On the positive square root of the fourth derivative operator

D. L. Russell
1988 Quarterly of Applied Mathematics  
It is only for a special subset of the natural boundary conditions for the operator d4w Aw=df 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 A1/2 in the form just noted, and we show that in the other cases -D2w = -4S = [/ + P]Ai/2w dxl where P is a bounded, but in general not compact, operator on the Hilbert space L2 [0,7i]. Possible
more » ... ossible applications to the theory of the partial differential equation d2w _ <?% d^ui n P~di2 ~ 7did.X2 + are indicated.
doi:10.1090/qam/973388 fatcat:jiapcifirbdjfiaiysgmocbpxu