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

Abstract

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 $L\left(y\right)=y^{″}+ky$ with $k\ne n\pi /L$ where n is a natural number or zero, the Green function is unique, whereas for $k={(n\pi /L)}^2$, 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.

1. Introduction

Green’s functions can be interpreted as the response of linear operators to impulse functions and play an important role in dealing with initial- and boundary-value problems governed by linear and nonlinear ordinary or partial differential equations in applied mathematics, physics and engineering, e.g., [1,2,3,4]. For example, Green’s functions have played a paramount role in heat and mass transfer [5], quantum physics [6], electromagnetism and electrodynamics [7], acoustics [8], fluid dynamics [9,10], solid mechanics [11,12], cosmology [13], etc., where the governing partial differential equations are of second-order in space and either first- or second-order in time. In addition, Green’s functions play a key role in boundary element formulations for the solution of problems arising, for example, in fluid [9,10] and solid mechanics [14,15].

Green’s functions have also been employed to obtain the solution of two-point nonlinear boundary-value problems and shown to be identical to dual-variational principles [16]. They have also been used to obtain solutions of third-, fourth- and higher-order boundary-value problems arising in, for example, bending of bars and plates, e.g., [17,18], where lower and upper solution techniques and Schauder’s fixed point theory have been used to determine the existence and uniqueness and the sign changes of the Green function as functions of the parameters that appear in the ordinary differential equations and their boundary conditions; these higher-order problems are, in general, much more difficult than those associated with second-order ones, such as the ones considered in this paper [19,20,21,22].

For two-point boundary-value problems governed by linear ordinary differential equations subject to Dirichlet boundary conditions, the differential equation for the Green function corresponds to that of the adjoint operator of the original one with a right-hand side equal to Dirac’s delta function subject to homogeneous boundary conditions, and the Green function is unique, e.g., [2,3,23]. This type of problem—and those corresponding to Robin’s boundary conditions, which in heat transfer are associated with Newton’s cooling law—have received some attention in the literature [3,4,5]. This, however, has not be the case for Neumann’s boundary conditions because if the nonhomogeneous term in the differential equation and the boundary conditions does not satisfy an integral constraint or compatibility condition, these problems do not have a solution [24,25,26]. But, even if such a constraint holds, and, therefore, the boundary-value problems admits an infinite number of solutions, the differential operator for the Green function may not be set equal to Dirac’s delta function, as is the case for Dirichlet or Robin boundary conditions, because this would result in an incompatibility, as shown in this paper. In order that the boundary-value problem for the Green function be compatible for Neumann’s boundary conditions, its ordinary differential equation must include not only the Dirac delta function but another integrable function that must, in turn, satisfy a compatibility condition, as reported herein.

Previous analytical studies on linear two-point boundary-value problems subject to Neumann’s boundary conditions include those of Stakgold and Holst [3] and Franklin [23]. Stakgold and Holst [3] considered Equation (1) below with $k=\alpha =\beta =0$ and left and right Green functions, whereas Franklin [23] dealt with nonhomogeneous Neumann’s boundary conditions and a self-adjoint differential equation that only included the first- and second-order derivatives. Moreover, Franklin [23] used eigenfunction expansions to determine the Green function. In contrast, in the present paper, we consider a self-adjoint differential equation that does not include the first-order derivative, make use of generalized functions, and provide a detailed study of the analytical compatibility condition that must be met in order to ensure the existence of both a solution and a Green function for a linear two-point boundary-value problem subject to Neumann’s boundary conditions.

The major contributions of this paper are two-fold. First, it is shown that, when there is an infinite number of Green functions, these depend on both a constant and an arbitrary but integrable function that must satisfy a compatibility condition of the integral type, and the solution to the boundary-value problem is governed by a Fredholm integral equation of the second kind. Second, it is shown that the second-order accurate finite difference discretization of the differential equation and three different discretizations of the Neumann boundary conditions result in numerical compatibility conditions that differ from well-known quadrature rules for the exact compatibility condition, except when the Neumann boundary conditions are discretized using second-order accurate finite differences and ghost points; in this case, the finite difference discretization results in the trapezoidal quadrature rule approximation to the exact compatibility condition.

This manuscript has been arranged as follows. In Section 2, a brief summary of well-known results for the solution of linear two-point boundary-value problems with constant coefficients is first presented. Green’s functions are also dealt with in Section 2 for two cases; the first one corresponds to boundary-value problems that have a unique solution and, therefore, have a unique Green’s function, while the second case corresponds to problems that satisfy an integral constraint and have an infinite set of solutions. For this second case, we show that the Green function is not unique and depends on an integrable function that must satisfy an integral constraint or compatibility condition. In Appendix A dealing with finite difference discretizations of the two-point boundary-value problem for $k=0$, discrete compatibility conditions are presented and compared with those corresponding to numerical quadrature approximations to the exact compatibility condition. A summary of the main results reported in the paper is presented in a concluding section.

2. Green’s Functions for Neumann’s Boundary Conditions

In this section, we shall be concerned with the determination of the Green function for the following linear two-point boundary-value problem subject to Neumann boundary conditions:

$$\mathcal{L}\left(y\left(x\right)\right)\equiv y^{″}\left(x\right)+ky\left(x\right)=f\left(x\right),y^{\prime }\left(0\right)=\alpha ,y^{\prime }\left(L\right)=\beta ,$$

(1)

where x and y denote the independent and dependent variables, respectively, the prime denotes differentiation with respect to x, k, $\alpha$ and $\beta$ are real constants, and $f\left(x\right)$ is Lebesgue integrable. Hereon, we shall refer to the ordinary differential appearing in the problem above as Equation (1).

For the sake of convenience, the cases $k>0$, $k=0$ and $k<0$ will be analyzed in the next three subsections.

Before obtaining the Green’s function, it is convenient to recall that the linear two-point boundary-value problem (1) has a unique solution if the corresponding homogeneous problem, i.e., Problem (1) with $f\left(x\right)=\alpha =\beta =0$, only has the trivial solution [24,25,26]. For $k>0$, the solution to the homogeneous problem is $y\left(x\right)=0$ if $\sin \left(\omega L\right)\ne 0$ and $y\left(x\right)=C\sin \left(\omega x\right)$ if $\sin \left(\omega L\right)=0$; therefore, Problem (1) has a unique solution for $k>0$ if $\sin \left(\omega L\right)\ne 0$, and an infinite number of solutions otherwise, where C is a constant. For $k=0$, the solution of the homogeneous problem is $y\left(x\right)=C$ and, therefore, Equation (1) does not have a unique solution. For $k=-{\Omega }^2<0$, the solution to the homogeneous problem is the trivial one and, therefore, the solution to Problem (1) is unique.

2.1. Green’s Function for $k>0$

In this subsection, we deal with the Green function for the two-point boundary-value problem in Equation (1) with $k\equiv {\omega }^2>0$. To that end, we multiply the differential Equation (1) by the Green function $G(x;\xi )$, integrate the result from $x=0$ to $x=L$, and use the boundary conditions for $y\left(x\right)$ to obtain

$$\begin{array}{c}{\int }_0^L(G^{″}(x;\xi )+kG(x;\xi ))y\left(x\right)dx=\alpha G(0;\xi )-\beta G(L;\xi )\hfill \\ +G^{\prime }(L;\xi )y\left(L\right)-G^{\prime }(0;\xi )y\left(0\right)+{\int }_0^{L}f\left(x\right)G(x;\xi )dx,\hfill \end{array}$$

(2)

where $y\left(0\right)$ and $y\left(L\right)$ are unknown and integration by parts has been employed. The terms including $y\left(0\right)$ and $y\left(L\right)$ may be eliminated from Equation (2) by setting

$$G^{\prime }(0;\xi )=0,G^{\prime }(L;\xi )=0.$$

(3)

If this choice is made and one sets

$$G^{″}(x;\xi )+kG(x;\xi )=\delta (x-\xi ),$$

(4)

where $\delta (x-\xi )$ is Dirac’s delta function, it is an easy exercise to show that the solution of Equation (4) subject to Equation (3) is

$$G(x;\xi )=\frac{1}{\omega \sin \left(\omega L\right)}(\cos \left(\omega (L-\xi )\right)\cos \left(\omega x\right)+\sin \left(\omega L\right)\sin \left(\omega (x-\xi )\right)H(x-\xi )),$$

(5)

provided that $\sin \left(\omega L\right)\ne 0$, and then the Green function is unique, where $H(x-\xi )$ is Heaviside’s step function.

The Green function thus determined may be used in Equation (2) to obtain $y\left(\xi \right)$. We have thus proved the following

Proposition 1. 

For $k={\omega }^2>0$ and $\sin \left(\omega L\right)\ne 0$, the Green function corresponding to the two-point boundary-value problem of Equation (1) is unique and given by the solution of Equations (3) and (4) as Equation (5), and the solution of the two-point boundary-value problem in Equation (1) is unique and is given by Equation (2), i.e.,

$$y\left(\xi \right)=\alpha G(0;\xi )-\beta G(L;\xi )+{\int }_0^{L}f\left(x\right)G(x;\xi )dx,\mathit{for}\sin \left(\omega L\right)\ne 0,$$

where $G(0;\xi )=\frac{\cos \left(\omega \right(L-\xi \left)\right)}{\omega \sin \left(\omega L\right)}$ and $G(L;\xi )=\frac{\cos \left(\omega \xi \right)}{\omega \sin \left(\omega L\right)}$ according to Equation (5).

Remark 1. 

For $k={\omega }^2>0$ and $\sin \left(\omega L\right)\ne 0$, the solution to Problem (1) can also be written as

$$y\left(x\right)=C\cos \left(\omega x\right)+\frac{\alpha }{\omega }\sin \left(\omega x\right)+\frac{1}{\omega }{\int }_0^{x}f\left(s\right)\sin \left(\omega (x-s)\right),$$

where

$$C=\frac{1}{\omega \sin \left(\omega L\right)}\left(\alpha \cos \left(\omega L\right)-\beta +{\int }_0^{L}f\left(s\right)\cos \left(\omega (L-s)\right)ds\right).$$

For $\sin \left(\omega L\right)=0$, Equation (4) is not valid because, upon integration of this equation, one obtains that $G^{\prime }(L;\xi )=\cos \left(\omega (L-\xi )\right)$, which is incompatible with the value of $G^{\prime }(L;\xi )=0$ (cf. Equation (3)), for the equality of these two expressions would demand that $\cos \left(\omega \xi \right)=0$ for $\xi \in (0,L)$.

In order to satisfy the boundary conditions of Equation (3) for $\sin \left(\omega L\right)=0$, one must solve the following differential equation for the (modified) Green function (cf. compare with Equation (4))

$$G^{″}(x;\xi )+kG(x;\xi )=\delta (x-\xi )+M(x;\xi ),$$

(6)

where $M(x;\xi )$ is Lebesgue integrable.

The solution to Equations (3) and (6) is

$$G(x;\xi )=C\cos \left(\omega x\right)+\frac{1}{\omega }\left(\sin \left(\omega (x-\xi )\right)H(x-\xi )+{\int }_0^{x}M(s;\xi )\sin \left(\omega (x-s)\right)ds\right),$$

(7)

where C is an integration constant and, upon making use of Equation (3), $M(x;\xi )$ must satisfy the following equation:

$$\cos \left(\omega \xi \right)+{\int }_0^{L}M(s;\xi )\cos \left(\omega s\right)ds=0.$$

(8)

Equation (7) indicates that the Green function is not unique due to the presence of C and $M(x;\xi )$. In addition, Equation (8) is a compatibility condition for $M(x;\xi )$ and, therefore, for $G(x;\xi )$.

Use of Equation (7) in Equation (2) yields

$$y\left(\xi \right)=\alpha G(0;\xi )-\beta G(L;\xi )+{\int }_0^{L}f\left(x\right)G(x;\xi )dx-{\int }_0^{L}M(x;\xi )y\left(x\right)dx,$$

(9)

which is a Fredholm’s integral equation of the second kind, where (cf. Equation (7))

$$G(0;\xi )=C,$$

(10)

$$G(L;\xi )=C\cos \left(\omega L\right)+\frac{1}{\omega }\left(\sin \left(\omega (L-\xi )\right)+{\int }_0^{L}M(s;\xi )\sin \left(\omega (L-s)\right)ds\right),$$

(11)

and $G(x;\xi )$ is given by Equation (7).

We have thus proved the following

Proposition 2. 

For $k={\omega }^2>0$ and $\sin \left(\omega L\right)=0$, there is an infinite number of Green functions governed by Equations (3) and (6) if and only if $M(x;\xi )$ is Lebesgue integrable and satisfies the compatibility condition of Equation (8) and, therefore, there is an infinite number of solutions to the boundary-value problem of Equation (1); if this compatibility condition is not satisfied, no Green function exists and there is no solution to the two-point boundary-value problem of Equation (1).

Remark 2. 

For $k={\omega }^2>0$ and $\sin \left(\omega L\right)=0$, it is an easy exercise to show that there is an infinite number of solutions to Problem (1) given by

$$y\left(x\right)=C\cos \left(\omega x\right)+\frac{\alpha }{\omega }\sin \left(\omega x\right)+\frac{1}{\omega }{\int }_0^{x}f\left(s\right)\sin \left(\omega (x-s)\right)ds,$$

provided that

$$\cos \left(\omega L\right){\int }_0^{L}f\left(s\right)\sin \left(\omega s\right)ds=\beta -\alpha \cos \left(\omega L\right),$$

which is a compatibility condition between the forcing and the boundary conditions for the boundary-value Problem (1); otherwise, Problem (1) has no solution.

For $M(x;\xi )=P\left(x\right)Q\left(\xi \right)$ with $P\ne 0$ and $Q\ne 0$, Equation (8) becomes

$$\cos \left(\omega \xi \right)+Q\left(\xi \right){\int }_0^{L}P\left(s\right)\cos \left(\omega s\right)ds=0,$$

(12)

which, upon assuming that $Q\left(\xi \right)=F\cos \left(\omega \xi \right)$, where F is a constant, becomes

$$1+F{\int }_0^{L}P\left(s\right)\cos \left(\omega s\right)ds=0.$$

(13)

Example 1. 

For $P\left(x\right)=B\cos \left(\omega x\right)$, where B is a constant, the compatibility condition of Equation (13) implies that $B=-\frac{2}{FL}$. Therefore, $M(x;\xi )=-\frac{2}{L}\cos \left(\omega x\right)\cos \left(\omega \xi \right)$, which has the following characteristics: $M(x;\xi )\le 0$ for $0<\omega L\le \frac{\pi }{2}$; $M(x;\xi )<0$ for $x\in [0,\frac{\pi }{2}]$ and $\xi \in [0,\frac{\pi }{2}]$ and for $x\in [\frac{\pi }{2},\pi ]$ and $\xi \in [\frac{\pi }{2},\pi ]$, and $0<\omega L\le \pi$; $M(x;\xi )>0$ for $x\in [0,\frac{\pi }{2}]$ and $\xi \in [\frac{\pi }{2},\pi ]$ and for $x\in [\frac{\pi }{2},\pi ]$ and $\xi \in [0,\frac{\pi }{2}]$, and $0<\omega L\le \pi$; and, the number of sign changes of $M(x;\xi )$ increases as $\omega L$ increases. This means that, for a fixed domain, i.e., fixed L, the number of sign changes of $M(x;\xi )$ increases as ω or $k={\omega }^2={\left(\frac{n\pi }{L}\right)}^2$ is increased. Moreover, use of the above expression for $M(x;\xi )$ in Equation (7) results in the following (modified) Green function:

$$G(x;\xi )=C\cos \left(\omega x\right)+\frac{1}{\omega }\left(\sin \left(\omega (x-\xi )\right)H(x-\xi )-\frac{x}{L}\sin \left(\omega x\right)cos\left(\omega \xi \right)\right),$$

which may also be written as

$$G(x;\xi )=C\cos \left(\omega x\right)-\frac{x}{\omega L}\sin \left(\omega x\right)\cos \left(\omega \xi \right),x\le \xi ,$$

and

$$G(x;\xi )=C\cos \left(\omega x\right)+\frac{1}{\omega }\left(\sin \left(\omega (x-\xi )\right)-\frac{x}{L}\sin \left(\omega x\right)\cos \left(\omega \xi \right)\right),x\ge \xi ,$$

which indicate that the Green function is not symmetric, i.e., $G(x;\xi )\ne G(\xi ;x)$.

2.2. Green’s Function for $k=0$

The Green function for $k=0$ can be obtained from those reported in the previous subsection by simply taking the limit $\omega \to 0$ in the corresponding equations and using L’Hôpital rule. In such a limit, Equation (7) becomes

$$G(x;\xi )=C+(x-\xi )H(x-\xi )+{\int }_0^{x}M(s;\xi )(x-s)ds,$$

(14)

and Equation (8), the compatibility condition for $M(x;\xi )$, reads

$${\int }_0^{L}M(s;\xi )ds+1=0,$$

(15)

which implies that $M(x;\xi )$ does not depend on $\xi$, i.e., $M(x;\xi )=m\left(x\right)$.

Since there is an infinite number of functions that satisfy Equation (15), there is an infinite number of Green’s functions on account of $M(x;\xi )$ and also on account of C in Equation (14).

We have thus proved the following

Proposition 3. 

For $k=0$, there is an infinite number of Green functions governed by Equations (3) and (6) if and only if $M(x;\xi )=m\left(x\right)$ is Lebesgue integrable and satisfies the compatibility condition of Equation (15) and, therefore, there is an infinite number of solutions to the boundary-value problem of Equation (1); if this compatibility condition is not satisfied, no Green function exists and there is no solution to the two-point boundary-value problem of Equation (1).

In the following paragraphs, some examples illustrating the Green function are presented for several $M(x;\xi )$ and $k=0$.

Example 2. 

For $M(x;\xi )=-\frac{1}{L}$, the compatibility condition of Equation (15) is satisfied, and Equation (14) becomes

$$G(x;\xi )=C+(x-\xi )H(x-\xi )-\frac{x^2}{2L}.$$

(16)

Equation (16) indicates that $G(x;\xi )=C-\frac{x^2}{2L}$ for $0\le x\le \xi$ and $G(x;\xi )=C+x-\xi -\frac{x^2}{2L}$ for $\xi \le x\le L$, thus indicating that the Green function is not symmetric, i.e., $G(x;\xi )\ne G(\xi ;x)$.

In order to determine a unique Green function, it has been suggested to impose the following normalization condition ${\int }_0^{L}G(x;\xi )dx=0$ [3]. If this condition is imposed on Equation (16), then $C=-\frac{L}{3}+\xi -\frac{{\xi }^2}{2L}$ in that equation. Substitution of Equation (16) into Equation (2) yields

$$y\left(\xi \right)=\frac{1}{L}{\int }_0^{L}y\left(x\right)dx-\beta \left(\frac{L}{2}-\xi \right)+{\int }_0^{L}f\left(x\right)\left((x-\xi )H(x-\xi )-\frac{x^2}{2L}\right)dx,$$

(17)

where use has been made of Equation (3). The first term on the right-hand side of Equation (17) is the (unknown) mean value of $y\left(x\right)$ in $[0,L]$.

Example 3. 

For $M(x;\xi )=\alpha x+\beta$, where α and β are constants, the compatibility condition of Equation (15) yields $\alpha =-\frac{2}{L^2}(\beta L+1)$ and the Green function, i.e., Equation (14), becomes

$$G(x;\xi )=C+(x-\xi )H(x-\xi )+\frac{x^2}{6}(\alpha x+3\beta ),$$

which may also be written as

$$G(x;\xi )=C+\frac{x^2}{6}(\alpha x+3\beta ),x\le \xi ,$$

and

$$G(x;\xi )=C+x-\xi +\frac{x^2}{6}(\alpha x+3\beta ),x\ge \xi ,$$

which indicate that the Green function is not symmetric, i.e., $G(x;\xi )\ne G(\xi ;x)$. Moreover, the cubic dependence of the Green function on x may result in changes in its sign depending on α and β. Note that since $G^{″}(0;\xi )=\beta$ and $G^{″}(L;\xi )=-\frac{1}{L}(\beta L+2)$, the one-sided curvatures at the left and right boundaries may have opposite signs depending on the length of the domain, L, and β, respectively.

Remark 3. 

For $k=0$, it is an easy exercise to show that there is an infinite number of solutions to Problem (1) given by

$$y\left(x\right)=C+\alpha x+{\int }_0^{x}f\left(s\right)(x-s)ds,$$

provided that

$$\beta -\alpha ={\int }_0^{L}f\left(s\right)ds,$$

which is a compatibility condition between the forcing and the boundary conditions for problem (1) and can also be obtained by integrating Equation (1) from $x=0$ to L. If this compatibility condition is not satisfied, problem (1) has no solution.

Remark 4. 

If $M(x;\xi )$ is not specified, integration of the normalization condition ${\int }_0^{L}G(x;\xi )dx=0$ [3] for $k=0$ yields

$$C=-\frac{1}{L}\left(\frac{1}{2}{\xi }^2-\xi L-{\int }_0^{L}x(L-x)M(x;\xi )dx\right),$$

(18)

which clearly depends on $M(x;\xi )$ and whose substitution in Equation (14) indicates that the Green function depends on $M(x;\xi )$ through ${\int }_0^{x}M(s;\xi )(x-s)ds$ and ${\int }_0^{L}x(L-x)M(x;\xi )dx$.

Remark 5. 

The compatibility condition reported in Remark 3 may be evaluated numerically in an equally spaced grid of step size equal to $h=\frac{L}{N}$ as

$$\beta -\alpha =h{\displaystyle \sum _{i=0}^{N-1}}{f}_i,$$

(19)

$$\beta -\alpha =h{\displaystyle \sum _{i=1}^N}{f}_i,$$

(20)

or

$$\beta -\alpha =\frac{h}{2}{\displaystyle \sum _{i=0}^{N-1}}(f_i+f_{i+1}),$$

(21)

which correspond to the left- and right-rectangular and trapezoidal quadrature rules, respectively, where $f_i=f\left(x_i\right)$, $x_i=0+ih$, $i=0,1,2,\cdots ,N$, and $N+1$ denotes the number of grid points [27]. In Appendix A, three approximations to the above exact compatibility condition are obtained by discretizing the ordinary differential Equation (1) by means of second-order accurate finite differences and using three different discretizations for the Neumann’s boundary conditions.

2.3. Green’s Function for $k<0$

For $k=-{\Omega }^2<0$, the linear two-point boundary-value problem of Equation (1) has a unique solution, and the Green function that is governed by Equations (3) and (4) is also unique and may be easily obtained from that presented in Section 2.1 by simply replacing $\omega$ with $\mathrm{i}\Omega$ where ${\mathrm{i}}^2=-1$ and using the identities between trigonometric and hyperbolic functions. As a consequence, the Green function for $k=-{\Omega }^2<0$ is unique and may be written as

$$G(x;\xi )=-\frac{1}{\Omega \sinh (\Omega L)}\cosh (\Omega (L-\xi ))\cosh (\Omega x)+\sinh (\Omega (x-\xi ))H(x-\xi ),$$

(22)

i.e.,

$$G(x;\xi )=-\frac{\cosh (\Omega (L-\xi \left)\right)}{\Omega \sinh (\Omega L)}\cosh (\Omega x),\mathrm{for}0\le x\le \xi ,$$

(23)

and

$$G(x;\xi )=-\frac{\cosh (\Omega (L-x\left)\right)}{\Omega \sinh (\Omega L)}\cosh (\Omega \xi ),\mathrm{for}\xi \le x\le L,$$

(24)

which clearly illustrate the symmetry of the Green function.

The solution to Problem (1) for $k<0$ is also unique and given by Equation (2) with $G(0;\xi )=-\frac{\cosh (\Omega (L-\xi \left)\right)}{\Omega \sinh (\Omega L)}$ and $G(L;\xi )=-\frac{\cosh (\Omega \xi )}{\Omega \sinh (\Omega L)}$, but it is not reported here.

3. Conclusions

The Green functions for linear two-point boundary-value problems governed by $y^{″}\left(x\right)+ky\left(x\right)=f\left(x\right)$ and subject to non-homogeneous Neumann’s boundary conditions have been derived for cases where these problems have either a unique solution or an infinite number of solutions for $k\ge 0$. When these problems have a unique solution, it has been shown that the Green function satisfies a differential equation which includes the Dirac delta function and is subject to homogeneous Neumann boundary conditions and is unique.

However, when the problems have an infinite number of solutions and, therefore, must satisfy a compatibility condition between the boundary conditions and $f\left(x\right)$, it has been shown that the differential equation for the Green function must include both the Dirac delta function and an integrable function that must satisfy an integral constraint, and the Green function is not unique.

For $k>0$, it has been shown that both the Green function and the solution of the boundary-value problem are unique.

For $k=0$, finite difference equations for the second-order discretization of the differential equations and three different approximations to the Neumann boundary conditions have been analyzed and used to derive discrete compatibility conditions. It has been shown that only second-order accurate finite differences that employ ghost points result in a trapezoidal quadrature approximation to the exact compatibility condition for $k=0$.