Uniform stability of the inverse spectral problem for a convolution integro-differential operator

The operator of double differentiation, perturbed by the composition differentiation and a convolution one, on finite interval with Dirichlet boundary conditions is considered. We obtain uniform stability recovering kernel from spectrum in weighted $L_2$-norm norm. For this purpose, we successively prove each step algorithm for solving inverse problem both norms. Besides justifying numerical computations, obtained results reveal some essential difference classical Sturm-Liouville problem.