Numerical solution of ordinary differential equations using Newton-Raphson method with Numpy and Autograd: Accuracy, convergence, and performance analysis

Abstract

The numerical solution of ordinary differential equations (ODEs) using the Newton-Raphson approach is investigated in this work. The aim is to evaluate, in solving first- and second-order ODEs, the accuracy, convergence, and limits of this approach. This Python-based approach uses Autograd for automatic differentiation and NumPy for effective array operations. Several case studies are analyzed, including various ODE challenges. The efficiency of the approach is assessed by comparing numerical findings with analytical solutions. In many situations, the Newton-Raphson method effectively and highly precisely approximates solutions for different ODEs. Some examples, however, show differences between numerical and analytical answers, suggesting possible problems with error accumulation or inherent constraints of the approach. Problem difficulty, step size, and initial guesses all affect convergence. Although the Newton-Raphson approach solves ODEs numerically quite well, it must be carefully validated against analytical solutions. The performance of the procedure depends on elements particular to the problem that must be taken into account in application. The need for choosing suitable numerical methods for solving ODEs in scientific and technical domains is underlined by this work. The results guide future research and useful implementations by offering an understanding of the strengths and constraints of Newton-Raphson-based solvers.