In this manuscript, we consider in detail numerical approach of solving Laplace’s equation in 2-dimensional region with given boundary values which is based on the Alternating Direction Implicit Method (ADI). This method was constructed using Taylor’s series expansion on the second order Laplace equation leading to a linear algebraic system. Solving the algebraic system, leads to the unknown coefficients of the basis function. The techniques of handling practical problems are considered in detail. The results obtained compared favorably with the results obtained from the Finite difference method constructed by Dhumal and Kiwne and the exact solution. Thus the ADI method can as well be used for the numerical solution of steady-state Laplace equations.

中文翻译：

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二阶二维Laplace方程数值的交替方向隐式求解

在本手稿中，我们详细考虑了基于交替方向隐式方法（ADI）的具有给定边界值的二维区域中拉普拉斯方程求解的数值方法。使用泰勒级数展开法对二阶Laplace方程建立线性代数系统。求解代数系统，导致基函数的未知系数。详细考虑了处理实际问题的技术。获得的结果与通过Dhumal和Kiwne构建的有限差分法获得的结果以及精确解相比具有优势。因此，ADI方法也可以用于稳态拉普拉斯方程的数值解。