Analysis of Two-Dimensional Heat Transfer Problem Using the Boundary Integral Equation

In this paper, we examine the problem of two-dimensional heat equations with certain initial and boundary conditions being considered. In a two-dimensional heat transport problem, the boundary integral equation technique was applied. The problem is expressed by an integral equation using the fundamental solution in Green ’ s identity. In this study, we transform the boundary value problem for the steady-state heat transfer problem into a boundary integral equation and drive the solution of the two-dimensional heat transfer problem using the boundary integral equation for the mixed boundary value problem by using Green ’ s identity and fundamental solution.


Introduction
For partial differential equations, the boundary integral equation is a basic method for analyzing boundary value problems [1]. Various schemes have emerged to discretize time domain boundary integral equations associated to parabolic problem [2]. In the inception of the boundary integral equation method, the thermal engineering community has been exploiting its potential in solving transient heat conduction problems [3,4]. Any approach for the approximate numerical solution of the boundary integral equations is referred to as a boundary element method [5]. The accurate solution of the differential equation of a two-dimensional heat transfer problem in the domain acquired by the boundary element method distinguishes the approximate solution of the boundary value problem produced by the boundary element method [6][7][8][9]. Only the domain's boundary needs to be discretized, notably in two-dimensional heat transfer problems with a simple circle boundary.
In some applications, the physical relevant data are provided by the boundary value of the solution or its derivatives rather than the solution in the domain boundary [10]. These data can be derived directly from the boundary integral equation's solution.
The advantage of using boundary integral formulation of partial differential equation problems is that we require only N d−1 unknown to discretize the boundary Γ, where N is the number of variables in each space dimension [5,6]. Many different formulations have been proposed for the treatment of heat conduction (diffusion) problems by the boundary integral equation BIE method, the most efficient of which is the one which employs a time-dependent fundamental solution. The formulation adopted for this analysis employs Green's identity to derive the boundary integral equation in [4,11]. A fundamental solution is generally not available if the coefficients of the original partial differential equation are not constant. One can use, in this case, a parametrix (Levi function), which is usually available, instead of fundamental solution Green formulae [3,12,13].
The solution exactly satisfies the differential equation inside the domain; nevertheless, approximate solutions exist because boundary conditions are only approximately satisfied. Because functions are defined globally, there is no need to divide the domain into elements [14][15][16].
The solution also meets the criterion at infinity, so dealing with infinite domains, where the finite element method must apply either truncation or approximate infinite elements, is not an issue [10,17,18]. As a result, the goal of this work is to use Green's identity and fundamental solution to transform the boundary value problem for steady-state heat transfer into a boundary integral equation and solve the boundary integral equation for the mixed boundary value problem [8,[19][20][21].
Using a boundary integral expression for a twodimensional heat transfer problem, we obtain a unique weak solution and a variational solution in the Sobolev space of order one, H 1 ðΩÞ [15,22]. The remainder of the current document is as follows: some basic definitions, theorems, and properties of the Laplace equation that arise as a steady-state problem for heat equation are mentioned in Section 2. Section 3 illustrates the details of the statement of the steady-state heat transfer problem, the boundary integral equation for the classical solution, and the boundary integral expression for the weak solution. Section 4 provides the conclusions of the paper.

Preliminaries
For a heat equation that does not change with time, the Laplace equation arises as a steady-state problem [20].
Equation (1) has no dependence on time, just on the spatial variables x and y. This means that the Laplace equation described steady state situated on the temperature distribution.
The steady-state solution satisfies Δu = 0 and boundary condition, u is prescribed on ∂Ω, and then, we consider the domain Ω that are circular [23].

Sobolev Space
Definition 1 (see [8]). Let 1 ≤ p ≤ ∞ and r ∈ ℕ 0 , and let Ω ⊆ ℝ n be a nonempty open set. The Sobolev space w r p order r based on L p ðΩÞ is defined by Remark 2. ∂ α u is viewed as a distribution on Ω, so the condition ∂ α u ∈ L p ðΩÞ means that there exists a function g α u ∈ L p ðΩÞ such that hu, ∂ α φi = ð−1Þ jαj hg α , φi, ∀φ ∈ D ′ ðΩÞ, such that a function g α is defined as a weak derivatives of u.
The complement of L p ðΩÞ implies that w r p ðΩÞ becomes a Banach space on putting the norm w r p ðΩÞ as For p = 2, W r 2 ðΩÞ = H r ðΩÞ is a Hilbert space with the inner product.
The norm induced by the inner product is Definition 3 (see [24]). In a particular case, the Sobolev space H 1 ðΩÞ is the set of all f ∈ L 2 ðΩÞ such that all the first partial derivative ∂f /∂x i belongs to L 2 ðΩÞ. The inner product in where ∇f :∇ g denotes ∑ n i=1 ð∂f /∂x i Þ:ð∂ g/∂x i Þ. This inner product clearly gives the norm Then, we denote the L 2 inner product by a subscript zero.
In particular, Then, the Cauchy sequence in H 1 ðΩÞ converges to the element of H 1 ðΩÞ. In other words, H 1 ðΩÞ is a Hilbert space. It is in fact the Hilbert space obtained by completing the set of smooth function with respect to the k,k 1 , in the same way that L 2 ðΩÞ is the Hilbert space obtained by completing the 2 Advances in Mathematical Physics set of smooth functions with respect to the L 2 norm [24].
2.3. Weak Solution [1,15]. Consider a partial differential Considering a differential equation LuðxÞ = f in the sense of distribution, then the following is true.
If the original problem was to find jαj-times differentiable function u defined on the open set Ω such that LuðxÞ = f for all x ∈ Ω, called the classical solution, then an integrable function u is said to be a weak solution if

Fundamental Solutions
Definition 4 (see [25]). A distribution D ′ is a fundamental solution of the differential operator L, if and only if The fundamental solution D of the differential operator L satisfies the equation; however, D need not fulfill the provided boundary conditions. A fundamental solution that satisfies the given boundary condition is known as Green's function [20,21,25].
Let Lu = f be Green's function Gðx, ξÞ; it satisfies the equation Physically, Green's function Gðx, ξÞ represents the effect at the point x of a Dirac delta function source at the point x = ξ [20].
Multiply equation (14) by f ðξÞ and integrate over the area A of the ξ circle so that Then, we have Since Lu = f , we have The fundamental solution for the Laplace operator is as follows.
Definition 5 (see [26]). Let E ∈ D ′ ðR 2 Þ such that with δ being the Dirac delta function. In general dimension, the D′ðR 2 Þ (distributional space in R 2 ) is a solution of equation (18) where c is the arbitrary constant and r is the distance from x to ξ.
It is also known as a heat kernel, which is a solution to the heat equation that corresponds to the initial condition of an initial point source at a specified place. This method can be used to discover a general solution to the heat equation for a given domain [21,25,26]. [16,26]. Let u, v ∈ C 1 ð ΩÞ ∩ C 2 ðΩÞ and Green's first identity for the pair u and v; then,

Green's Second
and again for the pair v and u, By subtracting equation (21) from equation (20), we get Green's second identity [23].
It is valid for the pair of functions u and v. The above integral is a line integral over the boundary curve of two-dimensional region Ω, and ds denotes the arc length of the boundary [16,26].

Boundary Integral Equation.
In a variety of applications, the efficient numerical solution of partial differential equations (PDE) using boundary integral formulation is critical [27,28].
Consider as an example a Laplace problem of the form In some domain Ω ⊂ ℝ 2 with piecewise smooth Lipschitz boundary Γ, Green's representation theorem 3 Advances in Mathematical Physics allows us to write the solution u as where n is the unit outward pointing normal at Γ and Gðx , yÞ is a fundamental solution defined as Hence, in principle, if either u or ∂u/∂n is known on G, we can recover the unknown quantity by restricting equation (24) to the boundary and solving to the unknown boundary (see, e.g., [6,16]).

Variational Formulation.
The variational approach to the problem not only lays the groundwork for mathematical proofs of existence and uniqueness but also strong numerical methods like the finite element method [15,29]. Using the boundary conditions mentioned above in an appropriate space of functions, we look for a unique weak solution u of the Laplace equation Δu = 0 in S [15,29].
The problem is written in a weak form as follows: (2) Apply integration by parts to arrive at

Statement of the Steady-State Heat Transfer Problem
Consider a heat-conducting body that is homogeneous and isotropic; Ω is a simple connected and bounded domain in ℝ 2 with a Lipschitz boundary Γ = Γ 1 ∪ Γ 2 ∪ Γ 3 ∪ Γ 4 when Γ 1 , Γ 2 , Γ 3 and Γ 4 are disjoint parts of Γ. Convection in the ambient medium is thought to occur at the boundary Γ 1 and Γ 2 , temperature is kept constant at the prescribed value T 3 on Γ 3 , and Γ 4 is insulated. The mixed boundary value problem describes the system's state equation as where θ = T − T 3 , θ ∞ = T ∞ − T 3 when T is temperature in the domain, T ∞ is the ambient temperature, n is the outward unit normal vector, h is the convection coefficient, and k is the conduction coefficient.
Since the classical solution θ ∈ C 1 ð ΩÞ ∩ C 2 ðΩÞ to the problem does not exist if x = ξ for equation (19), then ξ is a singular point [1,25,30], where Ω is closure of the domain Ω; we can be concerned with the variational solution H 1 ðΩÞ.

Boundary Integral Equation for the Classical Solution.
The boundary integral equation formulation for the heat transfer problem is based on Green's formula with the fundamental solution [20,28,31]. The simplest method for transforming variables to boundary variables is to use Green's second identity [1,25,32].
Let u and v be C 1 ð ΩÞ ∩ C 2 ðΩÞ function; then, Green's formula of equation (22) holds. If the classical solution θ ∈ C 1 ð ΩÞ ∩ C 2 ðΩÞ exists, we can substitute u by θ in equation (22). However, the singularity of E in equation (19) is preventing one from substituting v by E in equation (22). One way of overcoming the difficulty is to replace Ω by Ω − B ρ ð ξÞ where B ρ ðξÞ is a circle with the small radius ρ centered at a singular point ξ.
One can conclude from equation (22) that for where Ω − B ρ is the boundary of B ρ in equation (34). Since The first term in the integral over ∂B ρ in equation (34) becomes

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Since The second term in integration over ∂B ρ in equation (56) becomes where ω 2 = 2π is the boundary length of the unit circle in ℝ 2 and ω 2 ρ is the boundary of the circle with the radius ρ. If one chooses C = 1/ω 2 and substitutes equations (35), (36), (37), and (39) in equation (34), then holds as ρ goes to zero. If ξ is on Γ, equation (40) has a singularity. Then, we can divide the boundary Γ by Γ ε and Γ − Γ ε where Γ ε is half circle with small radius ε centered at a singular ξ. Then, equation (40) becomes The first term of the boundary integration over Γ ε in equation (41) becomes The second term becomes Since Assume C = 1/ω 2 ; then, equation (42) becomes By substituting equation (42) and equation (45) into equation (41) and let ε go to zero, then we obtain When we use a boundary element method for the problem with ξ ∈ Γ, θðξÞis obtained numerically from equation (46), while it is obtained from equation (40) when ξ ∈ Ω. By dividing the boundary into small segments, the classical solution, if it exists, can be approximated numerically using boundary integral equations (40) and (46) as illustrated above. However, in the mixed boundary value problem, the classical solution does not exist when x and ξ are at the same point; then, it has a singularity and ξ is singularity of the fundamental solution. Therefore, we cannot use equations (40) and (46) directly.

Boundary Integral Expression for the Weak Solution. The state equation of equations (29)-(32) is written in a variational form as
where K is the admissible set given by K = fv/v ∈ H 1 ðΩÞ, v = 0 on Γ 3 g. The weak solution of equation (47) is unique in H 1 ðΩÞ by using equations (27) and (28) and applying the Lax-Milgram theorem. For every u ∈ H 1 0 ðΩÞ, there exists a unique solution θ ∈ H 1 0 ðΩÞ. By using the Cauchy-Schwarz inequality, let us check the continuity of B½θ, v: ð On the boundary ᴦ 1 ∪ ᴦ 2 by the Cauchy-Schwarz inequality, ð Then, from equations (47) and (48), we have continuity The following is the bilinear form of B½θ, v:

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Poincare's inequality indicates that Then, we have Therefore, the condition of the Lax-Milgram theorem is satisfied, and there exists a unique weak solution on θ ∈ H 1 0 ðΩÞ [2,8,9,14].
To represent the boundary integral equation for the variational weak solution θ ∈ H 1 ðΩÞ, then we need the following theorem [8].
Theorem 6 (see [8,21]. Green's formula in the Sobolev space holds for the domain Ω with the Lipschitz boundary Γ if u, v ∈ H 1 ðΩ, ΔÞ whereH 1 ðΩ, ΔÞ = fu/u ∈ H 1 ðΩÞ such that Δ ∈ L 2 ðΩÞg. The variational solution θ is in H 1 ðΩ, ΔÞ, but the fundamental solution is not. In fact, it is in C ∞ ðℝ 2 − fξgÞ [8]. Then, u in equation (49) can be substituted by θ but v cannot by E. This difficulty is removed by replacing Ω by Ω − B ρ , since E is the H 1 ðΩ, ΔÞ in Ω − B ρ . Then, we can conclude from equation (54) that The left-hand side term of integration over Ω − B ρ is zero, and the first term in the integration over ∂B ρ of equation (55) becomes The second term in the integration over ∂B ρ of equation (55) becomes Then, by substituting equations (56) and (57) If we insert the boundary condition of equations (30)-(32) into equations (58) and (59), respectively, we can get The solution of equations (60) and (61) in H 1 ðΩÞ is equal to the variational solution, because it is unique in H 1 ðΩÞ. The solution of the problem in equations (60) and (61) can be approximated numerically by dividing the border into small parts, as shown by the previous results.

Conclusion
In this study, we present a two-dimensional heat transfer problem utilizing a boundary integral equation with specific initial and boundary conditions, and we discuss how a variational solution to a mixed boundary value problem can be obtained even though a classical solution does not exist. Also, we have transformed the boundary value problem for the steady-state heat transfer problem into boundary integral equation and the solution of boundary integral equation for the mixed boundary value problem by using Green's identity and fundamental solution. The boundary integral equation for the problem guided by the Laplace operator has a unique solution that is similar to the variational solution in H 1 ðΩÞ. As a result, a numerical approximation of the variational solution for the boundary integral problem can be obtained. Furthermore, the approach used in this study can be used for three-dimensional heat transfer problems as well as other elliptic problems.

Data Availability
No data were used to support the study. 6 Advances in Mathematical Physics