Symmetries of Fundamental Solutions and Their Application in Continuum Mechanics

An application of the symmetries of fundamental solutions in continuum mechanics is presented. It is shown that the Riemann function of a second-order linear hyperbolic equation in two independent variables is invariant with respect to the symmetries of fundamental solutions, and a method is proposed for constructing such a function. A fourth-order linear elliptic partial differential equation is considered that describes the displacements of a transversely isotropic linear elastic medium. The symmetries of this equation and the symmetries of the fundamental solutions are found. The symmetries of the fundamental solutions are used to construct an invariant fundamental solution in terms of elementary functions.