Dynamical Control of Accuracy using the Stochastic Arithmetic to Estimate the Solution of the Fuzzy Differential Equations via Homotopy Analysis Method

One of the important analytical-approximate schemes inorder to obtain the solution of a differential equation is the Homotopy Analysis Method (HAM). In this research, a novel technique is considered to validate the results of the algorithm obtained from the HAM to find the solution of a fuzzy differential equation with initial condition based on the generalized differentiablity. For this purpose, in place of the current floating-point arithmetic, a new arithmetic which is called the stochastic arithmetic is replaced. To this aim, the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs)method is applied which replaces the floating-point arithmetic by the stochastic arithmetic. Also, a numerical algorithm is presented to determine the steps of using the CESTAC method to find the numerical solution of a fuzzy differential equation at a given point by means of the HAM. In order to determine the accuracy of the proposed method, a theorem is proved. By using the proposed scheme, the optimal number of steps and the optimal auxiliary parameter in the HAM can be found and the results are computed in a valid way with their accuracy. Also, the stability of the method is verified and the results will be determined with their correct significant digits. Finally, two sample fuzzy differential equations are solved based on the mentioned algorithm to illustrate the importance, advantages and applicability of using the stochastic arithmetic in place of the floating-point arithmetic. The programs have been provided by MAPLE package.