The generalized Green’s function for an $n$th order linear differential operator

Abstract:

The generalized Green’s function $K(t,s)$ for an nth order linear differential operator L is characterized in terms of the 2nth order differential operators $L{L^\ast }$ and ${L^\ast }L$. The development is operator oriented and takes place in the Hilbert space ${L^2}[a,b]$. Two features of the characterization are a determination of the jumps occurring in the derivatives of orders n, $n + 1, \ldots ,2n - 1$ at $t = s$ and a determination of the boundary conditions satisfied by the functions $K(a, \cdot )$ and $K(b,\cdot )$. Several examples are given to illustrate the properties of the generalized Green’s function.