Newton's Method Applied to Nonlinear Boundary Value Problems: A Numerical Approach

Abstract

This work investigates the application of the Newton's method for the numerical solution of a nonlinear boundary value problem formulated through an ordinary differential equation (ODE). Nonlinear ODEs arise in various mathematical modeling contexts, where an exact solution is often unfeasible due to the intrinsic complexity of these equations. Thus, a numerical approach is employed, using Newton's method to solve the system resulting from the discretization of the original problem. The procedure involves the iterative formulation of the method, which enables the approximation of solutions and the evaluation of convergence with respect to the problem parameters. The results demonstrate that Newton's method provides a robust and efficient solution, highlighting its applicability to complex boundary value problems and reinforcing its relevance for the numerical analysis of nonlinear systems. It is concluded that the methodology discussed is suitable for solving a wide range of boundary value problems, ensuring precision and stability in the results.

Figures (2)

• Figure 1: Failure of the Newton Method due to a poor initial point kharab2011introduction.
• Figure 2: Plot of the vector $\overline{\mathbf{w}}$ over the discretized interval.