Application of the CESTAC Method to Find the Optimal Iteration of the Homotopy Analysis Method for Solving Fuzzy Integral Equations

The goal of this work is to apply the stochastic arithmetic (SA) in discrete form and the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method to validate the numerical results of solving fuzzy Fredholm and Volterra integral equations (IEs) by homotopy analysis method (HAM). The CADNA (Control of Accuracy and Debugging for Numerical Applications) library is applied for mathematical computations. Using the mentioned method, not only the optimal step and approximation of the HAM but also some of numerical instabilities can be found. Furthermore, the optimal convergence control of the HAM for solving fuzzy IEs is obtained. Main theorems are proved to show the accuracy of the HAM by applying the CESTAC method to evaluate the obtained results.