حل بعض المعادلات الإهليجية باستعمال طريقة الفروق المنتهية وطريقة العنصر المنتهي ومقارنتهما

Abstract

Abstract The majority of physical and engineering phenomena are described by partial differential equations. Solving these equations is essential for understanding such problems; however, obtaining their exact analytical solution is frequently challenging. Consequently, it becomes necessary to utilize numerical methods for approximating these equations. In this study, we focused on linear second-order elliptic partial differential equations, specifically employing two-dimensional Poisson's and Laplace's equations as representative models for the aforementioned phenomena. We successfully derived the analytical solutions for the equations under investigation: using the separation of variables method for homogeneous elliptic PDEs and the eigenfunction method for inhomogeneous ones. For the inhomogeneous equations, a comparative analysis was conducted between the analytical solution and the numerical solutions generated by applying both the Finite Difference Method and the Finite Element Method. This comparison was presented through numerical tables and graphical plots. It is worth noting that although some of these equations were found in existing literature and solved using similar or alternative techniques, their presentation often lacked detail and clarity for the reader. Our approach therefore involved re-formulating and presenting these solutions in a detailed, simplified manner to clearly illustrate each method's mechanism and enhance reader comprehension. Our findings from the examined examples indicate that both the Finite Differences Method and the Finite Element Method exhibit convergence towards the analytical solution when applied to a two-dimensional homogeneous elliptic (Laplace) equation defined on a regular domain (such as a rectangle or triangle) and subject to homogeneous and inhomogeneous Dirichlet boundary conditions. Conversely, when addressing a two-dimensional inhomogeneous elliptic (Poisson) equation under homogeneous and inhomogeneous Dirichlet boundary conditions, the Finite Element Method generally yields a superior approximate solution compared to the Finite Difference

Method, resulting in lower errors, particularly when the domain of the inhomogeneous elliptic (Poisson) equation possesses a regular geometric form (rectangle, triangle, etc.). Keywords: Finite Difference Method, Finite Element Method, Two-Dimensional Poisson's Equation, Two-Dimensional Laplace's Equation