DYNAMICAL CONTROL OF ACCURACY USING THE STOCHASTIC ARITHMETIC TO ESTIMATE THE SOLUTION OF THE ORDINARY DIFFERENTIAL EQUATIONS VIA ADOMIAN DECOMPOSITION METHOD

The Adomian decomposition method (ADM) is a well-known and powerful method to solve the linear and nonlinear differential equations. In this paper, a reliable implementation of the Adomian decomposition method based on the stochastic arithmetic to solve an ordinary differential equation with initial conditions is discussed. To this aim, the CESTAC (Control et Estimation STochastique des Arrondis de Calculs) method is applied which is a method based on a probabilistic approach of the round-o error propagation which replaces the oating-point arithmetic by the stochastic arithmetic. For this purpose, a numerical algorithm is presented to determine the steps of using the CESTAC method to nd the numerical solution of a differential equation at a given point by means of the ADM. Also, a theorem is proved to show the accuracy of the ADM in solving the linear or nonlinear di erential equations. According to this theorem, the common signi cant digits of two sequential results is also common with the exact solution at a given point in the domain when the number of iterations increases. By using the proposed scheme, the optimal number of iterations and the optimal auxiliary parameter in the ADM can be found and the results are computed in a valid way with their accuracy. Also, the stability of the method is veri ed and the results will be determined with their correct signi cant digits. Finally, some 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 oating-point arithmetic. The programs have been provided by MAPLE package.