Optimal Solution of Laplace's Equation Using Finite Differences

Ahmed T.  Ahmed

Ahmed T. Ahmed Department of Mathematics, College of Basic Education, Shirqat, University of Tirkit, Iraq

Keywords: Laplace's Equation, Finite Difference Methods, Five-point Method, Gaussian Method, Numerical Approximation

Abstract

In this study, we address the numerical approximation of Laplace's equation, a fundamental partial differential equation in physics and engineering, using finite difference methods. Specifically, we explore the application of the standard five-point and diagonal five-point methods to solve the equation under given boundary conditions. A grid network is established in the first quadrant of the coordinate plane, with values at grid points determined by either the standard five-point, diagonal five-point, or a combined approach. The resulting system of equations is formulated as a matrix and solved using the Gaussian method to obtain the values at each grid node. Our findings demonstrate the effectiveness of these methods in accurately approximating solutions to Laplace's equation, with potential implications for improving computational techniques in related fields.

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