Optimizing Linear Partial Differential Equation Solutions by Combining Whale Optimization Algorithm and Homotopy Analysis Method

This paper introduces a novel approach, denoted as HAM-WOA, for solving linear ordinary differential equations (LDEs) by amalgamating the symmetric analytical method (HAM) with the whale optimization algorithm (WOA). In recent times, swarm algorithms have gained widespread recognition and are increasingly adopted by researchers due to their expeditious problem-solving capabilities and efficacy in handling complex systems while minimizing errors. The parameter 'h,' integral to the Homotopy Analysis Method (HAM), is estimated employing WOA, resulting in a notably enhanced solution accuracy compared to the conventional HAM approach. The proposed methodology undergoes a comprehensive evaluation in terms of both reliability and efficiency, juxtaposed against the original HAM method, utilizing metrics such as absolute errors (AE) and mean errors. The findings unequivocally demonstrate the superior performance of the HAM-WOA method over the standard HAM technique, establishing it as a superior choice for tackling differential equations.