Abstract
The aim of this paper is to present a new method and the tool to validate the numerical results of the Volterra integral equation with discontinuous kernels in linear and non-linear forms obtained from the Adomian decomposition method. Because of disadvantages of the traditional absolute error to show the accuracy of the mathematical methods which is based on the floating point arithmetic, we apply the stochastic arithmetic and new condition to study the efficiency of the method which is based on two successive approximations. Thus the CESTAC method (Controle et Estimation Stochastique des Arrondis de Calculs) and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library are employed. Finding the optimal iteration of the method, optimal approximation and the optimal error are some of advantages of the stochastic arithmetic, the CESTAC method and the CADNA library in comparison with the floating point arithmetic and usual packages. The theorems are proved to show the convergence analysis of the Adomian decomposition method for solving the mentioned problem. Also, the main theorem of the CESTAC method is presented which shows the equality between the number of common significant digits between exact and approximate solutions and two successive approximations.This makes in possible to apply the new termination criterion instead of absolute error. Several examples in both linear and nonlinear cases are solved and the numerical results for the stochastic arithmetic and the floating-point arithmetic are compared to demonstrate the accuracy of the novel method.
Highlights
The numerical results are obtained based on the floating-point arithmetic (FPA) and the stochastic arithmetic (SA)
The CESTAC method is among applicable and important methods to validate the numerical results based on the SA
We introduced the stopping condition based on this method that it is independent from the exact solution
Summary
Many authors modelized the load leveling problem arising in the energy storages of the powering systems in the form of the linear and non-linear Volterra integral equation (VIE) with discontinuous kernel. They have been focused on solving this problem by numerical and semi-analytical methods. For introduction to the theory of the VIE of the first kind with discontinuous kernals readers may refer to the monograph [8] Such models belongs to the class of ill-posed problems
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