An Analysis of Double Laplace Equations on a Concave Domain

In the investigation, the complex geometric domain is a concave geometrical pattern. Due to the symmetric character, the left side of the geometric pattern, i.e. the L-shaped region is calculated in the study. The governing equation is expressed with Laplace equations. And the analysis is solved by eigenfunction expansion and point-match method. Besides, visual C helps obtain the results of numerical calculation. The local values and the mean values of the function are also discussed in this study.


Introduction
The Laplace equations show an important role in the applied mathematical researches and analysis. Some significant efforts, thus, have been directed towards researches into related fields. For example, Alliney [1] presented the two-dimensional potential flows to solve the Laplace's equation with appropriate regularity conditions at infinity. The problem is reduced to a finite domain by representing conditions at infinity by means of a boundary integral equation.
And Rangogni [2] presented the numerical solution of the generalized Laplace equation by coupling the boundary element method and the perturbation method. Besides, Zanger [3] presented the analysis of the boundary element method applied to Laplace's equation for the experiments involving solving the two-dimensional Laplace problem exterior to a circle and square, using both the direct and indirect methods. Furthermore, Bailey et al. [4] presented the generate grid points in two-dimensional simply connected spatial domains. As in many grid generation techniques, the solution of Laplace equation is involved.
lyzed the problem of Laplace equation with over-specified boundary conditions.
The results show that the unknown boundary potential can be reconstructed, and that both high wave-number content and divergent results can be avoided by using the proposed regularization technique. In addition, L. Gavete et al. [7] compared the GFD method with the element free Galerkin method (EFG). The EFG method with linear approximation and penalty functions to treat the essential boundary condition is used in his paper. Both methods are compared for solving Laplace equation. Nyambuya [8]  Although many researches about Laplace equation under different conditions have been discussed, the Laplace equation with concave domains is also worth discussing. The present paper, thus, will analyze a symmetric domain with complex Laplace equations under two kinds of boundary conditions in order to find local values and the mean values of the function. The analysis of two kinds of boundary conditions, case 1 and case 2, will be specified in the following mathematical formulation. Furthermore, in the present paper, visualization and image processing are obtained from mathematical formulation of the complex Laplace equations on a concave domain. It is hoped that the results can be further applied in engineering and technology, for example, the problem of fluid flow and heat conduction.

Mathematical Formulation
The geometric domain in Figure 1   The governing equation for the region is expressed with Laplace equation: The L-shape region is composed of two rectangles; the governing equation for the left one is: The boundary conditions for Equation (2) are: And the governing equation for the right rectangle is: H.-P. Hu World Journal of Engineering and Technology The boundary conditions for Equation (4) are: With the boundary conditions Equations (3) and (5), the analytical solution to , f x y , and to Equation (6) where the eigenvalues are: Other boundary conditions for governing equations are: We choose N points along the boundary at x = d, i y ih N = and truncate A n to N terms and B m to M terms, where Substitute the boundary condition into Equation (6), and the following equation can be obtained: along the common boundary conditions [9]. The conditions can be expressed as: Substitute the boundary condition into Equations (6)- (7) and can obtain the following equations: Integrating Equation (15) can obtain the following equation: The analysis of boundary conditions on Case 2: The geometric domain is also a concave domain. The outer dimension of the domain is 2w h × , and there is a concave, whose dimension is ( ) The boundary condition of the bottom is 0, and the condition of the top is 1. The L-shape region, in Figure 1(c) is composed of two rectangles.
The governing equation for the left rectangle is: The boundary conditions for Equation (17) are: And the governing equation for the right rectangle is: The boundary conditions for Equation (19) are: With the boundary conditions Equations (18)

Numerical Methods
The following steps of numerical methods are estimated by using Visual C ++ : 1) Give the constants h, b, w and d.
2) Set N = 29, 3) Equations (10), (13) and (14) are expressed as the linear system of (N + M) equations to solve coefficients n A and m B . 4) Substitute coefficients n A and m B into equations f 1 , (Equation (6)) and f 2 (Equation (7)). This process is repeated at all nodes within the range, i.e.

Results and Discussion
Results and discussion of Case 1: Figure 2(a) and Figure 2(b) show the contour plot for the domain.

Conclusion
The present paper can find the influence of the height and the width of the geo-