Numerical solution and stability of the inverse spectral problem for a convolution integro-differential operator

Abstract

In recent years, there has been growing interest in nonlinear inverse problems of spectral analysis for integro-differential operators. However, in spite of permanently increasing number of works, there are still no numerical results in this direction. The first aim of this paper is to fill this gap by developing an effective numerical approach to this class of inverse problems. The second aim is to prove the stability theorem, which theoretically justifies our numerical method. As a model situation, we consider one important and illustrative class of integro-differential operators, while the presented method can be extended to more complicated inverse problems. Our approach is based on reducing an inverse problem to some nonlinear integral equation and involves approximation of its solution by entire functions of exponential type. Concrete results of the numerical simulation are provided and discussed.