An effective wavelet neural network approach for solving first and second order ordinary differential equations

The development of efficient numerical methods for obtaining numerical solutions of first and second order ordinary differential equations (ODEs) is of paramount importance, given the widespread utilization of ODEs as a means of characterizing the behavior in various scientific and engineering disciplines. While various artificial neural networks (ANNs) approaches have recently emerged as potential solutions for approximating ODEs, the limited accuracy of existing models necessitates further advancements. Hence, this study presents a stochastic model utilizing wavelet neural networks (WNNs) to approximate ODEs. Leveraging the compact structure and fast learning speed of WNNs, an improved butterfly optimization algorithm (IBOA) is employed to optimize the adjustable weights, facilitating more effective convergence towards the global optimum. The proposed WNNs approach is then rigorously evaluated by solving first and second order ODEs, including initial value problems, singularly perturbed boundary value problems, and a Lane–Emden type equation. Comparative analyses against alternative training methods, other existing ANNs, and numerical techniques demonstrate the superior performance of the proposed method, affirming its efficiency and accuracy in approximating ODE solutions.