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.

1. 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 [69]. 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 unknown to discretize the boundary , where 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 [1416].

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, 1921].

Using a boundary integral expression for a two-dimensional heat transfer problem, we obtain a unique weak solution and a variational solution in the Sobolev space of order one, [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.

2. Preliminaries

2.1. Laplace Equation in Two Dimensions

Let open and . The Laplace equation for is

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 and . This means that the Laplace equation described steady state situated on the temperature distribution.

The steady-state solution satisfies and boundary condition, is prescribed on , and then, we consider the domain that are circular [23].

2.2. Sobolev Space

Definition 1 (see [8]). Let and , and let be a nonempty open set. The Sobolev space order based on is defined by

Remark 2. is viewed as a distribution on , so the condition means that there exists a function such that , such that a function is defined as a weak derivatives of .

The complement of implies that becomes a Banach space on putting the norm as

For , 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 is the set of all such that all the first partial derivative belongs to . The inner product in is where denotes . This inner product clearly gives the norm Then, we denote the inner product by a subscript zero. Then, equation (6) reads where is an abbreviation for . In particular,

Then, the Cauchy sequence in converges to the element of . In other words, is a Hilbert space. It is in fact the Hilbert space obtained by completing the set of smooth function with respect to the , in the same way that is the Hilbert space obtained by completing the set of smooth functions with respect to the norm [24].

2.3. Weak Solution [1, 15]

Consider a partial differential operator of order in variables where is a multi-index and are functions in .

Considering a differential equation in the sense of distribution, then the following is true.

Let ; then, in .

This implies where ; here, the operation is the adjoint operator of .

If the original problem was to find -times differentiable function defined on the open set such that for all , called the classical solution, then an integrable function is said to be a weak solution if

2.4. Fundamental Solutions

Definition 4 (see [25]). A distribution is a fundamental solution of the differential operator if and only if

The fundamental solution of the differential operator satisfies the equation; however, 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 be Green’s function ; it satisfies the equation

Physically, Green’s function represents the effect at the point of a Dirac delta function source at the point [20].

Multiply equation (14) by and integrate over the area of the circle so that

Then, we have

Since , we have

The fundamental solution for the Laplace operator is as follows.

Definition 5 (see [26]). Let such that with being the Dirac delta function. In general dimension, the (distributional space in ) is a solution of equation (18) which is called a fundamental solution of Laplace’s equation at . In the context of the heat equation, the fundamental solution of the Laplace equation is crucial to the heat kernel. In two dimensions, the fundamental radial solution of the Laplace equation is where is the arbitrary constant and is the distance from 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].

2.5. Green’s Second [16, 26]

Let and Green’s first identity for the pair and ; then, and again for the pair and ,

By subtracting equation (21) from equation (20), we get Green’s second identity [23].

It is valid for the pair of functions and .

The above integral is a line integral over the boundary curve of two-dimensional region , and denotes the arc length of the boundary [16, 26].

2.6. 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 with piecewise smooth Lipschitz boundary , Green’s representation theorem allows us to write the solution as where is the unit outward pointing normal at and is a fundamental solution defined as

Hence, in principle, if either or is known on , we can recover the unknown quantity by restricting equation (24) to the boundary and solving to the unknown boundary (see, e.g., [6, 16]).

2.7. 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 of the Laplace equation in [15, 29].

The problem is written in a weak form as follows: (1)Multiply on both sides of the Laplace equation by a function in and integrate over Ω(2)Apply integration by parts to arrive at

3. Statement of the Steady-State Heat Transfer Problem

Consider a heat-conducting body that is homogeneous and isotropic; is a simple connected and bounded domain in with a Lipschitz boundary when and are disjoint parts of . Convection in the ambient medium is thought to occur at the boundary and , temperature is kept constant at the prescribed value on , and is insulated. The mixed boundary value problem describes the system’s state equation as where , when is temperature in the domain, is the ambient temperature, is the outward unit normal vector, is the convection coefficient, and is the conduction coefficient.

Since the classical solution to the problem does not exist if for equation (19), then is a singular point [1, 25, 30], where is closure of the domain ; we can be concerned with the variational solution .

3.1. 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 and be function; then, Green’s formula of equation (22) holds. If the classical solution exists, we can substitute by in equation (22). However, the singularity of in equation (19) is preventing one from substituting by in equation (22). One way of overcoming the difficulty is to replace by where is a circle with the small radius centered at a singular point .

One can conclude from equation (22) that for where is the boundary of in equation (34). Since and on , we have

The first term in the integral over in equation (34) becomes


The second term in integration over in equation (56) becomes where is the boundary length of the unit circle in and is the boundary of the circle with the radius . If one chooses and substitutes equations (35), (36), (37), and (39) in equation (34), then holds as goes to zero. If is on , equation (40) has a singularity. Then, we can divide the boundary by and where is half circle with small radius centered at a singular . Then, equation (40) becomes

The first term of the boundary integration over in equation (41) becomes

The second term becomes


Assume ; then, equation (42) becomes By substituting equation (42) and equation (45) into equation (41) and let go to zero, then we obtain

When we use a boundary element method for the problem with , is obtained numerically from equation (46), while it is obtained from equation (40) when . 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 and 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.

3.2. Boundary Integral Expression for the Weak Solution

The state equation of equations (29)–(32) is written in a variational form as where is the admissible set given by . The weak solution of equation (47) is unique in by using equations (27) and (28) and applying the Lax-Milgram theorem. For every , there exists a unique solution .

By using the Cauchy-Schwarz inequality, let us check the continuity of :

On the boundary by the Cauchy-Schwarz inequality,

Then, from equations (47) and (48), we have continuity

The following is the bilinear form of :

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 [2, 8, 9, 14].

To represent the boundary integral equation for the variational weak solution , then we need the following theorem [8].

Theorem 6 (see [8, 21]. Green’s formula in the Sobolev space holds for the domain with the Lipschitz boundary if where.

The variational solution is in , but the fundamental solution is not. In fact, it is in [8]. Then, in equation (49) can be substituted by but cannot by . This difficulty is removed by replacing by , since is the in . Then, we can conclude from equation (54) that

The left-hand side term of integration over is zero, and the first term in the integration over of equation (55) becomes

The second term in the integration over of equation (55) becomes

Then, by substituting equations (56) and (57) in equation (54), we obtain as goes to zero.

With the similar way stated in Section 3.1 in equations (41), (42), (43), (44), and (45), if is on , equation (55) becomes

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 is equal to the variational solution, because it is unique in . 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.

4. 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 . 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.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the preparation of this research paper.


The author would like to thank Mizan Tepi University, College of Natural Sciences, Department of Mathematics, for providing the necessary resources during conducting this research and mathematics department staff members.