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.
Highlights
Exact solutions of Initial Value Problem (IVPs) and Boundary Value Problems (BVPs) play a significant role in understanding the unique properties and phenomena in science and engineering
The results revealed the performance of hybridizing the Genetic Algorithms (GAs) with the secant method speeding up the convergence, improving the performance, and avoiding the shortage or divergent condition for the overall process
HAMGS is classified as a heuristic algorithm to find the semi-analytical solution of higher-order IVP and BVP in different applications
Summary
Exact solutions of Initial Value Problem (IVPs) and Boundary Value Problems (BVPs) play a significant role in understanding the unique properties and phenomena in science and engineering. The approach is recommended for low-order homotopy iterations schemes Another way to determine the optimal value of the convergence control parameter is to minimize the squared residual error [23]–[25]. The proposed approach determines the parameter’s optimal value by minimizing the discretized residual function’s norm at each HAM approximation order. This approach could find an acceptable approximation to a class of nonlinear differential equations. A new algorithm combining the secant method and GA is proposed to solve the optimization problem yielded from the residual of the HAM method. As a result, both algorithms will integrate together to accelerate the convergence of the optimum value of the HAM control parameter.
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