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Klein-Gordon potential in characteristic coordinates

  • Tynysbek Kal’menov and Durvudkhan Suragan EMAIL logo
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

By the Klein-Gordon potential, we call a convolution-type integral with a kernel, which is the fundamental solution of the Klein-Gordon equation and also a solution of the Cauchy problem to the same equation. An interesting question having several important applications (in general) is what boundary condition can be imposed on the Klein-Gordon potential on the boundary of a given domain so that the Klein-Gordon equation with initial conditions complemented by this “transparent” boundary condition would have a unique solution within that domain still given by the Klein-Gordon potential. It amounts to finding the trace of the Klein-Gordon potential to the boundary of the given domain. In this article, we analyze this question and construct a novel initial boundary-value problem for the Klein-Gordon equation in characteristic coordinates.

Keywords: Klein-Gordon equation; wave equation; Cauchy problem; characteristic coordinates
MSC 2010: 35L05; 35A08; 47G40; 81Q05

1 Introduction

In a bounded domain Ω ⊂ R n with a piecewise smooth boundary ∂ Ω , consider the Newton potential

(1.1) u ( x ) = ε n ⁎ f = ∫ Ω ε n ( x − y ) f ( y ) d y , supp f ⊂ Ω ,

where

ε 2 ( x − y ) = − 1 2 π ln ∣ x − y ∣ , ε n ( x − y ) = 1 ( n − 2 ) σ n ∣ x − y ∣ 2 − n , n ≥ 3 ,

and σ n is the surface area of the unit n -sphere. It is known that the Newton potential satisfies the Poisson equation, i.e.,

(1.2) − Δ u ( x ) = f ( x ) , x ∈ Ω .

In [1], the boundary condition of the Newton potential (1.1) was constructed in the following form:

(1.3) 1 2 u ( x ) = − ∫ ∂ Ω ε n ( x − y ) ∂ u ( y ) ∂ n y d S y + ∫ ∂ Ω ∂ ε n ( x − y ) ∂ n y u ( y ) d S y , x ∈ ∂ Ω .

Hence, if we consider equation (1.2) with boundary condition (1.3), then we find a unique solution to this boundary-value problem in the form (1.1). This phenomenon is also called Kac’s principle of not feeling the boundary and history goes back to Kac’s celebrated work [2]. For elliptic and hypoelliptic operators, further results in answering these questions were investigated in [3–5]. For different applications (in spectral theory), recent works [6–10] may also be of interest to readers. We also refer to works [11–13] on related topics.

We shall note that following these ideas in [14] for the Schrödinger equation in a bounded domain proposed new formulations of boundary conditions that have the physical meaning, that is, the property of suppressing waves reflected from the boundary. Moreover, it was shown that in the bounded domain, this solution coincides with the solution of the problem posed in the unbounded domain with the Sommerfeld radiation condition at infinity. Hence, the open problem from [15] was solved in [14].

Motivated by the previous developments in the field, the main purpose of this article is to construct similar “transparent” boundary formulae for the Klein-Gordon potential and use these conditions to construct as well as study a nonlocal initial boundary-value problem for the Klein-Gordon equation.

In the characteristic coordinates without the dissipation term, the Klein-Gordon equation is stated in the form

(1.4) ∂ 2 u ( ξ , η ) ∂ ξ ∂ η + m 2 4 u ( ξ , η ) = f ( ξ , η ) .

When m = 0 , it gives the wave equation. The Klein-Gordon equation is widely used to model various phenomena such as scalar field theory, relativistic quantum mechanics, and wave propagation in elastic media, among others. The dynamics of scalar and spinor fields share remarkable similarities in terms of characteristic coordinates, as demonstrated in [16]. Moreover, the solution of (1.4) is obeyed by plane wave solutions of the Dirac equation.

Let Ω be a square restricted with A 0 B 0 : η = ξ , 0 < ξ < 1 ∕ 2 , A 0 A : η = − ξ , 0 < ξ < 1 ∕ 2 , A B : η = ξ − 1 , 1 ∕ 2 < ξ < 1 , and B 0 B : η = 1 − ξ , 1 ∕ 2 < ξ < 1 .

It is known that the Riemann function (see, for example, [17, p. 247]) R ( ξ , η , ξ 1 , η 1 ) of equation (1.4) can be represented by

(1.5) R ( ξ , η , ξ 1 , η 1 ) = J 0 ( m 2 ( ξ − ξ 1 ) ( η − η 1 ) ) ,

where J 0 is the order zero Bessel function.

A fundamental solution of the Cauchy problem for equation (1.4) is presented as

(1.6) ε ( ξ 1 , η 1 ) = θ ( ξ − ξ 1 ) θ ( η 1 − η ) R ( ξ , η , ξ 1 , η 1 ) .

By a Klein-Gordon potential in Ω , we call the integral

(1.7) u ( ξ , η ) = ∫ Ω ε ( ξ − ξ 1 , η 1 − η ) f ( ξ 1 , η 1 ) d ξ 1 d η 1 = ∫ η ξ d ξ 1 ∫ ξ 1 η R ( ξ , η , ξ 1 , η 1 ) f ( ξ 1 , η 1 ) d η 1 = ∫ η ξ d ξ 1 ∫ ξ 1 η J 0 ( m 2 ( ξ − ξ 1 ) ( η 1 − η ) ) f ( ξ 1 , η 1 ) d η 1 .

It is easy to check that the Klein-Gordon potential satisfies the Cauchy condition for η = ξ , 0 < ξ < 1 ∕ 2 , that is,

(1.8) u ∣ η = ξ = 0 , ∂ ∂ ξ − ∂ ∂ η u ∣ η = ξ = 0

and equation (1.4). Thus, in this article:

  • We establish trace formulae for the Klein-Gordon potential operator on A 0 A : η = − ξ , and B 0 B : η = 1 − ξ . We use this to introduce a new initial boundary-value problem for the Klein-Gordon equation in the square Ω . Our result shows that the unique solution of the new initial boundary-value problem coincides with the solution of the Cauchy problem for the Klein-Gordon equation. This means that the obtained boundary conditions have a property that the wave should pass through the lateral boundaries η = − ξ and η = 1 − ξ undisturbed, that is, it should remain undisturbed if the wave is carrying “specific information.”

In Section 2, we construct conditions on the characteristics and establish an initial boundary-value problem for the Klein-Gordon equation in Theorem 1.

2 Main results

Our main result for the Klein-Gordon potential operator is the following trace formulae, which in turn give a new initial boundary-value problem for the Klein-Gordon equation. Let Ω be square (domain) as in Section 1.

Theorem 1

Let f ∈ C 1 ( Ω ¯ ) , supp f ⊂ Ω then the Klein-Gordon potential u ∈ C 2 ( Ω ¯ ) satisfies the following boundary conditions:

(2.1) N [ u ] ∣ A 0 A ≔ N [ u ] ∣ η = − ξ = ∫ 0 ξ J 0 ( − m 2 ( ξ − ξ 1 ) 2 ) ∂ u ( ξ 1 , − ξ 1 ) ∂ η 1 d ξ 1 + m 2 2 ∫ 0 ξ J 1 ( − m 2 ( ξ − ξ 1 ) 2 ) − m 2 ( ξ − ξ 1 ) 2 ( ξ − ξ 1 ) u ( ξ 1 , − ξ 1 ) d ξ 1 = 0 ,

(2.2) N [ u ] ∣ B 0 B ≔ N [ u ] ∣ η = 1 − ξ = − ∫ 1 2 ξ J 0 ( − m 2 ( ξ − ξ 1 ) 2 ) ∂ u ( ξ 1 , 1 − ξ 1 ) ∂ ξ 1 d ξ 1 + m 2 2 ∫ 1 2 ξ J 1 ( ( − m 2 ( ξ − ξ 1 ) 2 ) ) − m 2 ( ξ − ξ 1 ) 2 ( ξ − ξ 1 ) u ( ξ 1 , 1 − ξ 1 ) d ξ 1 = 0 ,

where J 0 is the zero-order Bessel function and J 1 is the first-order Bessel function.

Conversely, if u ∈ C 2 ( Ω ¯ ) is a solution of the Klein-Gordon equation (1.4) satisfying the Cauchy condition (1.8) and the boundary conditions (2.1) and (2.2), then u coincides with the Klein-Gordon potential (1.7).

We shall note that for m = 0 , the lateral boundary conditions of the Klein-Gordon potential coincide with the ones of the one-dimensional wave potential which is given [18]. This also shows that the lateral boundary conditions (2.1) and (2.2) are local if and only if m = 0 . It follows from Theorem 1 that the kernel ε given by (1.6), which is a fundamental solution of the Klein-Gordon equation, is the Green function of the nonlocal initial boundary-value problem

∂ 2 u ( ξ , η ) ∂ ξ ∂ η + m 2 4 u ( ξ , η ) = f ( ξ , η ) , f ∈ C 1 ( Ω ¯ ) , supp f ⊂ Ω ,

with the Cauchy conditions

u ∣ η = ξ = 0 , ∂ ∂ ξ − ∂ ∂ η u ∣ η = ξ = 0

and nonlocal boundary conditions

N [ u ] ∣ A 0 A ≔ N [ u ] ∣ η = − ξ = ∫ 0 ξ J 0 ( − m 2 ( ξ − ξ 1 ) 2 ) ∂ u ( ξ 1 , − ξ 1 ) ∂ η 1 d ξ 1 + m 2 2 ∫ 0 ξ J 1 ( − m 2 ( ξ − ξ 1 ) 2 ) − m 2 ( ξ − ξ 1 ) 2 u ( ξ 1 , − ξ 1 ) d ξ 1 = 0 ,

N [ u ] ∣ B 0 B ≔ N [ u ] ∣ η = 1 − ξ = − ∫ 1 2 ξ J 0 ( − m 2 ( ξ − ξ 1 ) 2 ) ∂ u ( ξ 1 , 1 − ξ 1 ) ∂ ξ 1 d ξ 1 + m 2 2 ∫ 1 2 ξ J 1 ( ( − m 2 ( ξ − ξ 1 ) 2 ) ) − m 2 ( ξ − ξ 1 ) 2 ( ξ − ξ 1 ) u ( ξ 1 , 1 − ξ 1 ) d ξ 1 = 0 .

Therefore, this initial nonlocal boundary-value problem can serve as an example of an explicitly solvable initial boundary-value problem for the Klein-Gordon equation for any m 2 > 0 .

Proof

We continue f by zero outside of the square Ω , that is, f ≡ 0 in R 2 \ Ω . Then, the Klein-Gordon potential

u ( ξ , η ) = ∫ η ξ d ξ 1 ∫ ξ 1 η R ( ξ , η , ξ 1 , η 1 ) f ( ξ 1 , η 1 ) d η 1

is a solution of equation (1.4) for all ( ξ , η ) ∈ R 2 and u ( ξ , η ) ∈ C 2 ( Ω ¯ ) satisfying the homogeneous Cauchy condition (1.8) on the line ξ = η , − ∞ < ξ < + ∞ . Moreover, the value of the function u at the point ( ξ , η ) is determined by the value of f ( ξ 1 , η 1 ) in the characteristic triangle △ ξ , η = { η ≤ ξ 1 ≤ ξ , η ≤ η 1 ≤ ξ 1 } .

Thus, the function u ( ξ , η ) on A 0 A , that is, u ∣ A 0 A = u ( ξ , − ξ ) is defined by the formula

(2.3) u ( ξ , − ξ ) = ∫ − ξ ξ d ξ 1 ∫ ξ 1 − ξ R ( ξ , − ξ , ξ 1 , η 1 ) f ( ξ 1 , η 1 ) d η 1 .

Since f ≡ 0 outside of ξ 1 ≤ 0 , η 1 ≤ − ξ 1 , the integral (2.3) takes the form

(2.4) u ∣ A 0 A = u ( ξ , − ξ ) = ∫ 0 ξ d ξ 1 ∫ ξ 1 − ξ 1 R ( ξ , − ξ , ξ 1 , η 1 ) f ( ξ 1 , η 1 ) d η 1 = ∫ 0 ξ d ξ 1 ∫ ξ 1 − ξ 1 J 0 ( m 2 ( ξ − ξ 1 ) ( − ξ − η 1 ) ) f ( ξ 1 , η 1 ) d η 1 .

Now in (2.4) instead of f ( ξ 1 , η 1 ) let us plug in ∂ 2 u ( ξ 1 , η 1 ) ∂ ξ 1 ∂ η 1 + m 2 4 u ( ξ 1 , η 1 ) , that is,

(2.5) u ( ξ , − ξ ) = ∫ 0 ξ d ξ 1 ∫ ξ 1 − ξ 1 J 0 ( m 2 ( ξ − ξ 1 ) ( − ξ − η 1 ) ) ∂ 2 u ∂ ξ 1 ∂ η 1 + m 2 4 u d η 1 = I 1 , 0 − + I 1 − ,

where

I 1 , 0 − = m 2 ∫ 0 ξ d ξ 1 ∫ ξ 1 − ξ 1 J 0 ( m 2 ( ξ − ξ 1 ) ( − ξ − η 1 ) ) u ( ξ 1 , η 1 ) d η 1 , I 1 − = ∫ 0 ξ d ξ 1 ∫ ξ 1 − ξ 1 J 0 ( m 2 ( ξ − ξ 1 ) ( − ξ − η 1 ) ) ∂ 2 u ∂ ξ 1 ∂ η 1 ( ξ 1 , η 1 ) d η 1 .

Integrating by parts, we obtain

I 1 − = ∫ 0 ξ J 0 ( m 2 ( ξ − ξ 1 ) ( − ξ − η 1 ) ) ∂ u ∂ ξ 1 ξ 1 − ξ 1 d ξ 1 − ∫ 0 ξ d ξ 1 ∫ ξ 1 − ξ 1 ∂ ∂ η 1 J 0 ( m 2 ( ξ − ξ 1 ) ( − ξ − η 1 ) ) ∂ u ∂ ξ 1 d η 1 = ∫ 0 ξ J 0 ( m 2 ( ξ − ξ 1 ) ( ξ 1 − ξ ) ) ∂ u ( ξ 1 , − ξ 1 ) ∂ ξ 1 d ξ 1 − ∫ 0 ξ J 0 ( m 2 ( ξ − ξ 1 ) ( ξ 1 − ξ 1 ) ) ∂ u ( ξ 1 , ξ 1 ) ∂ ξ 1 d ξ 1 − ∫ 0 ξ d ξ 1 ∫ ξ 1 − ξ 1 ∂ ∂ η 1 J 0 ( m 2 ( ξ − ξ 1 ) ( − ξ − η 1 ) ) ∂ u ( ξ 1 , η 1 ) ∂ ξ 1 d η 1 .

By the Cauchy condition (1.8), we have ∂ u ( ξ 1 , ξ 1 ) ∂ ξ 1 = 0 . Using this the latter equality can be represented in the form

I 1 − = ∫ 0 ξ J 0 ( m 2 ( ξ − ξ 1 ) ( ξ 1 − ξ ) ) ∂ u ( ξ 1 , − ξ 1 ) ∂ ξ 1 d ξ 1 − I 1 , 2 − ,

where

(2.6) I 1 , 2 − = ∫ 0 ξ d ξ 1 ∫ ξ 1 − ξ 1 ∂ ∂ η 1 J 0 ( m 2 ( ξ − ξ 1 ) ( − ξ − η 1 ) ) ∂ u ( ξ 1 , η 1 ) ∂ ξ 1 d η 1 .

Changing the integration order, we obtain

I 1 , 2 = ∫ 0 ξ d ξ 1 ∫ ξ 1 − ξ 1 ∂ ∂ η 1 J 0 ( m 2 ( ξ − ξ 1 ) ( − ξ − η 1 ) ) ∂ u ( ξ 1 , η 1 ) ∂ ξ 1 d η 1 = ∫ − ξ 0 d η 1 ∫ ξ − η 1 ∂ ∂ η 1 J 0 ( m 2 ( ξ − ξ 1 ) ( − ξ − η 1 ) ) ∂ u ( ξ 1 , η 1 ) ∂ ξ 1 d ξ 1 + ∫ 0 ξ d η 1 ∫ ξ η 1 ∂ ∂ η 1 J 0 ( m 2 ( ξ − ξ 1 ) ( − ξ − η 1 ) ) ∂ u ( ξ 1 , η 1 ) ∂ ξ 1 d ξ 1 .

By using the formula ∂ ∂ z J 0 ( z ) = − J 1 ( z ) with z = m 2 ( ξ − ξ 1 ) ( − ξ − η 1 ) , we arrive at

(2.7) ∂ 1 ∂ η J 0 ( m 2 ( ξ − ξ 1 ) ( − ξ − η 1 ) ) = ∂ ∂ z J 0 ( z ) ∂ z ∂ η 1 = − J 1 ( z ) ∂ ∂ η 1 ( m 2 ( ξ − ξ 1 ) ( − ξ − η 1 ) ) = m 2 2 J 1 ( z ) ( ξ − ξ 1 ) m 2 ( ξ − ξ 1 ) ( − ξ − η 1 ) .

Hence integrating by parts, we obtain

(2.8) I 1 , 2 = m 2 2 ∫ − ξ 0 J 1 ( m 2 ( ξ − ξ 1 ) ( − ξ − η 1 ) ) m 2 ( ξ − ξ 1 ) ( − ξ − η 1 ) ( ξ − ξ 1 ) u ( ξ 1 , η 1 ) ξ − η 1 d η 1 + m 2 2 ∫ 0 ξ J 1 ( m 2 ( ξ − ξ 1 ) ( − ξ − η 1 ) ) m 2 ( ξ − ξ 1 ) ( − ξ − η 1 ) ( ξ − ξ 1 ) u ( ξ 1 , η 1 ) ξ η 1 d η 1 − ∫ 0 ξ d ξ 1 ∫ ξ 1 − ξ 1 ∂ 2 ∂ ξ 1 ∂ η 1 J 0 ( m 2 ( ξ − ξ 1 ) ( − ξ − η 1 ) ) u ( ξ 1 , η 1 ) d η 1 = m 2 2 ∫ − ξ 0 J 1 ( − m 2 ( ξ + η 1 ) ( ξ + η 1 ) ) − m 2 ( ξ + η 1 ) 2 ( ξ + η 1 ) u ( − η 1 , η 1 ) d η 1 − ∫ 0 ξ d ξ 1 ∫ ξ 1 − ξ 1 ∂ 2 ∂ ξ 1 ∂ η 1 J 0 ( m 2 ( ξ − ξ 1 ) ( − ξ − η 1 ) ) u ( ξ 1 , η 1 ) d η 1 .

Taking − η 1 = ξ 1 in the first integral of (2.8), we obtain

(2.9) I 1 , 2 = m 2 2 ∫ 0 ξ J 1 ( − m 2 ( ξ − ξ 1 ) 2 ) − m 2 ( ξ − ξ 1 ) 2 ( ξ − ξ 1 ) u ( ξ 1 , − ξ 1 ) d ξ − ∫ 0 ξ d ξ 1 ∫ ξ 1 − ξ 1 ∂ 2 ∂ ξ 1 ∂ η 1 J 0 ( m 2 ( ξ − ξ 1 ) ( η − η 1 ) ) u ( ξ 1 , η 1 ) d η 1 .

By using relations (2.6) and (2.9) from (2.5), it implies that

(2.10) u ∣ A 0 A = u ( ξ , − ξ ) = ∫ 0 ξ J 0 − m 2 ( ξ − ξ 1 ) 2 ∂ u ( ξ 1 , − ξ 1 ) ∂ ξ 1 d ξ 1 − m 2 2 ∫ 0 ξ J 1 ( − m 2 ( ξ − ξ 1 ) 2 ) − m 2 ( ξ − ξ 1 ) 2 ( ξ − ξ 1 ) u ( ξ 1 , − ξ 1 ) d ξ 1 + ∫ 0 ξ d ξ 1 ∫ ξ 1 − ξ 1 ∂ 2 ∂ ξ 1 ∂ η 1 J 0 ( m 2 ( ξ − ξ 1 ) ( η − η 1 ) ) + m 2 J 0 ( m 2 ( ξ − ξ 1 ) ( η − η 1 ) ) u ( ξ 1 , − η 1 ) d η 1 .

Since J 0 ( m 2 ( ξ − ξ 1 ) ( η − η 1 ) ) is the Riemann function of equation (1.4), we have

(2.11) ∂ 2 ∂ ξ 1 ∂ η 1 J 0 ( m 2 ( ξ − ξ 1 ) ( η − η 1 ) ) + m 2 4 J 0 ( m 2 ( ξ − ξ 1 ) ( η − η 1 ) ) = 0 .

Using equality (2.5) from (2.10), it follows that

(2.12) u ∣ A 0 A = u ∣ η = ξ = u ( ξ , − ξ ) = ∫ 0 ξ J 0 ( − m 2 ( ξ − ξ 1 ) 2 ) ∂ u ( ξ 1 , − ξ 1 ) ∂ ξ 1 d ξ 1 − m 2 2 ∫ 0 ξ J 1 ( − m 2 ( ξ − ξ 1 ) 2 ) − m 2 ( ξ − ξ 1 ) 2 ( ξ − ξ 1 ) u ( ξ 1 , − ξ 1 ) d ξ 1 .

It is easy to see that d u ( ξ 1 , − ξ 1 ) d ξ 1 = ∂ u ∂ ξ 1 − ∂ u ∂ η 1 η 1 = − ξ 1 and this implies

(2.13) ∂ u ( ξ 1 , η 1 ) ∂ ξ 1 η 1 = − ξ 1 = d u ( ξ 1 , − ξ 1 ) d ξ 1 + ∂ u ( ξ 1 , η 1 ) ∂ η 1 η 1 = − ξ 1 .

Now integrating by parts in (2.12), we obtain

∫ 0 ξ J 0 ( − m 2 ( ξ − ξ 1 ) 2 ) ∂ u ( ξ , − ξ 1 ) ∂ ξ 1 d ξ 1 = ∫ 0 ξ J 0 ( − m 2 ( ξ − ξ 1 ) 2 ) d u ( ξ 1 , − ξ 1 ) d ξ 1 + ∂ u ( ξ 1 , η 1 ) ∂ η 1 η 1 = − ξ 1 d ξ 1 = ∫ 0 ξ J 0 ( − m 2 ( ξ − ξ 1 ) 2 ) ∂ u ( ξ 1 , η 1 ) ∂ η 1 η 1 = − ξ 1 d ξ 1 + J 0 ( − m 2 ( ξ − ξ 1 ) 2 ) u ( ξ 1 , − ξ 1 ) ∣ ξ 1 = ξ − ∫ 0 ξ ∂ ∂ ξ 1 J 0 ( − m 2 ( ξ − ξ 1 ) 2 ) u ( ξ 1 , − ξ 1 ) d ξ 1 = m 2 ∫ 0 ξ J 1 ( − m 2 ( ξ − ξ 1 ) 2 ) − m 2 ( ξ − ξ 1 ) 2 ( ξ − ξ 1 ) u ( ξ 1 , − ξ 1 ) d ξ 1 + u ( ξ , − ξ ) + ∫ 0 ξ J 0 ( − m 2 ( ξ − ξ 1 ) 2 ) ∂ u ( ξ 1 , η 1 ) ∂ η 1 η 1 = − ξ 1 d ξ 1 ,

since J 0 ( 0 ) = 1 .

This equality implies that

(2.14) u ( ξ , − ξ ) = u ∣ A 0 A + N [ u ] ∣ A 0 A = u ( ξ , − ξ ) + N [ u ] ∣ η = − ξ ,

that is,

(2.15) N [ u ] ∣ A 0 A ≔ N [ u ] ∣ η = − ξ = ∫ 0 ξ J 0 ( − m 2 ( ξ − ξ 1 ) 2 ) ∂ u ( ξ 1 , − ξ 1 ) ∂ η 1 d ξ 1 + m 2 2 ∫ 0 ξ J 1 ( − m 2 ( ξ − ξ 1 ) 2 ) − m 2 ( ξ − ξ 1 ) 2 ( ξ − ξ 1 ) u ( ξ 1 , − ξ 1 ) d ξ 1 = 0 .

The boundary condition (2.15) is the lateral boundary condition of the Klein-Gordon potential on A 0 A : η = − ξ , 0 < ξ < 1 ∕ 2 .

Now it remains to find a lateral boundary condition on B 0 B .

Since f ≡ 0 outside of the square △ ξ , 1 − ξ = { 1 − ξ ≤ ξ 1 ≤ ξ , η 1 = 1 − ξ , η 1 = 1 − ξ 1 , 1 ∕ 2 ≤ ξ ≤ 1 , η 1 = ξ 1 , 1 − ξ ≤ ξ ≤ 1 ∕ 2 } , using (1.7) and changing the integration order in the following integral we obtain

(2.16) u ∣ B 0 B = u ∣ η = 1 − ξ = ∫ 1 − ξ 1 2 d η 1 ∫ 1 − η 1 η 1 R ( ξ , η , ξ 1 , η 1 ) η = 1 − ξ f ( ξ 1 , η 1 ) d ξ 1 = ∫ 1 − ξ 1 2 d η 1 ∫ 1 − η 1 η 1 J 0 ( m 2 ( ξ − ξ 1 ) ( 1 − ξ − η 1 ) ) ∂ 2 u ∂ ξ 1 ∂ η 1 d ξ 1 + m 2 4 ∫ 1 − ξ 1 2 d η 1 ∫ 1 − η 1 η 1 J 0 ( m 2 ( ξ − ξ 1 ) ( 1 − ξ − η 1 ) ) u ( ξ 1 , η 1 ) d ξ 1 = I 1 + + I 2 + ,

where

(2.17) I 2 + = ∫ 1 − ξ 1 2 d η 1 ∫ 1 − η 1 η 1 J 0 ( m 2 ( ξ − ξ 1 ) ( 1 − ξ − η 1 ) ) ∂ 2 u ∂ ξ 1 ∂ η 1 d ξ 1

and

(2.18) I 1 + = m 2 4 ∫ 1 − ξ 1 2 d η 1 ∫ 1 − η 1 η 1 J 0 ( m 2 ( ξ − ξ 1 ) ( 1 − ξ − η 1 ) ) u ( ξ 1 , η 1 ) d ξ 1 .

Integrating by parts in (2.17), we obtain

I 2 + = ∫ 1 − ξ 1 2 J 0 m 2 ( ξ − ξ 1 ) ( 1 − ξ − η 1 ) ∂ u ( ξ 1 , η 1 ) ∂ η 1 1 − η 1 η 1 d η 1 − ∫ 1 − ξ 1 2 d η 1 ∫ 1 − η 1 η 1 ∂ J 0 ∂ ξ 1 ∂ u ∂ η 1 d ξ 1 = ∫ 1 − ξ 1 2 J 0 ( m 2 ( ξ − η 1 ) ( 1 − ξ − η 1 ) ) ∂ u ( η 1 , η 1 ) ∂ η 1 − J 0 ( m 2 ( ξ − ( 1 − η 1 ) ) ( 1 − ξ − η 1 ) ) ∂ u ( 1 − η 1 , η 1 ) ∂ η 1 d ξ d η 1 − ∫ 1 − ξ 1 2 d η 1 ∫ 1 − η 1 η 1 ∂ J 0 ∂ ξ 1 ∂ u ∂ η 1 d ξ 1 .

From the Cauchy condition (1.8), it follows that ∂ u ( η 1 , η 1 ) ∂ η 1 = 0 .

Thus, the last integral can be represented in the form

(2.19) I 2 + = − ∫ 1 − ξ 1 2 J 0 ( m 2 ( ξ + η 1 − 1 ) ( 1 − η 1 − ξ ) ) ∂ u ( 1 − η 1 , η 1 ) ∂ η 1 d η 1 − ∫ 1 − ξ 1 2 d η 1 ∫ 1 − η 1 η 1 ∂ J 0 ∂ ξ 1 ∂ u ∂ η 1 d ξ 1 = I 2 , 1 + + I 2 , 2 + ,

where

(2.20) I 2 , 1 + = − ∫ 1 − ξ 1 2 J 0 ( m 2 ( ξ + η 1 − 1 ) ( 1 − η 1 − ξ ) ) ∂ u ( 1 − η 1 , η 1 ) ∂ η 1 d η 1

and

(2.21) I 2 , 2 + = − ∫ 1 − ξ 1 2 d η 1 ∫ 1 − η 1 η 1 ∂ J 0 ∂ ξ 1 ∂ u ∂ η 1 d ξ 1 .

Setting 1 − η 1 = ξ 1 , the integral I 2 , 1 + is formulated as

(2.22) I 2 , 1 + = − ∫ 1 − ξ 1 2 J 0 ( m 2 ( ξ + η 1 − 1 ) ( 1 − η 1 − ξ ) ) ∂ u ( 1 − η 1 , η 1 ) ∂ η 1 d η 1 = − ∫ 1 2 ξ J 0 ( − m 2 ( ξ − ξ 1 ) 2 ) ∂ u ( ξ 1 , 1 − ξ 1 ) ∂ η 1 d ξ 1 .

Changing the integration order in I 2 , 2 + , we obtain

(2.23) I 2 , 2 + = − ∫ 1 − ξ 1 2 d ξ 1 ∫ ξ 1 1 − ξ ∂ J 0 ∂ ξ 1 ∂ u ∂ η 1 d η 1 − ∫ 1 2 ξ d ξ 1 ∫ 1 − ξ 1 1 − ξ ∂ J 0 ∂ ξ 1 ∂ u ∂ η 1 d η 1 = − ∫ 1 − ξ 1 2 ∂ J 0 ∂ ξ 1 u ξ 1 1 − ξ d ξ 1 + ∫ 1 − ξ 1 2 d ξ 1 ∫ ξ 1 1 − ξ ∂ 2 J 0 ∂ ξ 1 ∂ η 1 u ( ξ 1 , η 1 ) d η 1 − ∫ 1 2 ξ ∂ J 0 ∂ ξ 1 u ( ξ 1 , η 1 ) 1 − ξ 1 1 − ξ d ξ 1 + ∫ 1 2 ξ d ξ 1 ∫ 1 − ξ 1 1 − ξ ∂ 2 J 0 ∂ ξ 1 ∂ η 1 u ( ξ 1 , η 1 ) d η 1 .

Using the equality u ( ξ 1 , ξ 1 ) = 0 and

∂ J 0 ∂ ξ 1 = ∂ J 0 ( z ) ∂ z ∂ z 0 ∂ ξ 1 = m 2 J 1 ( z ) z ( η − η 1 ) ∣ η 1 = 1 − ξ 1 , η = 1 − ξ = 0 ,

we obtain

(2.24) I 2 , 2 + = ∫ 1 − ξ 1 2 d ξ 1 ∫ ξ 1 1 − ξ ∂ 2 J 0 ∂ ξ 1 ∂ η 1 u ( ξ 1 , η 1 ) d η 1 − m 2 2 ∫ 1 2 ξ J 1 ( m 2 ( ξ − ξ 1 ) ( 1 − ξ − ( 1 − ξ 1 ) ) ) − m 2 ( ξ 1 − ξ ) 2 ( ξ − ξ 1 ) ⋅ u ( ξ 1 , 1 − ξ 1 ) d ξ + ∫ 1 2 ξ d ξ 1 ∫ 1 − ξ 1 1 − ξ ∂ 2 J 0 ∂ ξ 1 ∂ η 1 u ( ξ 1 , η 1 ) d η 1 .

Using (2.18), (2.19), (2.21), and (2.24) from (2.23) implies that

(2.25) u ∣ B 0 B = u ∣ η = 1 − ξ = − ∫ 1 2 ξ J 0 ( − m 2 ( ξ − ξ 1 ) 2 ) ∂ u ( ξ 1 , 1 − ξ 1 ) ∂ η 1 d ξ 1 + m 2 2 ∫ 1 2 ξ J 1 ( − m 2 ( ξ − ξ 1 ) 2 ) − m 2 ( ξ − ξ 1 ) 2 ( ξ − ξ 1 ) u ( ξ 1 , 1 − ξ 1 ) d ξ 1 + ∫ 1 − ξ 1 2 d η 1 ∫ 1 − η 1 η 1 ∂ 2 J 0 ∂ ξ 1 ∂ η 1 + m 2 4 J 0 u ( ξ 1 , η 1 ) d ξ 1 .

Since J 0 ( m 2 ( ξ − ξ 1 ) ( η − η 1 ) ) is the Riemann function, we have

∂ 2 ∂ ξ 1 ∂ η 1 J 0 ( m 2 ( ξ − ξ 1 ) ( η − η 1 ) ) + m 2 4 J 0 ( m 2 ( ξ − ξ 1 ) ( η − η 1 ) ) ≡ 0 .

It implies

(2.26) u ∣ B 0 B = u ∣ η = 1 − ξ = − ∫ 1 2 ξ J 0 ( m 2 ( ξ − ξ 1 ) 2 ) ∂ u ∂ η 1 ( ξ 1 , 1 − ξ 1 ) d ξ − m 2 2 ∫ 1 2 ξ J 1 ( − m 2 ( ξ − ξ 1 ) 2 ) − m 2 ( ξ − ξ 1 ) 2 ( ξ − ξ 1 ) u ( ξ 1 , 1 − ξ 1 ) d ξ 1 .

It is easy to see that

d d ξ 1 u ( ξ 1 , 1 − ξ 1 ) = ∂ u ∂ ξ 1 − ∂ u ∂ η 1 η 1 = 1 − ξ 1

is the full derivative on the line η 1 = 1 − ξ 1 . This gives

(2.27) − ∂ u ∂ η 1 η 1 = 1 − ξ 1 = d u d ξ 1 − ∂ u ∂ ξ 1 η 1 = 1 − ξ 1 .

Putting the right-hand side of (2.27) in (2.26) and integrating by parts, we obtain

u ∣ B 0 B = u ∣ η = 1 − ξ = ∫ 1 2 ξ J 0 ( − m 2 ( ξ − ξ 1 ) 2 ) d u d ξ 1 − ∂ u ∂ ξ 1 d ξ 1 + m 2 2 ∫ 1 2 ξ J 1 ( − m 2 ( ξ − ξ 1 ) 2 ) − m 2 ( ξ − ξ 1 ) 2 ( ξ − ξ 1 ) u ( ξ 1 , 1 − ξ 1 ) d ξ 1 = u ( ξ , 1 − ξ ) − ∫ 1 2 ξ J 0 ( m 2 ( ξ − ξ 1 ) 2 ) ∂ u ( ξ 1 , 1 − ξ 1 ) ∂ ξ 1 d ξ 1 − ∫ 1 2 ξ u ( ξ 1 , 1 − ξ 1 ) ∂ ∂ ξ 1 J 0 ( m 2 ( ξ − ξ 1 ) 2 ) d ξ 1 + m 2 2 ∫ 1 2 ξ J 1 ( − m 2 ( ξ − ξ 1 ) 2 ) − m 2 ( ξ − ξ 1 ) 2 ( ξ − ξ 1 ) u ( ξ 1 , 1 − ξ 1 ) d ξ = u ( ξ , 1 − ξ ) + N [ u ] B 0 B ,

that is,

(2.28) u ∣ B 0 B = u ( ξ , 1 − ξ ) + N [ u ] ∣ B 0 B ,

where

N [ u ] ∣ B 0 B = − ∫ 1 2 ξ J 0 ( m 2 ( ξ − ξ 1 ) 2 ) ∂ u ( ξ 1 , 1 − ξ 1 ) ∂ ξ 1 d ξ 1 + m 2 2 ∫ 1 2 ξ J 1 ( − m 2 ( ξ − ξ 1 ) 2 ) − m 2 ( ξ − ξ 1 ) 2 ( ξ − ξ 1 ) u ( ξ 1 , 1 − ξ 1 ) d ξ 1 .

From (2.28), we see that the lateral boundary condition of the Klein-Gordon potential on B 0 B is given in the form

(2.29) N [ u ] ∣ B 0 B ≔ − ∫ 1 2 ξ J 0 ( m 2 ( ξ − ξ 1 ) 2 ) ∂ u ( ξ 1 , 1 − ξ 1 ) ∂ ξ 1 d ξ 1 + m 2 2 ∫ 1 2 ξ J 1 ( − m 2 ( ξ − ξ 1 ) 2 ) − m 2 ( ξ − ξ 1 ) 2 ( ξ − ξ 1 ) u ( ξ 1 , 1 − ξ 1 ) d ξ 1 = 0 .

Thus, the lateral boundary conditions of the Klein-Gordon potential on A 0 A and B 0 B are given by formulae (2.15) and (2.29), respectively.

Let ϑ ∈ C 2 ( Ω ¯ ) satisfy equation (1.4) with the initial condition (1.8) and lateral boundary conditions (2.15) and (2.29). Let u be the Klein-Gordon potential defined by formula (1.7), then ω = u − ϑ satisfies the homogeneous Klein-Gordon equation and the initial condition (1.8).

In this case the left-hand sides of (2.10) and (2.28) are equal to zero, that is,

ω + N [ ω ] ∣ A 0 A = ω ∣ A 0 A = 0 ,

ω + N [ ω ] ∣ B 0 B = ω ∣ B 0 B = 0 .

Since ω ∣ t = 0 = 0 from the uniqueness of a solution of the mixed Cauchy problem for the Klein-Gordon equation follows that ω = u − ϑ ≡ 0 , that is, u = ϑ . □

3 Conclusion

In conclusion, our research has focused on investigating the Klein-Gordon potential, which is defined as a convolution-type integral with a kernel representing the fundamental solution of the Klein-Gordon equation. This kernel also serves as a solution to the Cauchy problem for the same equation. The primary question we have addressed in this study pertains to the imposition of boundary conditions on the Klein-Gordon potential at the boundary of a specified domain. These boundary conditions are designed to ensure the existence of a unique solution to the Klein-Gordon equation within the given domain while maintaining the solution in the form of the Klein-Gordon potential.

This research is of particular significance due to its potential applications across various fields. Understanding how to define these “transparent” boundary conditions and determine the trace of the Klein-Gordon potential at the domain’s boundary opens up possibilities in areas such as signal processing and wave propagation.

To tackle this question, we have conducted an in-depth analysis and introduced an approach in the form of an initial boundary-value problem for the Klein-Gordon equation within characteristic coordinates. Our findings contribute to advancing the theoretical foundations of boundary-value problems for the Klein-Gordon equation and provide practical tools for addressing real-world applications where such equations are relevant. Our future work in this direction will be about the multidimensional case.

  1. Funding information: This research was funded by the Committee of Science of the Ministry of Science and Higher Education of Kazakhstan (Grant number AP14871460). This research is also funded by Nazarbayev University under CRP grant 20122022CRP1601.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Ethical approval: The conducted research is not related to either human or animal use.

  5. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2023-04-30
Revised: 2024-01-31
Accepted: 2024-05-10
Published Online: 2024-08-05

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/dema-2024-0015
Keywords for this article
Klein-Gordon equation; wave equation; Cauchy problem; characteristic coordinates
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  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
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