A Review of Different Approaches on Determination of Convergence Control Parameter in Homotopy Analysis Method

One of the principal components of the Homotopy Analysis Method (HAM) is the determination of the convergence control parameter, which plays a pivotal role in ensuring the accuracy and efficiency of solutions obtained using HAM. The convergence control parameter directly impacts the rate of convergence and the precision of the method, making its proper determination essential for solving nonlinear problems. This study aims to systematically compare the performance of several approaches for determining the convergence control parameter in HAM. By examining different methods, the paper highlights their respective strengths, weaknesses, and applicability to a range of nonlinear problems. Numerical experiments and theoretical analysis are conducted to assess the accuracy and convergence rate associated with each approach. Particular attention is given to identifying strategies that achieve a balance between computational efficiency and solution precision. The results provide valuable insights into the impact of the convergence control parameter on HAM's performance and offer guidelines for selecting the most suitable approach for various types of problems. This study contributes to advancing the application of HAM in solving nonlinear equations, enhancing its utility in scientific and engineering contexts.

Keywords:

Homotopy Analysis Method , Optimal Control Parameter , Nonlinear Problems