The evolutionary integral dynamical models of storage systems are addressed. Such models are based on systems of weakly regular nonlinear Volterra integral equations with piecewise smooth kernels. These equations can have non-unique solutions that depend on free parameters. The objective of this paper was two-fold. First, the iterative numerical method based on the modified Newton–Kantorovich iterative process is proposed for a solution of the nonlinear systems of such weakly regular Volterra equations. Second, the proposed numerical method was tested both on synthetic examples and real world problems related to the dynamic analysis of microgrids with energy storage systems.

The aim of this study is to present a novel method to find the optimal solution of the reverse osmosis (RO) system. We apply the Sinc integration rule with single exponential (SE) and double exponential (DE) decays to find the approximate solution of the RO. Moreover, we introduce the stochastic arithmetic (SA), the CESTAC method (Controle et Estimation Stochastique des Arrondis de Calculs) and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library instead of the mathematical methods based on the floating point arithmetic (FPA). Applying this technique, we would be able to find the optimal approximation, the optimal error and the optimal iteration of the method. The main theorems are proved to support the method analytically. Based on these theorems, we can apply a new stopping condition in the numerical procedure instead of the traditional absolute error. These theorems show that the number of common significant digits (NCSDs) of exact and approximate solutions are almost equal to the NCSDs of two successive approximations. The numerical results are obtained for both SE and DE Sinc integration rules based on the FPA and the SA. Moreover, the number of iterations for various $\epsilon$ are computed in the FPA. Clearly, the DE case is more accurate and faster than the SE for finding the optimal approximation, the optimal error and the optimal iteration of the RO system.

Thermal research of printed circuit assemblies (PCA) can be carried out field tests (thermal control in the different operating modes) and numerical simulation. The basic data for numerical calculation are the parameters of the environment and the characteristics of heat sources. For newly designing equipment (or PCA), these data can be calculated using circuit simulation packages or getting from element specifications (typically maximum power). Using the maximum power dissipated in the crystal is possible, but results this thermal research may be far from reality. For the already existing equipment and assemblies can only distribution of thermal field. For research of the different operation modes it is advisable to perform a numerical simulation. This paper proposes a technique to obtain the values of the heat source by thermal picture PCA. The initial data used by the temperatures at several time steps. As a result we obtain the value of heat source at each point of board/assembly (on the base of user defined variable grid). The availability of these data allows make a more detailed analysis of the PCA by computer-aided engineering (CAE) software, and taking into account the individual characteristics of each board. To solve the proposed scheme for solving the problem of determining the power density internal heat sources of PCA. It is based on the numerical solution of the Fourier heat equation by finite differences method.

In this paper we construct complexity order optimal numerical methods for Volterra integral equations with different types of weakly singular kernels. We show that for Volterra equations (in contrast to Fredholm integral equations) using the ”block-by-block” technique it is not necessary to employ the additional iterations to construct complexity optimal methods.