The inverse problem of recovering an unsteady linear load for an elastic rod of finite length

The main purpose of the paper is to obtain solutions for new non-stationary inverse problems for elastic rods. The objective of this study is to develop and implement new methods, approaches and algorithms for solving non-stationary inverse problems of rod mechanics. The direct non-stationary problem for an elastic rod consists in determining elastic displacements, which satisfies a given equation of non-stationary oscillations in partial derivatives and some given initial and boundary conditions. The solution of inverse retrospective problems with a completely unknown space-time law of load distribution is based on the method of influence functions. With its application, the inverse retrospective problem is reduced to solving a system of integral equations of the Volterra type of the first kind in time with respect to the sought external axial load of the elastic rod. To solve it, the method of mechanical quadratures is used in combination with the Tikhonov regularisation method.