Improved Solution Methods for Initial Value Problems of Ordinary Differential Equations Using Advanced Optimization Techniques

Qasim Abd Ali Tayyeh^1,*

¹Department of Mechanical Techniques,Al-Nasiriya Technical Institute, Southern Technical University,Thi-Qar, Al-Nasiriya 64001, Iraq

Email: qassim.tayih@stu.edu.iq

Abstract

Initial value problems (IVPs) of ordinary differential equations (ODEs) are ubiquitous in science and engineering applications, and the classical fourth-order Runge–Kutta (RK4) method is by far the most popular solver due to its good accuracy-to-cost ratio. Among all four-stage fourth-order explicit RK methods there are two free node parameters left after satisfying the eight B-series order conditions, thus allowing further systematic enhancements. Here we employ a hybrid multi-seed Particle Swarm Optimization (PSO)-Nelder–Mead algorithm to search for optimal RK node parameters (c₂, c₃) with respect to a minimax normalized objective over eight commonly used nonlinear benchmark problems. The resulting PSO-RK4 method with (c₂ = 0.323665, c₃ = 0.653527) retains both the exact same order of convergence and absolute stability region as the classical RK method called the 3/8-rule, but exhibits reduced maximum global error on each of the eight benchmarks when N = 300; average improvement of 27.9% with gains up to 46.0% on the Bernoulli equation, 29.7% on logistic growth, and 21.3% on exponential. Robustness of these gains with respect to multi-step-size (N = 100, 200, 300, 500) is demonstrated.

Keywords: Initial value problems; Runge–Kutta methods; Particle Swarm Optimization; Nelder–Mead simplex; hybrid metaheuristic optimization; Butcher tableau; Absolute stability; Nonlinear ODEs