Analytical Solution of Laplace's Equation via Separation of Variables

Abstract: Laplace's equation is a pivotal partial differential equation (PDE) in mathematical physics, with applications spanning electrostatics, steady-state heat transfer, fluid mechanics, and gravitational fields. It models systems in equilibrium, where quantities like temperature or electric potential remain constant over time in the absence of sources. In two dimensions, Laplace's equation is written as: where denotes the potential at coordinates . In three dimensions, it includes an additional term for the -coordinate: To solve this equation analytically, appropriate boundary conditions must be defined, such as Dirichlet (prescribed values) or Neumann (prescribed derivatives) conditions. Among the most effective methods for tackling Laplace's equation in regular geometries is separation of variables, which transforms the PDE into simpler ordinary differential equations (ODEs). This article delves into the separation of variables technique, focusing on its application to a two-dimensional Laplace's equation in a rectangular domain, with a detailed derivation, key assumptions, and a practical example. Keywords: Laplace's equation, partial differential equation (PDE), mathematical physics. Title: Analytical Solution of Laplace's Equation via Separation of Variables Author: Ahmad M. D. Al-Eybani International Journal of Mathematics and Physical Sciences Research ISSN 2348-5736 (Online) Vol. 13, Issue 1, April 2025 - September 2025 Page No: 50-54 Research Publish Journals Website: www.researchpublish.com Published Date: 08-May-2025 DOI: https://doi.org/10.5281/zenodo.15366723 Paper Download Link (Source) https://www.researchpublish.com/papers/analytical-solution-of-laplaces-equation-via-separation-of-variables