Non-linear longitudinal oscillations of a viscoelastic rod in Kelvin's model

Non-linear longitudinal oscillations of a viscoelastic rod are investigated in Kelvin's model. All the second-order conservation laws for the differential equation describing these oscillations see obtained, which, using non-local variables, generate two non-linear systems of differential equations, equivalent to this equation. A group analysis of these systems is carried out. All their essentially different invariant solutions (unconnected with point transformations) are obtained, which are either found in explicit form or the search for them is reduced to solving non-linear integro-differential equations, which opens up new possibilities for analytical and numerical investigations. The presence of arbitrary constants in these equations enables them to be used to investigate various boundary value problems. With additional conditions, the existence and uniqueness of the solutions of certain boundary value problems, describing non-linear longitudinal oscillations of a visco-elastic rod in Kelvin's model, are established.