The paper presents an application of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind. In this method a series is created, sum of which (if the series is convergent) gives the solution of discussed equation. Conditions ensuring convergence of this series are presented in the paper. Error of approximate solution, obtained by considering only partial sum of the series, is also estimated. Examples illustrating usage of the investigated method are presented as well, including the example having practical application for calculating the charge in supply circuit of flash lamps used in cameras.

This article reviews the most popular energy storage technologies and hybrid energy storage systems. With the dynamic development of the sector of renewable energy sources, it has become necessary to design and implement solutions that enable the maximum use of the energy obtained; for this purpose, an energy storage device is suggested. The most popular methods of electric energy storage are described, with an indication of the features of each technology, along with the presentation of the advantages and disadvantages of a given storage reservoir. Next, hybrid energy storage systems are presented along with their suggested applications and advantages resulting from the hybridization of technologically diverse energy storage systems.

In this paper, we present the application of the homotopy analysis method for solving integrodifferential equations. In this method, a series is created, the successive elements of which are determined by calculating the appropriate integral of the previous element. In this elaboration, we prove that, if this series is convergent, then its sum is the solution of the objective equation. We formulate and prove the sufficient condition of this convergence, and we give also the estimation of error of an approximate solution obtained by taking the partial sum of the considered series. Moreover, we present in this paper the example of using the investigated method for determining the vibrations of the freely supported reinforced concrete beam as well as for solving the equation of movement of the electromagnet jumper mechanical system.

The aim of this article is mathematical modelling and investigation of chosen parameters of a small energy harvesting system designed for energy harvesting from car tire mechanical vibrations to provide power supply for various sensors, e.g., in tire pressure monitoring system. The energy harvester consists of three permanent magnets inserted into a tube made from polyamide material. Comsol program has been used to calculate the force between the magnets, the stiffness of the magnetic spring, and the natural frequency of the system. MATLAB program has been used to simulate the movement of the moveable magnet to compare it with the measurements. Finally, the parameters of the mathematical model of the energy harvester were investigated and validated on a specially prepared laboratory test bench.