Nonuniqueness of the Green’s Function for Neumann’s Boundary-Value Problems

Green’s functions for linear two-point boundary-value problems subject to non-homogeneous Neumann’s boundary conditions are derived. It is shown that, for the differential operator L(y)=y′′+ky with k≠nπ/L where n is a natural number or zero, the Green function is unique, whereas for k=(nπ/L)2, a nonlocal integral compatibility condition that involves the nonhomogeneous term and the boundary conditions must be satisfied in order that the boundary-value problem has a solution. In this case, the ordinary differential equation for the Green function must include the Dirac delta function and an integrable function that must satisfy an integral constraint, and there is an infinite number of Green’s functions and solutions to the boundary-value problem. If the integral compatibility condition is not met, it is shown that there is no solution to the problem, and no Green’s function exists. For k=0, three numerical approximations to the compatibility condition have been derived by discretizing both the differential equation and the boundary conditions, and it is shown that only the discretization of the boundary conditions that employs ghost points agrees with the trapezoidal quadrature approximation to the exact compatibility condition.