Study and Comparison of Finite Difference Methods for the Numerical Solution of Partial Differential Equations

This paper presents a comprehensive study of Partial Differential Equations (PDEs), beginning with a general definition and classification into three primary types: parabolic, hyperbolic, and elliptic equations. We focus on the development of finite difference schemes for numerically solving specific PDEs and Ordinary Differential Equations (ODEs), demonstrating their effectiveness through selected examples implemented in Matlab. A general study of some finite difference schemes for the numerical resolution of specific PDEs and ODEs has been conducted. Well-selected numerical examples have been proposed, and numerical implementation of different examples has been carried out using MATLAB. Our study concluded that the numerical methods of finite difference schemes for numerical accuracy of definite PDEs and ODEs have had good effectiveness and this has been verified by the numerical examples that have been implemented using MATLAB software.