On a Finite Difference Scheme for an Inverse Integro-Differential Problem Using Semigroup Theory: A Functional Analytic Approach

ABSTRACT

In this article, the problem of reconstructing an unknown memory kernel from an integral overdetermination in an abstract linear (of convolution type) evolution equation of parabolic type is considered. After illustrating some results of the existence and uniqueness of a solution for the differential problem, we study its approximation by Rothe's method. We prove a result of stability and another concerning the order of approximation of the solution in dependence of its regularity. The main tool is a maximal regularity result for solutions to abstract parabolic finite difference schemes. Two model problems to which the results are applicable are illustrated.

KEYWORDS:

• Convolution kernel
• inverse problem
• Rothe's method
• semigroup theory

MATHEMATICS SUBJECT CLASSIFICATION:

• 35A01
• 65M06
• 37L05
• 65L09

Acknowledgment

The first author acknowledges the support of the Research Foundation, Flanders (FWO15/PDO/076). The second author is a member of GNAMPA of CNR (Italian National Council of Research).