Finite Difference Methods for Solving Linear and Nonlinear Ellipse Equations

The study aims to determine the numerical solution of partial differential equations of the ellipse type, both linear and non-linear. One of the most popular methods for solving partial differential equations is the finite differences method, which provides accurate approximations in comparison to the actual solution of the problem. The study provided clarification by utilizing the Laplace equation's numerical and analytical solution as one example of a partial differential equation of the linear ellipse kind. Regarding the study of solving partial differential equations of the type of nonlinear ellipse that is difficult to solve theoretically, we were satisfied with finding the formula only and programming it to obtain the required results. We then used the MATLAB system to find the solution to these equations. The results showed that the numerical solution is superior to the analytical solution because it can solve complex problems numerically, which is what the analytical solution cannot master accurately.