Stability of an elastic rod on a fractional derivative type of foundation

The lateral vibration of an axially loaded elastic rod positioned on a fractional derivative type of foundation is studied. It is shown that the dynamics of the problem is governed by a system of two coupled linear differential equations with fractional derivatives. For this system of equations the questions of existence, regularity and the stability of solution are analysed. The results are compared with the stability bound for an elastic rod on Winkler (elastic) type of foundation.