Homotopy Analysis-Based Hybrid Genetic Algorithm and Secant Method to Solve IVP and Higher-Order BVP

Abstract

The analytical solution of the Initial Value Problem (IVP) and the Boundary Value Problem (BVP) is an essential issue in numerous engineering applications. However, it cannot usually be realized by simple methods. Accordingly, utilizing semi-analytical solution methods such as the homotopy analysis method (HAM) is an alternative. However, it depends on calculating a dedicated convergence parameter. Different algorithms were proposed to realize this parameter. Thus, converting the residual function into an optimization problem represented a promising solution. The resulted optimization problem can be solved either using traditional analytical methods or meta-heuristic algorithms. In this paper, a new algorithm called Homotopy Analysis Method-based hybrid Genetic algorithm and Secant method (HAMGS) is proposed for solving such optimization problems. The proposed hybridization is based on utilizing the Secant method in the crossover process of the genetic algorithm to improve and accelerate the convergence. Moreover, this hybridization will protect the secant method from divergence in searching for new individuals. As a result, the convergence control parameter can be detected algebraically without drawing the h-curves. This facilitates identifying the minimum number of HAM terms to reach the solution, which reduces the computational burden remarkably. The presumed improvement of pairing the genetic algorithm and Secant method for improving the HAM solver is verified via four IVPs and three higher-order BVPs. The results corroborate the competence of the proposed algorithm and its ability to solve such problems efficiently.