Theoretical foundations for application of the direct and inverse Laplace transform to the telegrapher equation

In this article, we have considered the application of the inversion of the Laplace transform to problems of the telegrapher equation of hyperbolic type and parabolic type with an instantaneous source and a flat boundary. The application of the Laplace transform to the solution of hyperbolic and parabolic problems has a number of advantages over the classical methods of integrating the above problems. In this article, the application of the direct transformation to the coefficient inverse problem of a parabolic equation and the inverse transformation to the coefficient inverse problem of the hyperbolic type are theoretically investigated. The uniqueness and stability of the solution of these two inverse problems is substantiated and they are mutually equivalent.