Optimizing ODE solutions: application of Nelder-Mead algorithm for solving mixed boundary value problems

Abstract

This abstract introduces a novel application of the Nelder-Mead algorithm in optimizing solutions to Ordinary Differential Equations (ODEs) with derivative boundary conditions. The study presents a refined methodology that leverages the Nelder-Mead algorithm’s adaptability to tackle ODEs characterized by intricate derivative constraints. With the aim of enhancing the toolkit for solving complex ODEs, the objective of this research is to showcase the algorithm’s efficacy in optimizing solutions under derivative boundary conditions. The methodology involves adapting the Nelder-Mead algorithm to navigate the param-eter space while satisfying both the ODEs and their derivative constraints. Experimental results demonstrate the algorithm’s capability to identify solutions that meet these stringent requirements, marking a significant advancement in addressing ODEs with derivative boundary conditions. The study concludes by emphasizing the algorithm’s potential for advancing ODE-solving techniques, particularly in scenarios where gradient-based methods struggle, thus widening the scope of applications across various scientific and engineering domains.