1. Introduction
Boundary value problems (BVPs) for ordinary and partial differential equations have appeared in widespread applications ranging from cognitive science to engineering. Some examples include a vibrating string with time depending upon external force under the Dirichlet boundary conditions [
1], Laplace’s equation in polar coordinates with the Neumann boundary conditions [
2], or the diffusion equation with the Robin boundary conditions [
3]. Finally, the heat flow in a nonuniform rod without sources accompanied with initial—boundary conditions [
4]. These types of problems inevitably associate with the partial differential operators—for example, the Laplace operator [
5,
6], the ultrahyperbolic operator [
7,
8], and the biharmonic operator [
9,
10].
One common choice to tackle such problems analytically is by using the method of separation of variables, which is somewhat limited. For instance, it must be applied to lower-order linear partial differential equations with a small number of variables. More sophisticated treatment for the BVPs was proposed by F. John [
11], who utilizes the Laplace operator using the following Green’s identity:
where
is the exterior normal vector to a boundary
and △ is the Laplace operator defined by
The solution,
, then becomes
where
is the Green’s function of the Laplace operator.
C. Bunpog [
12] subsequently studied BVPs of the diamond operator
in which it was originally investigated by A. Kananthai [
13] and later explored in more detail in [
14,
15]. It is denoted by
where the Laplace operator iterated
k-times,
, can be expressed as
and the ultrahyperbolic operator iterated
k-times,
, is represented by
The solution,
, can be formulated with the following expression:
where
is the Green’s function of the operator
. The functions
F and
G involve some boundary conditions on
.
The partial differential operator
has some qualitative properties which can be found in [
16,
17,
18,
19,
20]. It associates with the operators
and
such that
where
is defined by Equation (
1) and
is the biharmonic operator iterated
k-times:
In this paper, the Green’s identity of the operator will be presented. Furthermore, the solution’s existence under some suitable boundary conditions of the operator is manifested by using Green’s identity of the operators ♡ and , as well as the BVP solution of the diamond operator ♢. Finally, applications connected to the BVP of the linear partial differential operators are shown.
2. Preliminaries
Let us begin by introducing some functions and lemmas that are occasionally referred to in the paper.
Let
be a point of
and
The elliptic kernel of Marcel Riesz defined by Riesz [
21] has the following expression
where
is any complex number and
is the Gamma function. It is an ordinary function if
and is a distribution of
if
In addition,
is the Green’s function of the operator
defined by Equation (
2) (see [
13]).
Let
be a nondegenerated quadratic form. The interior of the forward cone is denoted by
. The ultrahyperbolic kernel of Marcel Riesz presented by Nozaki [
22] is expressed as
where
and
is a complex number. Note that
is an ordinary function if
and is a distribution of
if
Furthermore,
is the Green’s function of the operator
in the form of Equation (
3) (see [
23]).
Let
and
, where
. Functions
and
are defined by
for any complex numbers
and
. The convolution
is a tempered distribution (or a distribution of slow growth, [
24]) and the Green’s function of the operator
defined by Equation (
5), that is,
where
is the Dirac delta distribution [
18].
We modify these functions by introducing the following definitions.
Let
be a point of
and
We define
where
,
,
and
are defined by Equations (
6)–(
9), respectively. We let
Note that functions
and
are tempered distributions [
13,
16], which can be written in the form of functions
,
,
, and
. Equations (
11) and (
12) can thus be computed via [
25]:
Moreover, the function
satisfies
Lemma 1 (Gauss divergence theorem)
. Let Ω
be a bounded open subset of , is the boundary of Ω
, and , . Thenwhere denotes a differentiation in the direction of the exterior unit normal of , and is a surface element with integration on x. Lemma 2 (Green’s identity of the biharmonic operator)
. Let Ω
be a bounded open subset of , be the boundary of Ω
and , . Then, the Green’s identity of the biharmonic operator ♡
iswhere is given by denotes the complex-transversal to , and denotes the derivative in the complex-transversal direction. Proof of Lemma 2. From Equation (
5) with
, we can write
where
and
are defined by
and
By Equation (
14), we obtain
and
From Equations (
17) and (
18), we derive
By Equations (
19) and (
20), it can be concluded that
where
is defined by Equation (
16). The proof is completed. ☐
Lemma 3. Let Ω
be a bounded open subset of the Euclidian space , be the boundary of Ω
, and be a function which is given by Equation (
11)
with . Accordingly, the BVP solution of the diamond operator ♢
becomeswhere denotes the transversal to , and denotes the derivative in the transversal direction [
27].
3. Results
In this section, the Green’s identity along with the solution of the BVP of the operator are described. The results stated in the previous section are used to show the existence of a solution.
Theorem 1 (Green’s identity of the operator ⊕
k)
. Let Ω
be a bounded open subset of , be the boundary of Ω
and , . Then, the Green’s identity of the operator defined by Equation (4) is Proof of Theorem 1. Since
, replacing it by
in Equation (
15), we have
Likewise, since
, replacing it by
in Equation (
15), we have
Equation (
23) can be obtained according to Equations (
24) and (
25). ☐
Theorem 2. Let Ω
be a bounded open subset of , be the boundary of Ω
, , and be a function which is given by Equation (
12)
. Consequently (1) the BVP solution of the operator ⊕
becomes (2) the BVP solution of the operator , for , iswhere H and F are defined by Equations (16) and (22), respectively. Proof of Theorem 2. (1) By Equation (
15),
u and
v are replaced by
and
, respectively. It follows that
By Equations (
10)–(
12), with
, we have
. Therefore
According to Equations (
21) and (
26), the solution
becomes
(2) By Equation (
23)
k,
u, and
v are replaced by
,
, and
, respectively. This leads to
We have
and
resulting from Equation (
13). Thus
From Equations (
27) and (
28), we obtain
Our claim is now completely proved. ☐
3.1. Example 1
To illustrate the results, let us consider an equation
where
f is any tempered distribution on
. The boundary conditions on
are given by
From Equation (
30) and [
18] (p. 226), we have
By taking the convolution operator
on both sides of Equation (
31), it follows that
By Equations (
31) and (
32), we get
According to Equations (
22), (
32), and (
37), this leads to
By substituting Equations (
30), (
33)–(
36), (
38), and (
39) into Equation (
27), the solution becomes
Since all terms within the integrand are tempered distribution, the solution therefore exists.
Generally speaking, if we consider
where
and
are nonnegative integers. The operator
can reduce to the diamond operator iterated
k-times, the Laplace operator iterated
k-times, the ultrahyperbolic operator iterated
k-times and the biharmonic operator iterated
k-times, defined by Equations (
1), (
2), (
3) and (
5), respectively. For example, if we put
, the operator
becomes the Laplace operator iterated
k-times
.
3.2. Example 2 (Potential on Sphere with Dirichlet Boundary)
In the case that the operator
reduces to the Laplace operator iterated
k-times
,
where
f is any tempered distribution and
is a ball of radius
a. The boundary conditions on
are given by
and
where
g is a given tempered distribution. The solution of Equation (
42) is
The sphere is the locus of point x for which the ratio of distances and from certain points is constant. Here we can choose any point , then is the point obtained from by reflection with respect to the sphere .
That is,
, such that
and
are the Green’s functions of the Laplace operator with poles
and
respectively. Thus, for
,
Define the function
we have that
is the Green’s function of Laplace operator and
for
.
By substituting Equations (
42)–(
44) and (
46) into Equation (
45), the solution becomes
In the special case when
it is the potential on the sphere
of the problem (
42) with the Dirichlet boundary condition (
44).
3.3. Remark
In general, suppose that we consider equation where L is any linear partial differential operator. The solution to this problem can be found provided that L can be written in terms of two linear operators M and N (i.e., ). Moreover, the solution to the equation as well as the Green’s identity of the operator N are required.