Abstract
Load leveling problems and energy storage systems can be modeled in the form of Volterra integral equations (VIE) with a discontinuous kernel. The Lagrange–collocation method is applied for solving the problem. Proving a theorem, we discuss the precision of the method. To control the accuracy, we apply the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library. For this aim, we apply discrete stochastic mathematics (DSA). Using this method, we can control the number of iterations, errors and accuracy. Additionally, some numerical instabilities can be identified. With the aid of this theorem, a novel condition is used instead of the traditional conditions.
Highlights
A =(n+1)(n+1), Y = ( y0, y1, . . . , y n ) T, F = ( f ( x0 ), f ( x1 ), . . . , f) T, the n-th order approximate solution of the Volterra integral equations (VIE) with discontinuous kernel (1) based on the Lagrange polynomials can be obtained in the form of yn ( x ) = ∑nj=0 y j l j ( x )
Assume that B is a set of representable values by a computer; for s∗ ∈ R, we can produce S∗ ∈ B with α mantissa bits of the binary floating-point arithmetic (FPA) as the following: S∗ = s∗ − ρ2E−α φ, (10)
Comparing the results between the Lagrange–collocation method using the FPA and the SA, we can see that the SA has more advantages in comparison with the FPA
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. In order to avoid theses problems, we apply the DSA and a novel termination criterion instead of (2) To this aim, the CESTAC method and the CADNA library are handled. Laporte and Vignes [36,37] applied this method, and a research group in France developed it [38] By using this technique, we can find the optimal results, iterations and error of the presented scheme. Based on this procedure, we are able to find the optimal iteration, approximation and error, which are the main novelties of this study.
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