra-type integral equation of the second kind with respect to the unknown function. A priori estimates for the desired function and for its second-order derivatives are obtained. Next, two inverse problems are studied. The first is determining the memory kernel of a wave process with an integral overdetermination condition. In the second inverse problem, the kernel of the integral term is found from the known solution of the direct problem at some fixed point. In both cases, the inverse problem is reduced to a nonlinear convolution-type Volterra integral equation of the second kind. The method of contracting mappings is used to prove the unique solvability of the posed inverse problems in the space of continuous functions with weighted norms, and an estimate for the conditional stability of the solution is obtained.