On Cauchy problem for first-order partial differential equation and its applications in inversion theory

The inverse coefficient problem for the second-order operator in the simply connected domain with the piecewise-smooth boundary arising in the theory of the deformable system vibrations is investigated. The solution method for the inverse coefficient problem based on studying Cauchy problem for the first-order differential equation with variable coefficients is offered. Both direct and inverse problems are solved on the ground of the difference approximations method. The reconstruction of the variable shear modulus of various types, obtained at both accurate and noisy input data, is resulted.