Boundary Value Problem of the Operator ⊕ k Related to the Biharmonic Operator and the Diamond Operator

This paper presents an alternative methodology for finding the solution of the boundary value problem (BVP) for the linear partial differential operator. We are particularly interested in the linear operator ⊕k, where ⊕k = ♥k♦k, ♥k is the biharmonic operator iterated k-times and ♦k is the diamond operator iterated k-times. The solution is built on the Green’s identity of the operators ♥k and ⊕k, in which their derivations are also provided. To illustrate our findings, the example with prescribed boundary conditions is exhibited.


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 η is the exterior normal vector to a boundary ∂Ω and is the Laplace operator defined by The solution, u(ξ), then becomes where K(x, ξ) is the Green's function of the Laplace operator.
C. Bunpog [12] subsequently studied BVPs of the diamond operator ♦ k in which it was originally investigated by A. Kananthai [13] and later explored in more detail in [14,15].It is denoted by where D k (x, ξ) is the Green's function of the operator ♦ k .The functions F and G involve some boundary conditions on ∂Ω.The partial differential operator ⊕ k has some qualitative properties which can be found in [16][17][18][19][20].It associates with the operators ♦ k and ♥ k such that where ♦ k is defined by Equation ( 1) and ♥ k is the biharmonic operator iterated k-times: In this paper, the Green's identity of the operator ⊕ k will be presented.Furthermore, the solution's existence under some suitable boundary conditions of the operator ⊕ k is manifested by using Green's identity of the operators ♥ and ⊕ k , as well as the BVP solution of the diamond operator ♦.Finally, applications connected to the BVP of the linear partial differential operators are shown.

Preliminaries
Let us begin by introducing some functions and lemmas that are occasionally referred to in the paper.
Let x = (x 1 , x 2 , . . ., x n ) be a point of R n and v(x The elliptic kernel of Marcel Riesz defined by Riesz [21] has the following expression where α is any complex number and Γ is the Gamma function.It is an ordinary function if Re(α) ≥ n and is a distribution of α if Re(α) < n.In addition, (−1) k R e 2k (v) is the Green's function of the operator k defined by Equation (2) (see [13]).
n be a nondegenerated quadratic form.The interior of the forward cone is denoted by F = {x ∈ R n | x 1 > 0, y > 0}.The ultrahyperbolic kernel of Marcel Riesz presented by Nozaki [22] is expressed as where and β is a complex number.Note that R H β (y) is an ordinary function if Re(β) ≥ n and is a distribution of β if Re(β) < n.Furthermore, R H 2k (y) is the Green's function of the operator 2 k in the form of Equation (3) (see [23]). Let Functions S γ (w) and T η (z) are defined by for any complex numbers γ and κ.The convolution S 2k (w) * T 2k (z) is a tempered distribution (or a distribution of slow growth, [24]) and the Green's function of the operator ♥ k defined by Equation ( 5), that is, where δ(x) is the Dirac delta distribution [18].We modify these functions by introducing the following definitions.
Proof of Lemma 2. From Equation ( 5) with k = 1, we can write where L 1 and L 2 are defined by By Equation ( 14), we obtain and Therefore Hence From Equations ( 17) and ( 18), we derive Similarly, we find that By Equations ( 19) and ( 20), it can be concluded that where H(u, v) is defined by Equation ( 16).The proof is completed.

Results
In this section, the Green's identity along with the solution of the BVP of the operator ⊕ k are described.The results stated in the previous section are used to show the existence of a solution.

Theorem 2.
Let Ω be a bounded open subset of R n , ∂Ω be the boundary of Ω, u ∈ C 8k (Ω), Ω = Ω ∪ ∂Ω and M k (x, ξ) be a function which is given by Equation (12).Consequently (1) the BVP solution of the operator ⊕ becomes where 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 ♦u and M 1 , respectively.It follows that .
Generally speaking, if we consider where k 1 , k 2 and k 3 are nonnegative integers.The operator ⊕ k 1 ,k 2 ,k 3 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 k 1 = k, k 2 = k 3 = 0, the operator ⊕ k 1 ,k 2 ,k 3 becomes the Laplace operator iterated k-times k .

Example 2 (Potential on Sphere with Dirichlet Boundary)
In the case that the operator ⊕ k reduces to the Laplace operator iterated k-times k , where f is any tempered distribution and Ω = B(0, a) = {ξ, |ξ| < a} is a ball of radius a.The boundary conditions on Ω are given by where g is a given tempered distribution.The solution of Equation (42) is The sphere ∂Ω is the locus of point x for which the ratio of distances r = |x − ξ| and r * = |x − ξ * | from certain points is constant.Here we can choose any point ξ ∈ Ω, then ξ * is the point obtained from ξ by reflection with respect to the sphere ∂Ω.

That is, ξ
are the Green's functions of the Laplace operator with poles ξ and ξ * respectively.Thus, for x ∈ ∂Ω, Define the function we have that G k (x, ξ) is the Green's function of Laplace operator and G k (x, ξ) = 0 for x ∈ ∂Ω.By substituting Equations ( 42)-( 44 In the special case when k = 1, it is the potential on the sphere ∂Ω of the problem (42) with the Dirichlet boundary condition (44).

Remark
In general, suppose that we consider equation Lu = f 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., L = MN).Moreover, the solution to the equation Mu = f as well as the Green's identity of the operator N are required.

Conclusions
This paper focuses on finding the Green's identity together with the solution of the BVP for the operator ⊕ k which can be formulated in terms of the biharmonic and diamond operators.We first consider the solution for the case where k = 1 by employing the solution of the diamond operator and the Green's identity of the biharmonic operator.The solution for k > 2 is subsequently derived using the solution for the case k = 1 as well as the Green's identity for the operator ⊕ k .The solution for all k consists of the boundary terms satisfying Equations ( 16) and (22).