EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 2, 2018, 390-399
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

On the elementary solution for the partial differential
operator }kc related to the wave equation

Sudprathai Bupasiri

Department of Mathematics, Sakon Nakhon Rajabhat University, Sakon Nakhon 47000,
Thailand

Abstract. In this article, we study an elementary solution of the operator }k
c , iterated k-times

and is defined by

}k
c =

( 1

c2

p∑
i=1

∂2

∂x2i
+m2

)2

−

 p+q∑
j=p+1

∂2

∂x2j

2


k

where p + q = n, k is a nonnegative integer, c is a positive real number, m is a nonnegative
real number and n is the dimension of Rn. In this work we study an elementary solution of the
operator }k

c . After that, we apply such an elementary solution to solve the solution of the equation
}k

cu(x) = f(x), where f is generalized function and u(x) is unknown function for x ∈ Rn.

2010 Mathematics Subject Classifications: 46F10

Key Words and Phrases: Elementary solution, Dirac-delta distribution, Temper distribution

1. Introduction

Trione [10] has showed that the generalized function RH2k,1(x) defined by (13) is the

unique elementary solution of the operator �k1, that is �k1R
H
2k,1(x) = δ where x ∈ Rn,

with n-dimensional Euclidean space. Also, Tellez ([7], p.147-149) has proved that RH2k,1(x)
exists only if n is an odd with p odd and q even, or only n is an even with p odd and
q odd. Later, Bupasiri [9] has showed that the solution of the convolution form u(x) =
(−1)kRe2k,c(x) ∗RH2k,c(x) is an elementary solution of the ♦kcu(x) = δ , where the operator

♦kc is defined by

♦kc =

 1

c4

(
p∑
r=1

∂2

∂x2r

)2

−

 p+q∑
j=p+1

∂2

∂x2j

2k

, (1)

where p + q = n is the dimension of the Euclidean space Rn, c is a positive real number
and k is a nonnegative integer. Otherwise, the operator ♦kc can be expressed in the form

Email address: sudprathai@gmail.com

http://www.ejpam.com 390 c© 2018 EJPAM All rights reserved.



S. Bupasiri / Eur. J. Pure Appl. Math, 11 (2) (2018), 390-399 391

♦kc = �kc4k
c = 4k

c�
k
c , where �kc is the operator related to the ultra-hyperbolic operator

iterated k-times, defined by

�kc =

(
1

c2

(
∂2

∂x21
+

∂2

∂x22
+ · · ·+ ∂2

∂x2p

)
− ∂2

∂x2p+1

− ∂2

∂x2p+2

− · · · − ∂2

∂x2p+q

)k
, (2)

and 4k
c is the operator related to the Laplace operator iterate k-times, defined by

4k
c =

(
1

c2

(
∂2

∂x21
+

∂2

∂x22
+ · · · ∂

2

∂x2p

)
+

∂2

∂x2p+1

+
∂2

∂x2p+2

+ · · ·+ ∂2

∂x2p+q

)k
. (3)

Next, Tellez [8] has studied the convolution product of Wα(u,m) ∗Wβ(u,m). Now in
this paper, the operator }kc can be expressed in the form

}kc =

( 1

c2

p∑
i=1

∂2

∂x2i
+m2

)2

−

 p+q∑
j=p+1

∂2

∂x2j

2k

=

 1

c2

p∑
i=1

∂2

∂x2i
−

p+q∑
j=p+1

∂2

∂x2j

+m2

k 1

c2

p∑
i=1

∂2

∂x2i
+

p+q∑
j=p+1

∂2

∂x2j
+m2

k

. (4)

Thus equation (4) can be written as

}kc =
(
�c +m2

)k (4c +m2
)k

=
(
4c +m2

)k (
�c +m2

)k
, (5)

where
(
4c +m2

)k
is the operator related to the Helmholtz operator iterated k-times which

is denoted by

(
4c +m2

)k
=

(
1

c2

(
∂2

∂x21
+

∂2

∂x22
+ · · · ∂

2

∂x2p

)
+

(
∂2

∂x2p+1

+ · · ·+ ∂2

∂x2p+q

)
+m2

)k
(6)

and
(
�c +m2

)k
is the operator related to the Klein-Gordon operator iterated k-times

which is denoted by

(
�c +m2

)k
=

(
1

c2

(
∂2

∂x21
+

∂2

∂x22
+ · · ·+ ∂2

∂x2p

)
−

(
∂2

∂x2p+1

+ · · ·+ ∂2

∂x2p+q

)
+m2

)k
, (7)

p+ q = n and from (4) with q = 0, c = 1 and k = 1, we obtain

}1 =
(
4p +m2

)2
(8)

where (
4p +m2

)
=

(
∂2

∂x21
+

∂2

∂x22
+ · · ·+ ∂2

∂x2p
+m2

)
. (9)



S. Bupasiri / Eur. J. Pure Appl. Math, 11 (2) (2018), 390-399 392

By putting p = 1,m = 0, c = 1 and x1 = t (time) in (7) then we obtain the wave operator

�1 =
∂2

∂x2t
−
n−1∑
j=1

∂2

∂x2j
(10)

and from (8) with q = 0,m = 0, c = 1 and k = 1, we obtain Laplace operator iterated
2-times of p-dimension

}1 = 42
p. (11)

In this paper, we study an elementary solution for the operator }kc , that is

}kcG(x) = δ,

where G(x) is an elementary solution , δ is the Dirac - delta distribution, k is a nonnegative
integer, c is a positive real number and m is a nonnegative real number.

We then also apply such an elementary solution to solve the solution of the equation
}kcu(x) = f(x), where f(x) is a given generalized function and u(x) is an unknown function
for x ∈ Rn.

2. Preliminaries

Definition 1. Let x = (x1, x2, . . . , xn) be a point of the n - dimensional space Rn,

u = c2
(
x21 + x22 + · · ·+ x2p

)
− x2p+1 − x2p+2 − · · · − x2p+q, (12)

where c is a positive real number, p + q = n. Define Γ+ = {x ∈ Rn : x1 > 0 and u > 0}
which designates the interior of the forward cone and Γ+ designates its closure and the
following functions introduce by Nozaki ([12], p.72) that

RHα,c(x) =

{
u
α−n
2

Kn(α)
if x ∈ Γ+

0 if x 6∈ Γ+,
(13)

RHα,1(x) is called the ultra-hyperbolic kernel of Marcel Riesz. Here α is a complex parameter
and n the dimension of the space. The constant Kn(α) is defined by

Kn(α) =
π
n−1
2 Γ

(
2+α−n

2

)
Γ
(
1−α
2

)
Γ(α)

Γ
(
2+α−p

2

)
Γ
(p−α

2

) (14)

and p is the number of positive terms of

u = c2
(
x21 + x22 + · · ·+ x2p

)
− x2p+1 − x2p+2 − · · · − x2p+q, p+ q = n

and let supp RHα,c(x) ⊂ Γ+. Now RHα,c(x) is an ordinary function if Re (α, c) ≥ n and is a
distribution of α if Re (α, c) < n.



S. Bupasiri / Eur. J. Pure Appl. Math, 11 (2) (2018), 390-399 393

Now, if p = 1 then (13) reduces to the function Mα,c(u) say, and defined by

Mα,c(u) =

{
u
α−n
2

Hn(α)
if x ∈ Γ+

0 if x 6∈ Γ+,
(15)

where u = c2x21 − x22 − · · · − x2n and Hn(α) = π
(n−1)

2 2α−1Γ(α−n+2
2 ). The function

Mα,1(u) is called the hyperbolic kernel of Marcel Riesz.

Definition 2. Let x = (x1, x2, . . . , xn) ∈ Rn and

v = c2
(
x21 + x22 + · · ·+ x2p

)
+ x2p+1 + x2p+2 + · · ·+ x2p+q, p+ q = n. (16)

For any complex number β , we define the function

Reβ,c(v) = 2−βπ−n/2Γ

(
n− β

2

)
v(β−n)/2

Γ(β/2)
. (17)

The function Reβ,1(v) is called the elliptic kernel of Marcel Riesz. It is an ordinary function
if Re(β, c) ≥ n and a distribution of β if Re(β, c) < n.

Lemma 1. Given the equation 4k
cu(x) = δ for x ∈ Rn, where 4k

c is the operator related
to the Laplace operator iterated k-times defined by (3). Then u(x) = (−1)kRe2k,c(v) is an

elementary solution of the operator 4k
c ,with β = 2k.

Proof. See [2].

Lemma 2. If �kcu(x) = δ for x ∈ Γ+ = {x ∈ Rn : x1 > 0 and u > 0}, where �kc is
the operator related to the ultra-hyperbolic operator iterated k-times defined by (2). Then
u(x) = RH2k,c(u) is the unique elementary solution of the operator �kc , with α = 2k.

Proof. See [10].

Lemma 3. Given the equation
(
�c +m2

)k
u(x) = δ for x ∈ Rn, where

(
�c +m2

)k
is

the operator related to the Klein-Gordon operator iterated k-times defined by equation (7),
δ is the Dirac-delta distribution, k is a nonnegative integer and m is a nonnegative real

number, then u(x) = W2k,c(u,m) is an elementary solution of the operator
(
�c +m2

)k
,

where

W2k,c(u,m) =
∞∑
r=0

(
−k
r

)
m2rRH2k+2r,c(u), (18)

RH2k,c(u) is defined by (13).

Proof. See [6].



S. Bupasiri / Eur. J. Pure Appl. Math, 11 (2) (2018), 390-399 394

Lemma 4. Let �c be the operator related to the ultra-hyperbolic operator, defined by (2)
and δ is the Dirac delta distribution for x ∈ Rn, then(

�c +m2
)k
δ = W−2k,c(u,m),

where W−2k,c(u,m) is the inverse of W2k,c(u,m) in the convolution algebra.

Proof. Let

V (x) =
(
�c +m2

)k
δ,

convolving both sides by W2k,c(u,m), then

W2k,c(u,m) ∗ V (x) = W2k,c(u,m) ∗
(
�c +m2

)k
δ

=
(
�c +m2

)k
W2k,c(u,m) ∗ δ

= δ. (19)

Since W2k,c(u,m) is lie in S′, where S′ is a space of tempered distribution, choose S′ ⊂ D′R,
where D′R is the right-side distribution which is a subspace of D′ of distribution. Thus
W2k,c(u,m) ∈ D′R, it follow that W2k,c(u,m) is an element of convolution algebra, thus by
([1], p.150-151), we have that the equation (19) has a unique solution

V (x) = W−2k,c(u,m) ∗ δ = W−2k,c(u,m). (20)

That complete the proof.

Lemma 5. Given the equation
(
4c +m2

)k
u(x) = δ for x ∈ Rn, where

(
4c +m2

)k
is

the operator related to the Helmholtz operator iterated k-times defined by equation (6), δ
is the Dirac-delta distribution, k is a nonnegative integer, then u(x) = Y2k,c(v,m) is an

elementary solution of the operator
(
4c +m2

)k
, where

Y2k,c(v,m) =
∞∑
r=0

(
−k
r

)
m2r(−1)k+rRe2k+2r,c(v), (21)

Re2k,c(v) is defined by (17).

Proof. See [6].

Lemma 6. Let 4c be the operator related to the Laplace operator, defined by (3) and δ is
the Dirac delta distribution for x ∈ Rn, then(

4c +m2
)k
δ = Y−2k,c(v,m),

where Y−2k,c(v,m) is the inverse of Y2k,c(v,m) in the convolution algebra.

Proof. The proof of this lemma similar lemma 4.



S. Bupasiri / Eur. J. Pure Appl. Math, 11 (2) (2018), 390-399 395

Lemma 7. The convolution W2k,c(u,m)∗Y2k,c(v,m) exists and is a tempered distribution
where W2k,c(u,m) and Y2k,c(v,m) be defined by (18) and (21), respectively.

Proof. From (18) and (21), we have

W2k,c(u,m) ∗ Y2k,c(v,m) =

( ∞∑
r=0

(
−k
r

)
m2rRH2k+2r,c(u)

)

∗

( ∞∑
r=0

(
−k
r

)
m2r(−1)k+rRe2k+2r,c(v)

)

=

∞∑
r=0

∞∑
s=0

(
−k
r

)(
−k
s

)
m2r+2s(−1)k+rRe2k+2r,c(v) ∗RH2k+2s,c(u).

Since the function Re2k+2r,c(v) and RH2k+2s,c(u) are tempered distributions, see([3], p.34,
[5], p.302 and [4], p.97) and the convolution of functions

(−1)k+rRH2k+2r,c(u) ∗Re2k+2s,c(v)

exists and is also a tempered distribution, see ([11], p.152). Thus, W2k,c(u,m)∗Y2k,c(v,m)
exists and also is a tempered distribution.

3. Main results

Theorem 1. Given the equation
}kcG(x) = δ (22)

for x ∈ Rn, where }kc is the operator related to the Helmhotz operator and Klein-Gordon
operator iterated k-times defined by (4), then

G(x) = W2k,c(u,m) ∗ Y2k,c(v,m) (23)

is an elementary solution of (22), where W2k,c(u,m) and Y2k,c(v,m) are defined by (18)
and (21), respectively, k is a nonnegative integer and m is a nonnegative real number.
Moreover, from (23) we obtain

W−2k,c(u,m) ∗G(x) = Y2k,c(v,m) (24)

as the elementary solution of the operator (4c + m2)k related to the Helmholtz operator
iterated k-times defined by (6) and in particular, for q = 0 and c = 1 then }kc reduces

to the Helmhotz operator
(
4p +m2

)2k
of p-dimension iterated 2k-times and is defined by

(9), where

4p =
1

c2

(
∂2

∂x21
+

∂2

∂x22
+ · · ·+ ∂2

∂x2p

)
,

thus (22) becomes (
4p +m2

)2k
G(x) = δ, (25)



S. Bupasiri / Eur. J. Pure Appl. Math, 11 (2) (2018), 390-399 396

we obtain
G(x) = Y4k,1(v,m) (26)

is an elementary solution of (25) and from (23). Moreover,

Y−2k,c(u,m) ∗G(x) = W2k,c(u,m) (27)

is an elementary solution of operator related to the Klein-Gordon operator. In particular,
we obtain

(−1)kRe−2,1(v) ∗G(x) = M2,1(u)

is an elementary solution of the wave operator defined by (10) where u = t2 − x21 − x22 −
· · · − x2n−1. Also, for m = 0 , q = 0 and c = 1 then (25) becomes

42k
p G(x) = δ (28)

where 42k
p is the Laplacian of p-dimension iterated 2k-times. We have

G(x) = Re4k,1(v)

is an elementary solution of (28) where

v = c2
(
x21 + x22 + · · ·+ x2p

)
.

Proof. From (5) and (22) we have

}kcG(x) =
((
�c +m2

)k (4c +m2
)k)

G(x) = δ.

Convolving both sides of the above equation by the convolution W2k,c(u,m) ∗ Y2k,c(v,m)
and the properties of convolution with derivatives, we obtain(

�c +m2
)k
W2k,c(u,m) ∗

(
4c +m2

)k
Y2k,c(v,m) ∗G(x)

= W2k,c(u,m) ∗ Y2k,c(v,m) ∗ δ. (29)

Thus

G(x) = δ ∗ δ ∗G(x) = W2k,c(u,m) ∗ Y2k,c(v,m) (30)

by Lemma 3 and 5. Now from (23) and by Lemma 3 and Lemma 4 and properties of
inverses in the convolution algebra, we obtain

W−2k,c(u,m) ∗G(x) = δ ∗ Y2k,c(v,m) = Y2k,c(v,m)

is an elementary solution of operator related to the Helmhotz operator iterated k-times
defined by (6). In particular, for q = 0 and c = 1 then (22) becomes(

4p +m2
)2k

G(x) = δ (31)



S. Bupasiri / Eur. J. Pure Appl. Math, 11 (2) (2018), 390-399 397

where
(
4p +m2

)2k
is the Helmholtz operator of p-dimension, iterated 2k-times and is

defined by (9). By Lemma 5, we have

G(x) = Y4k,1(v,m) (32)

is an elementary solution of (31). Moreover, from (23) and by Lemma 6 and Lemma 5
and properties of inverses in the convolution algebra, we obtain

Y−2k,c(u,m) ∗G(x) = W2k,c(u,m) ∗ δ = W2k,c(u,m)

is an elementary solution of operator related to the Klein-Gordon operator. In particular,
by putting p = 1, q = n− 1, k = 1, x1 = t, c = 1 and m = 0 in (23) and (27), W2,1(u,m =
0) = RH2,1(u) reduces to M2,1(u) where M2,1(u) is defined by (15) with α = 2. Thus we
obtain

(−1)kRe−2,1(v) ∗G(x) = M2,1(u)

is an elementary solution of the wave operator defined by (10) where u = t2 − x21 − x22 −
· · · − x2n−1. Also, for m = 0, c = 1 and q = 0 then (25) becomes

42k
p G(x) = δ (33)

where 42k
p is the Laplacian of p-dimension iterated 2k-times. By Lemma 1, we have

G(x) = (−1)2kRe4k,1(v) = Re4k,1(v)

is an elementary solution of (33) where

v = c2(x21 + x22 + · · ·+ x2p).

On the other hand, we can also find G(x) from (23), since q = 0, c = 1 and m = 0, we have
W2k,1(u,m = 0) = RH2k,1(u) reduces to (−1)kRe2k,1(v), where v = c2(x21 + x22 + · · · + x2p).
Thus, by (23) for q = 0, c = 1 and m = 0, we obtain

G(x) = (−1)kRe2k,1(v) ∗ (−1)kRe2k,1(v)

= (−1)2kRe2k+2k,1(v)

= Re4k,1(v) by W.F. Donoghue ([11],p 158).

That complete the proofs.

Theorem 2. Given the equation

}kcu(x) = f(x), (34)

where f is a given generalized function and u(x) is an unknown function, we obtain

u(x) = G(x) ∗ f(x)

is a solution of the equation (34), where G(x) is an elementary solution for }kc operator.



REFERENCES 398

Proof. Convolving both sides of (34) by G(x), where G(x) is an elementary solution
of }kc in Theorem 1, we obtain

G(x) ∗}kcu(x) = G(x) ∗ f(x)

or,
}kcG(x) ∗ u(x) = G(x) ∗ f(x)

applying the Theorem 1 , we have

δ ∗ u(x) = G(x) ∗ f(x).

Therefore,
u(x) = G(x) ∗ f(x).

Acknowledgements

The author would like to thank the referee for his suggestions which enhanced the pre-
sentation of the paper. The author was supported by Sakon Nakhon Rajabhat University
.

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