Visualization of Complex Laplace’s Equations on a Hollow Rectangle

Abstract In the investigation, the complex geometry domain is a hollow rectangle. The governing equations are expressed with complex Laplace’s equations. And the analysis is solved by point-matching method. Besides, the results of numerical calculation are obtained by using Visual C++. In the present paper, visualization and image processing obtained from mathematical formulation of the complex Laplace’s equations on a hollow rectangle. Then, the local values and the mean values of the function are also discussed in the article. We hope the results can further apply in the problem of fluid flow and heat conduction.


Introduction
The partial differential equations show an important role in the mathematical researches and analysis. Some significant efforts, thus, have been directed towards researches into related fields. For example, about the Laplace equation, Bungartz et al. [1] presented a simple model problem, the Laplace equation on the unit square with a Dirichlet boundary function. They present a proof of convergence for the so-called combination technique, a modern, efficient, and easily parallelizable sparse grid solver for elliptic partial differential equations. And Abouchabaka, et al. [2] presented the numerical approach of the free boundary by using shape optimization method. The numerical simulation is possible with the Laplace-Poisson model, which introduces two regions.
Besides, John G. Fikioris [3] used the method of Watson's transformation to apply to a two-dimensional, orthogonal eigenfunction series of rectangular harmonic functions, and he provided the solution to a typical boundary value problem of Laplace's equation. Furthermore, Wang [4] solved the diffusion across a corrugated sawtooth plate with the Laplace equation. The transport properties and the theoretical increase in total flux due to corrugations were discussed. Recently, Antoine et al. [5] described a Monte Carlo method for the numerical computation of the principal eigenvalue of the Laplace operator in a bounded domain with Dirichlet conditions. Various tools of statistical estimation and different simulation schemes are developed to optimize the method. In the other related research, Reutskiy [6] presented the meshless radial basis function method for 2D steady-state heat conduction problems in anisotropic and inhomogeneous media. Next, Gal and Gal [7] solved the heat and Laplacetype equations with complex spatial variables in weighted bergman spaces and studied the classical heat and Laplace equations with real-time variable and complex spatial The L-shape region is composed of two rectangles; the governing equation for the left is one: The boundary conditions for Equation (2) are: And the governing equation for the right rectangle is: The boundary conditions for Equation (4) are: With the boundary conditions Equations (3) and (5), the analytical solution to Equation (2) is f 1 (x, y), and to Equation (6) is f 2 (x, y). They are as follows: where the eigenvalues are: variable by the semigroup theory methods. Chang et al. [8] present a review of the current computational methods and applications of inverse heat conduction problems in different fields. In the present paper, there are two major solving categories of the issue: mesh methods and meshless algorithms of their strengths and weaknesses are also discussed in the study. Although many researches about Laplace equation under different conditions had been discussed, the Laplace equation with complex 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.

Mathematical formulation
The geometry domain in Figure 1a is a rectangle. The outer dimension of the rectangle is 2w × 2 h, and there is a hollow part in the center, whose dimension is 2(w -d) × 2b.

The analysis of boundary conditions on Case 1
The boundary condition of the outer part is 0, and the condition of the hollow part is 1. The local function distribution in the outer solid part of the rectangle is f(x, y).
Decompose the rectangle into four parts. Due to the symmetric character, only the left-bottom quarter of the geometry, i.e. the L-shaped region in Figure 1b, needs calculating.
The governing equation for the region is expressed with Laplace equation:  Other boundary conditions for governing equations are: Substitute the boundary condition into Equation (6), and the following equation can be obtained: Next, the solutions to the two regions of the L-shape domain can be matched along the common boundary conditions [9]. The conditions can be expressed as: Substituting the boundary condition into Equations (6) and (7) can obtain the following equations: The mean value for f(x, y) is expressed as: Integrating Equation (15) can obtain the following equation:

The analysis of boundary conditions on Case 2
The geometry domain is also a rectangle. The outer dimension of the rectangle is 2w × 2 h, and there is a hollow part in the center, whose dimension is 2(w -d) × 2b. The boundary condition of the outer part is 1, and the condition of the hollow part is 0. The L-shape region is composed of two rectangles; the governing equation for the left rectangle is: The boundary conditions for Equation (17)      The mean value for f(x, y) is expressed as: Integrating Equation (15) can obtain the following equation:

Numerical methods
The following steps of numerical methods are estimated by using Visual C ++ : (1) Give the constants h, b, w and d.

Results and discussion
Case 1: Figures 2a and 2b show the contour plot for the domain. Figure 2a is a rectangle (w < h), and Figure 2b  (30)

Conclusions
The following conclusions may be drawn from the results of this study: