Finding optimal convergence control parameter in the homotopy analysis method to solve integral equations based on the stochastic arithmetic

摘要

The goal of this paper is to present a new scheme based on the stochastic arithmetic (SA) to find the optimal convergence control parameter, the optimal iteration and the optimal approximation of the homotopy analysis method (HAM). This scheme is called the CESTAC(1) method. Also, the CADNA(2) library is applied to implement the CESTAC method on the proposed algorithms. CADNA is able to present a new, robust and valid environment to implement the HAM and optimize the results. By using this method, not only the optimal auxiliary control parameter can be computed but also the unnecessary iterations can be neglected and optimal step of this method is achieved. The main theorems are presented to guarantee the validity and accuracy of the HAM. Different kinds of integral equations such as singular and first kind are considered to find the optimal results by applying the proposed algorithms. The numerical results show the importance and efficiency of the SA in comparison with the floating-point arithmetic (FPA).