Abstract
In this article, we deal with an inverse problem concerning the two-dimensional Laplace equation with local boundary conditions on a bounded region. In this problem, the goal is to reduce it into a system of Fredholm integral equations of the second kind involving kernels with weakly/or no singularities (Fredholm property) by considering some additional conditions on the parameters of the problem. Finally, the method is carried out on an example, to show the simplicity and efficiency of the method.
1. Introduction
The inverse problems arise in modeling of many physical and geophysical phenomena, such as elastography and medical imaging, seismology, potential theory, ion transport problems or chromatography, and finances (see, for example, [1–4]). In many cases of these problems which have been studied for the first time by Lavrentiev [5], the goal is to determine the coefficients or the right hand side of the differential equations for some known data about its solutions. There are many numerical methods to solve these problems. Among them, we cite the method of fundamental solutions (MFS) by Marin and Lesnic [6] and Wang et al. [7] and by Chen et al. [8] and Sun and He [9] for the two-dimensional and three-dimensional inverse problems, respectively, the boundary function method (BFM) by Wang et al. [10], the boundary particle method (BPM) by Chen and Fu [11], the variational iteration method (VIM) by Canon and Tatari [12], the globally convergent numerical method by Baysal [13], the weighted homotopy analysis method (WHAM) by Shidfar et al. [14], and the conjugate gradient method (CGM) by Lu et al. [15]. In some special cases, these problems are solved by the analytical methods by Liua and Tatar [16] or Liu [17].
In 1997, Aliev and Jahanshahi [18] proposed a new approach for reducing the direct problem concerning the mixed PDE with nonlocal boundary conditions into a system of Fredholm integral equations with weak singularities, i.e., the boundary values of the unknown function and their derivatives satisfy some of the Fredholm integral equations with weak singularities. The system obtained can be solved by some of the numerical methods, such as semiorthogonal B-spline wavelet collocation method by Sahau and Saha Ray [19], the method based on Bernstein polynomial by Basit and Khan [20], triangular functions method by Almasieh and Roodaki [21], homotopy perturbation method by Javidi and Golbabai [22], and Taylor-series expansion method by Maleknejad et al. [23].
In this article, based on the method presented in [18], we study an inverse boundary value problem for a two-dimensional Laplace equation which is a logical continuation of the paper [24] concerning the inverse problem of Cauchy-Riemann equation with nonlocal boundary conditions.
In the bounded domain (see Figure 1) where its boundary is given as where are known functions satisfying we consider an inverse boundary value problem of the two-dimensional Laplace equation subject to the boundary conditions where and are known functions on . The interest is then to recover the unknown function based on a Fredholm integral equation of the second kind with respect to boundary values of the unknown function and its first-order partial derivatives.
2. Main Results
Our first main result is stated as follows:
Theorem 1. The solution of Equation (3) at internal points of the domain can be expressed by Green’s formula: where is the exterior normal to the boundary and is the fundamental solution of 2D Laplace operator satisfying which is given by where is the Dirac measure at point [25]. Moreover, the boundary values and should satisfy the following boundary integral equations: where is the angle between the exterior normal vector to the boundary and coordinate axes .
Proof. The first-order partial derivatives of Equation (8) are as follows: Multiplying both sides of Equation (3) by (8) and (12), respectively, integrating over , using the divergence theorem [25] (similar to [24]), the proof is completed.
Let and be the angles between exterior normal and unit tangent vectors on the boundaries with coordinate axes , respectively. Then
Taking into account arcs of the boundary as and , plugging (12)–(14) in (9)–(11), we obtain the following boundary integral equations: where
As we see, the kernels in (20) contain singularities. To remove them, we state the following theorem.
Theorem 2. Let be a bounded domain with the boundary as Lyapunov curve in Cartesian coordinates which is convex in direction and assume that (for ), and (for all ). Then the singularities in kernels are regularized. Moreover, the unknown function in (5) can be expressed as the following boundary integral equation with no singularities where
Proof. Applying the mean value theorem twice on , we get the kernels with no singularities, where and are between and , respectively.
From the boundary condition (4), we get
Taking into account that , for all , plugging the boundary condition (5) and relations (27) in Equation (15) (for ) and considering (20) and (26) with and , respectively, the integral equation (21) can be resulted.
Similar to Equation (26), we obtain where is between and .
For regularization and removing singularities in , we construct the following two linear combinations with unknown coefficients from the boundary integral equations (16)–(17): where .
The notation in the right-hand side of (29)–(30) stands for all integrals without singularities.
Plugging and in (29)–(30), respectively, and considering boundary conditions (4), we get
Plugging (27) in (31)–(32), respectively, we obtain where
Finally, plugging the kernels (20) (for ), (26), and (28) in kernels (37)–(41), we get
According to what explained above, the following theorems can be resulted:
Theorem 3. Under the assumptions of Theorem 2, and also the unknown function in (5) can be recovered as a system of boundary integral equations (21) and (33)–(34) subject to kernels (25) and (45)–(49), respectively, with no singularities.
Theorem 4. Under the assumptions of Theorem 3, the main problem (3)–(5) is reduced to a system of boundary integral equations, i.e., the unknown function can be written as the integral equation Equation (6) where the existing boundary values can be expressed as a system of boundary integral Equations (21), (33)–(34) and relations (5) and (27) with no singularities.
The application of the method described will be illustrated in the following example.
Example 1. Consider the Laplace equation subject to the following boundary conditions where the boundary of the elliptic domain has the following polar equation (see Example 1 in [26]) or in the Cartesian coordinates, and the boundary has the following form
Moreover, the functions and in (53) are known.
The exact solution of (51) and also the unknown boundary value in (53) are given by
Comparing the boundary conditions (53) with (4)–(5) yields that
As we see, the domain as well as and satisfies the hypotheses of Theorem 4. Therefore, from (25) and (45)–(49), we obtain
Plugging the kernels (66) in (33)–(34), a system of second kind Fredholm integral equations with respect to is obtained which can be solved by some of numerical methods.
In addition, after substituting the kernels (62) as well as into the integral equation (21), the unknown function can be resulted.
Finally, plugging the boundary values in (27) as well as in (6), the unknown function in (51) is obtained.
3. Conclusion
In this paper, we studied an inverse boundary value problem including Laplace equation with nonlocal boundary conditions in a two-dimensional convex bounded domain with the boundary as Lyapunov curve. By considering some of the extra conditions, the problem is reduced to a system of second kind Fredholm integral equations with no singularities. This method can be used to any inverse boundary value problem where its fundamental solution is known.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All authors read and approved the final form of the manuscript. Also, they carried out the proofs and conceived of the study.