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.

In this article the inverse heat conduction problem and the inverse Stefan problem with the third kind of boundary condition are solved by applying the immune algorithm. This method has been introduced in recent years and belongs to the group of optimization algorithms inspired by natural processes. In this case the applied algorithm is based on the rules of immune system functioning in vertebrate bodies. It is used for minimizing a functional playing a crucial role in the solution of the problem posed. The algorithm considered is investigated with respect to the parameters which should be chosen in order to provide the most efficient algorithm performance.

Recently, a lot of attention has been paid to the field of research connected with the wireless sensor network and industrial internet of things. The solutions found by theorists are next used in practice in such area as smart industries, smart devices, smart home, smart transportation and the like. Therefore, there is a need to look for some new techniques for solving the problems described by means of the appropriate equations, including differential equations, integral equations and integro-differential equations. The object of interests of this paper is the method dedicated for solving some integro-differential equations with a retarded (delayed) argument. The proposed procedure is based on the Taylor differential transformation which enables to transform the given integro-differential equation into a respective system of algebraic (nonlinear, very often) equations. The described method is efficient and relatively simple to use, however a high degree of generality and complexity of problems, defined by means of the discussed equations, makes impossible to obtain a general form of their solution and enforces an individual approach to each equation, which, however, does not diminish the benefits associated with its use.