Construction of Green's functions for the two-dimensional static Klein ...

JOURNAL OF PARTIAL DIFFERENTIAL EQUATIONS
J. Part. Diff. Eq., Vol. 24, No. 2, pp. 114-139
doi: 10.4208/jpde.v24.n2.2
May 2011
Construction of Green’s Functions for the Two-Dimensional
Static Klein-Gordon Equation
MELNIKOV Yu. A. ∗
Department of Mathematical, Sciences Middle Tennessee State University,
Murfreesboro, Tennessee 37132, USA.
Received 29 December 2009; Accepted 10 March 2011
Abstract. In contrast to the cognate Laplace equation, for which a vast number of
Green’s functions is available, the field is not that developed for the static Klein-Gordon
equation. The latter represents, nonetheless, a natural area for application of some of
the methods that are proven productive for the Laplace equation. The perspective
looks especially attractive for the methods of images and eigenfunction expansion.
This study is based on our experience recently gained on the construction of Green’s
functions for elliptic partial differential equations. An extensive list of boundary-value
problems formulated for the static Klein-Gordon equation is considered. Computerfriendly
representations of their Green’s functions are obtained, most of which have
never been published before.
AMS Subject Classifications: 35J08, 65N80
Chinese Library Classifications: O175.2, O241
Key Words: Static Klein-Gordon equation; Green’s function.
1 Introduction
To prevent a possible confusion as to the subject of the present study, note that our focus
is not on the hyperbolic Klein-Gordon equation representing a classical mathematical
model in quantum field theory and for which the Green’s function formalism is well developed.
This project deals instead with the elliptic two-dimensional static Klein-Gordon
equation (SKGE)
∇ 2 u(P)−k 2 u(P)=0 (1.1)
with ∇ 2 representing the Laplace operator written in the coordinates of point P and the
parameter k is a real constant.
∗ Corresponding author. Email address: ymelniko@mtsu.edu (Y. A. Melnikov)
http://www.global-sci.org/jpde/ 114

Construction of Green’s Functions for Static Klein-Gordon Equation 115
To our best knowledge, [1] and [2] represent the only book-format publications covering,
to a certain extend, the Green’s function topic for the SKGE. Just a limited number of
boundary-value problems has been reviewed in those books, with the method of eigenfunction
expansion being used. The present study aims at obtaining computer-friendly
forms of Green’s functions for a significant list of problems stated for the SKGE.
The Green’s function G(P,Q) for the SKGE will be introduced, in this study, by setting
up the homogeneous boundary-value problem
for the nonhomogeneous equation
M i u(P)≡α i (P) ∂u(P)
∂n i
+β i (P)u(P)=0, P∈Γ i , (1.2)
∇ 2 u(P)−k 2 u(P)=− f(P), P∈Ω, (1.3)
where: Ω represents a simply connected region in two-dimensional Euclidean space with
Γ= ⋃ m
i=1 Γ i denoting a piecewise smooth contour of Ω; α i (P) and β i (P) represent given
functions defined on Γ in such a way that at least one of them is nonzero for every piece
Γ i of Γ; and n i represents the normal direction to Γ i at the point P. The right-hand side
function f(P) in (1.3) is supposed to be integrable on Ω.
Assume that the boundary-value problem in (1.2) and (1.3) is well-posed. This implies,
in fact, that it has a unique solution or, in other words, the corresponding homogeneous
problem, with f(P)≡0, has only the trivial u(P)≡0 solution. With this, we define
the Green’s function for the SKGE. That is, if, for any integrable on Ω right-hand side
term f(P) in (1.3), the solution to the boundary-value problem in (1.2) and (1.3) is found
in the form
∫ ∫
u(P)= G(P,Q) f(Q)dΩ(Q), (1.4)
Ω
then the kernel G(P,Q) of the above representation is said to be the Green’s function for
the homogeneous problem corresponding to that of (1.2) and (1.3).
A standard terminology will apply, according to which P and Q in (1.4) are referred
to as the field (observation) point and the source point, respectively.
For any location of the source point Q∈Ω, the Green’s function, as a function of the
coordinates of the observation point P, holds the following properties (being referred to
herein as the defining properties):
1. at any point P∈Ω, except at P=Q, G(P,Q) satisfies the homogeneous equation in
(1.1), that is
(∇ 2 −k 2 )G(P,Q)=0, P̸=Q.
2. For P→Q, G(P,Q) approaches infinity like the modified cylindrical Bessel (or Macdonald)
function K 0 (k|P−Q|) of the second kind of order zero.
3. G(P,Q) satisfies the boundary conditions in (1.2), that is
M i G(P,Q)=0,
P∈Γ i , i=1,m.

116 Y. A. Melnikov / J. Partial Diff. Eq., 24 (2011), pp. 114-139
Due to the properties of the Macdonald function (see, for example, [3–5], the Green’s
function G(P,Q) of the SKGE possesses the type of logarithmic singularity of Green’s
function for the Laplace equation. To support this claim, explore the nature of the Macdonald
function K 0 (x). For this we present its standard series expansion
(
K 0 (x)=− C+ln x ) ∞
x
2 ∑ 2j ∞
j=0
2 2j (j!) 2+ ∑
m=1
x 2m
2 2m (m!) 2 ( m∑
n=1
)
1
, (1.5)
n
where C≈0.5772157 is referred to as the Euler’s constant.
The representation in (1.5) appears, in fact, computer-friendly. This is so because both
of its infinite series components uniformly converge for any value of x∈(0,∞); and their
convergence rate is fairly high.
From the definition just introduced, the Green’s function of the homogeneous boundary-value
problem corresponding to (1.2) and (1.3) can be expressed as
G(P,Q)= 1
2π K 0(k|P−Q|)+R(P,Q), (1.6)
where R(P,Q), as a function of P, satisfies the homogeneous SKGE everywhere on Ω,
regardless of a mutual location of P and Q.
A special comment is appropriate as to the use of the terms regular component and
singular component of the Green’s function. Upon explicitly expressing the logarithmic
term in (1.5) as
−ln x 2
∞
∑
j=0
x 2j
2 2j (j!) 2 =−ln x 2
(1− x2
4 + x4
64 −···)
,
one realizes that the singularity in (1.6) is only associated with the j=0 term of the series.
With this in mind, we are going to refer to the additive terms K 0 (k|P−Q|)/2π and
R(P,Q) in (1.6) as the singular component and the regular component of the Green’s function,
respectively.
2 Method of images
We turn now to the construction of Green’s functions for the SKGE. The method of images
scheme [3, 6–9] will be applied to a variety of boundary-value problems.
The singular component[K 0 (k|P−Q|)]/2π of G(P,Q) in (1.6) is interpreted, similarly
to the term[ln|P−Q|]/2π in the case of Laplace equation, as the response at a field point P
to a unit source placed at an arbitrary point Q. With this, the regular component R(P,Q)
of G(P,Q) is intended, in the method of images, to be expressed as a sum of a number of
unit sources and sinks placed at points Q1 ∗, Q∗ 2 , ··· , Q∗ m outside the region Ω. This makes
the regular component
m
R(P,Q)= ∑± 1
2π K 0(k|P−Q ∗ j |)
j=1

Construction of Green’s Functions for Static Klein-Gordon Equation 117
of G(P,Q) a function satisfying the homogeneous SKGE at any point P in Ω (since all the
source points Q ∗ j
are outside Ω).
The algorithm of the method of images appears instrumental in obtaining a number
of Green’s functions for the SKGE.
Consider a trivial case of the Dirichlet problem stated for the SKGE on the upper
half-plane
Ω={(x,y) | −∞< x0}.
Influence of the unit source at a point Q(ξ,η)∈ Ω represents the singular component
( )
1
2π K 0 k
√(x−ξ) 2 +(y−η) 2
of the Green’s function. The above can be compensated with a single unit sink placed at
the point Q ∗ (ξ,−η) located at the lower half-plane and symmetric to Q(ξ,η) about the
boundary y=0. Having the influence of this sink given as
− 1 ( )
2π K 0 k
√(x−ξ) 2 +(y+η) 2 ,
the Green’s function for the upper half-plane is ultimately found as
where the complex variable notation
G(x,y;ξ,η)= 1 [ ( )]
K0 (k|z−ζ|)−K 0 k|z−ζ| , (2.1)
2π
z= x+iy and ζ= ξ+iη
is used for the observation and the source point, respectively.
The method of images allows us to prove the following three statements.
Theorem 2.1. Green’s function of the Dirichlet problem stated on the infinite circular sector
{
Ω= (r,ϕ) ∣ π
}
0

118 Y. A. Melnikov / J. Partial Diff. Eq., 24 (2011), pp. 114-139
Proof. Since the distance between two points(r 1 ,ϕ 1 ) and(r 2 ,ϕ 2 ) is defined in polar coordinates
as √
r 2 1 −2r 1r 2 cos(ϕ 1 −ϕ 2 )+r 2 2 ,
the singular component of the Green’s function G(r,ϕ;̺,ψ) reads as
( √ )
1
2π K 0 k r 2 −2r̺cos(ϕ−ψ)+̺2 , (2.3)
which represents response of the field at an observation point(r,ϕ)∈Ω to the unit source
acting at (̺,ψ).
In order to compensate the trace of the function in (2.3) (or, in other words, to support
the Dirichlet condition) on the boundary segment y=0, we place the unit sink at the point
(̺,2π−ψ). The influence of this sink is given by
− 1 ( √
)
2π K 0 k r 2 −2r̺cos(ϕ−(2π−ψ))+̺2 . (2.4)
Similarly, with the unit sink at (̺,π−ψ), whose influence is defined as
− 1 ( √
)
2π K 0 k r 2 −2r̺cos(ϕ−(π−ψ))+̺2 , (2.5)
we compensate the trace of (2.3) on the boundary segment x=0, while to compensate the
traces of the functions in (2.4) and (2.5) on x=0 and y=0, respectively, the unit source is
required at (̺,π+ψ), with the influence
( √
)
1
2π K 0 k r 2 −2r̺cos(ϕ−(π+ψ))+̺2 . (2.6)
Hence, the Green’s function of the Dirichlet problem stated on the quarter-plane represents
the sum of the components shown in (2.3)–(2.6), which converts to the form
shown in (2.2).
Theorem 2.2. Green’s function of a mixed boundary-value problem for the infinite circular sector
with the opening angle of π/2, with Dirichlet and Neumann boundary conditions imposed on the
boundary segments y=0 and x=0, respectively, is given as
G(r,ϕ;̺,ψ)= 1
2π
2
∑
n=1
( √
)
[(−1) n K 0 k r 2 −2r̺cos(ϕ−(nπ−ψ))+̺2
√
)]
+(−1) n K 0
(k r 2 −2r̺cos(ϕ−((n−1)π+ψ))+̺2 . (2.7)

Construction of Green’s Functions for Static Klein-Gordon Equation 119
Proof. To support the Dirichlet condition on the boundary segment y=0, we compensate
the trace of the function
( √ )
1
2π K 0 k r 2 −2r̺cos(ϕ−ψ)+̺2
(2.8)
with the unit sink at (̺,2π−ψ), the influence of which is given by
− 1 ( √
)
2π K 0 k r 2 −2r̺cos(ϕ−(2π−ψ))+̺2 . (2.9)
The Neumann condition on the boundary segment x=0 will be supported if the unit
source is put at (̺,π−ψ), with the influence defined as
( √
)
1
2π K 0 k r 2 −2r̺cos(ϕ−(π−ψ))+̺2 , (2.10)
while to support both the boundary conditions imposed on x=0 and y=0, the unit sink
is required at (̺,π+ψ), with the influence
− 1 ( √
)
2π K 0 k r 2 −2r̺cos(ϕ−(π+ψ))+̺2 . (2.11)
Summing up the components in (2.8) through (2.11), one obtains the Green’s function
of the mixed problem under consideration in the form shown in (2.7).
Theorem 2.3. Green’s function of the Dirichlet problem set up on the infinite circular sector
{
Ω= (r,ϕ) ∣ 0

120 Y. A. Melnikov / J. Partial Diff. Eq., 24 (2011), pp. 114-139
Recall the Green’s function exhibited in (2.7) for the mixed (Dirichlet-Neumann) problem
stated on the infinite circular sector with the opening angle of π/2. We can readily
present, however, an example of the same type of mixed problem, for which the method
of images procedure fails. This claim will be justified by considering the Dirichlet-Neumann
problem stated for the infinite circular sector with the opening angle of π/3.
Clearly, the Dirichlet condition on ϕ=0 is supported with the unit sink placed at
(̺,2π−ψ). To allow this sink to support the Neumann condition on ϕ=π/3, the unit
sink is also required at (̺,2π/3+ψ). As to the Neumann condition on ϕ=π/3, the unit
source at(̺,ψ) ought to be supported with the unit source at(̺,2π/3−ψ), which, in turn,
ought to be accompanied with the unit sink placed at (̺,4π/3+ψ). The latter sink has to
be accompanied with the unit sink at(̺,4π/3−ψ) for the Neumann condition supported
on ϕ=π/3. If we now take a look at the two sinks at(̺,2π/3+ψ) and(̺,4π/3−ψ), they
do not, unfortunately, support the Dirichlet condition on the boundary fragment ϕ=0.
And this is what indicates, in fact, the method’s failure for the mixed problem under
consideration.
Although the method of images might fail for some problems stated on an infinite
circular sector, the range of problems, for which it appears efficient, is not limited to
those considered so far. Indeed, considering the case of Dirichlet problem stated on the
infinite circular sector Ω={(r,ϕ) | 0

Construction of Green’s Functions for Static Klein-Gordon Equation 121
placed unit sources and sinks. Omitting details, we present just its ultimate expression
G(r,ϕ;̺,ψ)= 1
2π
6
∑
n=1
( √ [K 0 k r 2 −2r̺cos
(ϕ−((n−1) π ) )
3 +ψ) +̺2
√
− K 0
(k r 2 −2r̺cos
(ϕ−(n π )
3 −ψ)
+̺2
)]
. (2.15)
Certain generalizations can be drawn from the analysis of the forms of Green’s functions
derived thus far for the boundary-value problems stated on an infinite circular sector.
Observing the expressions derived earlier for the Green’s function of the Dirichlet
problem stated on the half-plane, on the circular sector with the opening angles of π/4
and π/2, one arrives at the following generalization
G(r,ϕ;̺,ψ)= 1
2π
2 m
∑
n=1
( √ [K 0 k r 2 −2r̺cos
(ϕ−(2(n−1) π ) )
2 m+ψ) +̺2
√
− K 0
(k r 2 −2r̺cos
(ϕ−(2n π )
2 m−ψ)
+̺2
)]
, (2.16)
which represents the Green’s function of the Dirichlet problem stated on the set of infinite
circular sectors with the opening angle of π/2 m , where m=0,1,2, ··· .
Note that the case of m=0, which corresponds to the circular sector with the opening
angle of π or, in other words, to the upper half-plane y>0, reads from (2.16) as
G(r,ϕ;̺,ψ)= 1
2π
( √ [K 0 k r 2 −2r̺cos(ϕ−ψ)+̺2
√ )]
− K 0
(k r 2 −2r̺cos(ϕ+ψ)+̺2 , (2.17)
representing the Green’s function derived earlier in (2.1) and expressed here in polar
coordinates.
Another significant generalization can be drawn out upon analyzing the expressions
in (2.12) and (2.15), from which one derives a Green’s function of the Dirichlet problem
stated for the set of circular sectors with the angle of π/(3·2 m ) where m=0,1,2, ··· . It is
found in the form
G(r,ϕ;̺,ψ)= 1 3·2 m ( √ (
2π
∑
[K 0 k r 2 −2r̺cos ϕ−(2(n−1) π )
3·2 m+ψ) n=1
√ (
− K 0
(k r 2 −2r̺cos ϕ−( 2nπ )
3·2 m−ψ) +̺2
)
)
+̺2
)]
, (2.18)
with the case of m=0 staying for the sector of π/3, while the case of m=1 stays for the
sector of π/6.

122 Y. A. Melnikov / J. Partial Diff. Eq., 24 (2011), pp. 114-139
Speaking of the success or failure of the method of images in constructing Green’s
functions for infinite circular sectors, remind that earlier we have shown that the method
fails for a mixed (Dirichlet-Neumann) problem stated on the sector with the opening
angle of π/3. But it is worth noting that the method is not necessarily successful for
Dirichlet problems either. The following problem is presented to illustrate the point.
Consider the Dirichlet problem on the infinite sector Ω={(r,ϕ)|0

Construction of Green’s Functions for Static Klein-Gordon Equation 123
and y=b, respectively. The responses to these sinks at (x,y) evidently are
G − 1,0 (x,y;ξ,−η)=− 1
)
2π K 0
(√(x−ξ) 2 +(y+η) 2
and
G − 1,b (x,y;ξ,2b−η)=− 1
)
2π K 0
(√(x−ξ) 2 +(y−(2b−η)) 2 .
The functions G − 1,0 (x,y;ξ,−η) and G− 1,b
(x,y;ξ,2b−η) do not vanish on the boundary
lines y=0 and y=b; and their traces can, in turn, be compensated with the unit sources
S + 2,0 and S+ 2,b
which are located at (ξ,−2b+η) and (ξ,2b+η). The responses to these at
(x,y) are given as
G + 2,0 (x,y;ξ,−2b+η)= 1
)
2π K 0
(√(x−ξ) 2 +(y−(−2b+η)) 2 ,
and
G + 2,b (x,y;ξ,2b+η)= 1
)
2π K 0
(√(x−ξ) 2 +(y−(2b+η)) 2 .
To properly compensate the traces of the functions G + 2,0 (x,y;ξ,−2b+η) and G+ 2,b (x,y;
ξ,2b+η) on y=0 and y=b, we place unit sinks S − 3,0 and S− 3,b
at(ξ,−2b−η) and(ξ,4b−η),
respectively.
Upon following the described procedure of properly placing compensatory unit sources
that alternate with compensatory unit sinks, one expresses the Green’s function G=
G(x,y;ξ,η) that we are looking for in the form
G=G + 0 + ∞
∑
i=1
( )
G − 2i−1,0 +G− 2i−1,b
+
∞
∑
i=1
( )
G + 2i,0 +G+ 2i,b
,
which, after a trivial algebra, transforms ultimately into the single infinite series exhibited
in (2.19).
Since the Green’s function in (2.19) is expressed in a series form, convergence of the
latter ought to be specifically addressed to ensure its computability. Note first that the
singularity in (2.19) is provided by the component which is a part of a single term (n=0),
and cannot, in any way, affect the series convergence.
A close analysis reveals convergence of the series in (2.19) at a high rate. This makes
it suitable for immediate computer implementations. This assertion is supported by the
observation √
(x−ξ)2 +(y±η±2nb) 2
lim
n→∞
2nb
which implies that the arguments of the Macdonal functions in (2.19) are asymptotically
close to 2nb and converge to zero, when n increases, at the same rate as terms of the
= 1

124 Y. A. Melnikov / J. Partial Diff. Eq., 24 (2011), pp. 114-139
sequence {K 0 (2nb)}. But the Macdonald function K 0 (x) is [10, 11] continuous, positive,
decreasing (at a very high rate) and bounded below by zero. For the strip of a unit width
(b=1), for example, the first few terms of the sequence{K 0 (2nb)} are
K 0 (2)≈.11389, K 0 (4)≈.01116, K 0 (6)≈.00124, K 0 (8)≈.00015
approaching zero at a rate close to that of a geometric sequence whose ratio is of the order
of 10 −1 and which therefore rapidly converges.
The brief analysis just completed brings a confidence in a practicality of the representation
in (2.19). One might, however, call in question the rigor of the analysis. Indeed, it
in no ways proves the convergence.
Theorem 2.5. Green’s function of a mixed problem stated on the infinite strip Ω={(x,y)|−∞<
x

Construction of Green’s Functions for Static Klein-Gordon Equation 125
The trace of the function G − 2,b
(x,y;ξ,2b+η) on y=0 can, in turn, be compensated with
a unit source S + 3,0
placed at (ξ,−2b−η), with the response
G + 3,0 (x,y;ξ,−2b−η)= 1
)
2π K 0
(√(x−ξ) 2 +(y+(2b+η)) 2 ,
while the Neumann condition on y=b can be supported with a unit sink S − 3,b at(ξ,4b−η),
whose response at(x,y) is given as
G − 3,b (x,y;ξ,4b−η)=− 1
)
2π K 0
(√(x−ξ) 2 +(y−(4b−η)) 2 .
Proceeding with this pattern in compliance with the approach described in the proof
of Theorem 2.4, the Green’s function that we are looking for is ultimately obtained in
the form of (2.20) which convergence at a high rate comparable to that of the series in
(2.19).
The pattern used in the proof of Theorems 2.4 and 2.5 appears also workable for the
Dirichlet problem set up on the semi-infinite strip Ω={(x,y)| 0

126 Y. A. Melnikov / J. Partial Diff. Eq., 24 (2011), pp. 114-139
imposed on the boundary fragments y=0 and y=b, and Neumann condition imposed on
x=0
Similarly to the Dirichlet problem considered above, the traces of the fundamental
solution of the SKGE (the field generated by the unit source acting at an arbitrary point
(ξ,η) in Ω) on the edges y=0 and y=b are compensated with unit sources and sinks
placed at the set of exterior to Ω points(ξ,−η), (ξ,2b−η), (ξ,−2b+η), (ξ,2b+η), (ξ,−2b−
η), (ξ,4b−η),··· .
As to the Neumann condition imposed on the boundary fragment x=0, it can be
supported by the aggregate influence of the sources and sinks acting at the set of points
just specified. The latter can, similarly to the Dirichlet problem, be compensated with
unit sources and sinks placed at the set of exterior to Ω points(−ξ,η), (−ξ,−η), (−ξ,2b−
η),(−ξ,−2b+η), (−ξ,2b+η), (−ξ,−2b−η), (−ξ,4b−η),··· . The order of sources and sinks
is, however, different, in this case, of that suggested earlier for the Dirichlet problem.
Combining influence of all the sources and sinks in compliance with our procedure,
one arrives at the rapidly convergent series representation
G(x,y;ξ,η)= 1
2π
∞
∑
n=−∞
[K 0
(√(x−ξ) 2 +(y−η+2nb) 2 )−K 0
(√(x−ξ) 2 +(y+η−2nb) 2 )
+K 0
(√
(x+ξ) 2 +(y−η+2nb) 2 )
−K 0
(√
(x+ξ) 2 +(y+η−2nb) 2 )]
of the Green’s function for the mixed problem under consideration.
(2.22)
3 Method of eigenfunction expansion
As it was illustrated in Section 2, the method of images appears successful for a number
of problem settings with Dirichlet and Neumann conditions imposed. We turn now to the
method of eigenfunction expansion, which is also applicable to all the problems workable
with the method of images, but whose application range is notably wider.
We start with a trivial problems already reviewed in the previous section. An alternative
to (2.19) representation will be obtained, in what follows, for the Green’s function
G(x,y;ξ,η) to the homogeneous Dirichlet problem corresponding to
(∇ 2 −k 2 )u(x,y)=− f(x,y), (x,y)∈Ω, (3.1)
u(x,0)=u(x,b)=0, (3.2)
stated on the infinite strip Ω={(x,y)|−∞

Construction of Green’s Functions for Static Klein-Gordon Equation 127
Express the solution u(x,y) to the problem in (3.1) and (3.2), and the right-hand side
function f(x,y) of (3.1) by the following Fourier sine-series
and
u(x,y)=
∞
∑
n=1
f(x,y)=
u n (x)sinνy, ν= nπ b , (3.3)
∞
∑
n=1
This yields the set of self-adjoint boundary-value problems
f n (x)sinνy. (3.4)
d 2 u n (x)
dx 2 −(ν 2 +k 2 )u n (x)=− f n (x), n=1,2,3,··· , (3.5)
|u n (−∞)|

128 Y. A. Melnikov / J. Partial Diff. Eq., 24 (2011), pp. 114-139
where the first series uniformly converges and its numerical implementations bring no
problem. The second series in (3.9) does not converge uniformly but allows a complete
summation. The summation is conducted with the aid of the classical [11] summation
formula
∞
p
∑
n
n cosnϑ=−1 2 ln(1−2pcosϑ+p2 ), (3.10)
n=1
that holds for p

Construction of Green’s Functions for Static Klein-Gordon Equation 129
Accounting for the fact that ν

130 Y. A. Melnikov / J. Partial Diff. Eq., 24 (2011), pp. 114-139
and integrate it, along with its sum. This yields
∞
∑
n=1
which translates, in our terms, into
∞
∑
n=1
1
( π
)
n expn b (x−ξ)
This ultimately yields the following form
1
n expnp=−ln(1−expp),
=−ln
(1−exp( π )
b (x−ξ)) .
|R N (x,y;ξ,η)|<
π{ b [1−exp(k(x−ξ)) ] [ ( ( π
))
ln 1−exp
b (x−ξ)
−
N
∑
n=1
1
( nπ
) ]
n exp b (x−ξ) +k b π
[
π 2
]}
N
6 − 1
∑
n=1
n 2
(3.15)
for the estimate in (3.14).
Hence, the series in (3.11) converges uniformly and can be accurately computed by
a direct truncation, and the estimate in (3.15) helps appropriately find a value of the
truncating parameter N required for attaining a desired accuracy.
For the next example on the use of the method of eigenfunction expansion, we set up
the mixed boundary-value problem
u(x,0)= ∂u(x,b)
∂y
= 0 (3.16)
posed for the nonhomogeneous SKGE on the infinite strip Ω={(x,y)|−∞

Construction of Green’s Functions for Static Klein-Gordon Equation 131
ultimately converts, in the current setting, into a quite compact and computer-friendly
form as
G(x,y;ξ,η)= 1
2π ln E 1(z−ζ)E 2 (z−ζ)
E 2 (z−ζ)E 1 (z−ζ)
− 1 b
∞
∑
n=1
[ exp(ν(x−ξ))
ν
− exp(h(x−ξ)) ]
sinνysinνη, (3.18)
h
where the real-valued functions E 1 (w) and E 2 (w) of a complex variable w are defined as
(
∣ πw
)
(
∣ ∣ πw
)
∣
E 1 (w)= ∣exp +1∣, E 2 (w)= ∣exp −1∣.
2b
2b
Notice that the coefficient
exp(ν(x−ξ))
− exp(h(x−ξ))
ν
h
of the series in (3.18) is valid for x≤ ξ, whereas its expression for x≥ ξ can be obtained
from the above by interchanging x with ξ.
The series in (3.18) is uniformly convergent in Ω. Its analysis can be accomplished in
exactly same fashion as for the Dirichlet problem considered earlier.
Omitting details, we present an ultimate expression for the Green’s function of the
Dirichlet problem
u(0,y)=u(x,0)=u(x,b)=0
stated for the SKGE on the semi-infinite strip Ω={(x,y)| 0

132 Y. A. Melnikov / J. Partial Diff. Eq., 24 (2011), pp. 114-139
Proceeding in compliance with our approach, the Green’s function for the formulation
in (3.20) is ultimately obtained in the form
where
G(x,y;ξ,η)= 1
2π ln E 1(z−ζ)E 2 (z+ζ)E 1 (z+ζ)E 2 (z−ζ)
E 1 (z+ζ)E 2 (z−ζ)E 1 (z−ζ)E 2 (z+ζ)
∞
( coshνx
− 2 b
− 2β b
∑
n=1
∞
∑
n=1
νexpνξ − coshhx
hexphξ
)
sinνysinνη
exp(−h(x+ξ))
sinνysinνt, x≤ ξ, (3.21)
h(h+β)
h= √ ν 2 +k 2 , ν=(2n−1) π 2b . (3.22)
The coefficient of the first series in (3.21) is valid for x≤ξ, while its expression for x≥ξ
can be obtained by interchanging x with ξ. Clearly, the coefficient of the second series in
(3.21) remains invariant to the interchange.
The series in (3.21) are uniformly convergent, and one can readily estimate the remainder
of the first of them by using the approach that has already been implemented to
the series of (3.11). To estimate the remainder R N (x,y;ξ,η) of the second series in (3.21),
we proceed as follows
|R N (x,y;ξ,η)|=
∣
≤
<
∞
∑
n=N+1
∞
∑
n=N+1
∞
∑
n=N+1
∣
exp(−h(x+ξ))
h(h+β)
exp(−h(x+ξ))
h(h+β)
sinνysinνη
∣
∞ ∣ = ∑
n=N+1
exp(−h(x+ξ))
h(h+β)
exp(−ν(x+ξ))
. (3.23)
ν(ν+β)
Recalling from (3.22) the expression for ν in terms of n, we rewrite the estimate in (3.23)
in the explicit form
( )
|R N (x,y;ξ,η)|< 4b2 exp − π(2n−1)
2b
(x+ξ)
(2n−1)[(2n−1)+β 0 ] ,
∞
π ∑ 2
n=N+1
where β 0 =2bβ/π. The above can be rewritten as
|R N (x,y;ξ,η)|< 4b2
π 2
= 4b2
π 2 p
∞
∑
n=N+1
∞
∑
n=N+1
[ (
exp
− π(x+ξ)
2b
)] 2n−1
(2n−1)[(2n−1)+β 0 ]
p 2n
(2n−1)2n , (3.24)

Construction of Green’s Functions for Static Klein-Gordon Equation 133
where we introduce for compactness
p=exp
(− π 2b (x+ξ) )
.
To further proceed with the estimate obtained in (3.24), we will derive the summation
formula for the series
∞
∑
n=1
p 2n
(2n−1)2n ,
whose N-th remainder appears in (3.24). In doing so, let us take advantage of the standard
[11] relation
∞
p
∑
2n−1
2n−1 = 1 ( ) 1+p
2 ln (3.25)
1−p
n=1
valid for p 2

134 Y. A. Melnikov / J. Partial Diff. Eq., 24 (2011), pp. 114-139
We focus now on the Dirichlet problem set up on the rectangle Ω={(x,y)| 0< x<
a,0< y

Construction of Green’s Functions for Static Klein-Gordon Equation 135
valuated by an appropriate truncation. The second series in (3.30) is not, however, uniformly
convergent. It possesses the logarithmic singularity which shows up if both the
field and the source point go to the edge x=a.
For another example on the success of the eigenfunction expansion method in a case
when the method of images fails, consider the boundary-value problem
set up for the nonhomogeneous SKGE
1
r
(
∂
∂r
r ∂u(r,ϕ)
∂r
u(r,0)=u(r,α)=0 (3.31)
)
+ 1 r 2 ∂ 2 u(r,ϕ)
∂ϕ 2 −k 2 u(r,ϕ)=− f(r,ϕ) (3.32)
on the infinite wedge Ω(r,ϕ)={(r,ϕ)|0

136 Y. A. Melnikov / J. Partial Diff. Eq., 24 (2011), pp. 114-139
This yields the set of boundary-value problems
d 2 u n (r)
dr 2 + 1 )
du n (r)
−
(k 2 + ν2
r dr r 2 u n (r)=− f n (r), n=1,2,3,··· , (3.37)
lim|u n (r)|

Construction of Green’s Functions for Static Klein-Gordon Equation 137
representing the Green’s function for the Dirichlet problem stated on the quarter-plane
Ω={(r,ϕ)|0

138 Y. A. Melnikov / J. Partial Diff. Eq., 24 (2011), pp. 114-139
Green’s function for this problem is found by means of our approach as
G(r,ϕ;̺,ψ)= 1 ( √ ) ( √ )]
[K 0 k r
2π
2 −2r̺cos(ϕ−ψ)+̺2 −K 0 k r 2 −2r̺cos(ϕ+ψ)+̺2
− 2 I n (kr)I n (k̺)[K n(kR)+βK ′ n (kR)]
π I n(kR)+βI ′ sinnϕsinnψ. (3.46)
n (kR)
∞
∑
n=1
Green’s functions for two other boundary-value problems can be obtained from the
above representation. First, the form in (3.46) reduces to the expression of (3.44) (Dirichlet
problem for the half-disk), when the parameter β in the boundary condition of (3.45)
approaches infinity. Second, as β goes to zero, the representation in (3.46) reduces to the
Green’s function
G(r,ϕ;̺,ψ)= 1
2π
( √ ) ( √ )]
[K 0 k r 2 −2r̺cos(ϕ−ψ)+̺2 −K 0 k r 2 −2r̺cos(ϕ+ψ)+̺2
− 2 π
∞
∑
n=1
for the mixed boundary-value problem
stated on the half-disk.
I n (kr)I n (k̺)K n(kR)
′
I n(kR) ′ sinnϕsinnψ (3.47)
∂u(R,ϕ)
∂r
= u(r,0)=u(r,π)=0
4 Concluding remarks
Summarizing the results of the present study, point out that a variety of boundary-value
problems for the static Klein-Gordon equations, for which Green’s functions can potentially
be constructed by using the techniques developed herein, is not limited to the cases
just considered. Many other problems can also be successfully tackled. The presented
work especially aimed at those researchers who might be interested in a numerical implementation
of Green’s functions, in which case the practical computability is a critical
issue.
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Construction of Green’s Functions for Static Klein-Gordon Equation 139
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