On the positive square root of the fourth derivative operator

Abstract

It is only for a special subset of the natural boundary conditions for the operator \[ A w = d 4 w d x 4 Aw = \frac {{{d^4}w}}{{d{x^4}}} \] that its positive square root is the negative second derivative operator. In this paper we develop a procedure for parametric description of all natural boundary conditions, we show which ones admit A 1 / 2 {A^{1/2}} in the form just noted, and we show that in the other cases \[ − D 2 w ≡ − d 2 w d x 2 = [ I + P ] A 1 / 2 w - {D^2}w \equiv - \frac {{{d^2}w}}{{d{x^2}}} = \left [ {I + P} \right ]{A^{1/2}}w \] where P P is a bounded, but in general not compact, operator on the Hilbert space L 2 [ 0 , π ] {L^2}\left [ {0, \pi } \right ] . Possible applications to the theory of the partial differential equation \[ ρ ∂ 2 w ∂ t 2 − 2 γ ∂ 3 w ∂ t ∂ x 2 + E I ∂ 4 w ∂ x 4 = 0 \rho \frac {{{\partial ^2}w}}{{\partial {t^2}}} - 2\gamma \frac {{{\partial ^3}w}}{{\partial t\partial {x^2}}} + EI\frac {{{\partial ^4}w}}{{\partial {x^4}}} = 0 \] are indicated.