The paper discusses the method of solution of an inverse problem of one-dimensional temperature and stress fields for a sphere, a circular cylinder and an infinite plate. The inverse problem describes the dependance of the boundary conditions of different types on the prescribed temperature state or stress state within the body under consideration, in contrast with the direct problem which relates the temperature and stress states to known boundary conditions. To obtain a function describing the temperature of a heating medium and/or the Biot number in a simple form use has been made of the Laplace transformation. The numerical examples for both types of the inverse problems are presented.