Edyta Hetmaniok

Solution of the inverse heat conduction problem by using the ABC algorithm

In this paper, a numerical method of solving the inverse heat conduction problem based on the respectively new tool for combinational optimization, named the Artificial Bee Colony algorithm (ABC), is presented. In the first step, the direct heat conduction problem, associated to the considered inverse heat conduction problem, is solved by using the finite difference method. In the second step, the proper functional, based on the least squares method, is minimized by using the ABC algorithm, giving the solution of the considered problem. An example illustrating the precision and effectiveness of the method is also shown. The proposed approach is original and promising.

Determination of Optimal Parameters for the Immune Algorithm Used for Solving Inverse Heat Conduction Problems with and without a Phase Change

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.

Convergence and error estimation of homotopy perturbation method for Fredholm and Volterra integral equations

Abstract In this paper an application of the homotopy perturbation method for solving Fredholm and Volterra integral equations of the second kind is presented. Discussed method consists in constructing the functional series, sum of which determines the function giving the solution of considered problem. Conditions under which the constructed series is convergent are formulated and proved in the paper. In general, the composed series is rather fast convergent, thanks to which calculation of few first terms ensures a very good approximation of the sought solution. Estimation of errors of approximate solution obtained by taking the partial sum of the series is also elaborated in the paper.

Comparison of the Adomian decomposition method and the variational iteration method in solving the moving boundary problem

In this paper, a comparison between two methods: the Adomian decomposition method and the variational iteration method, used for solving the moving boundary problem, is presented. Both of the methods consist in constructing the appropriate iterative or recurrence formulas, on the basis of the equation considered and additional conditions, enabling one to determine the successive elements of a series or sequence approximating the function sought. The precision and speed of convergence of the procedures compared are verified with an example.

Usage of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind

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.

Comparison of ABC and ACO algorithms applied for solving the inverse heat conduction problem

In this paper we present the comparison of numerical methods applied for solving the inverse heat conduction problem in which two algorithms of swarm intelligence are used: Artificial Bee Colony algorithm (ABC) and Ant Colony Optimization algorithm (ACO). Both algorithms belong to the group of algorithms inspired by the behavior of swarms of insects and they are applied for minimizing the proper functional representing the crucial part of the method used for solving the inverse heat conduction problems. Methods applying the respective algorithms are compared with regard to their velocity and precision of the received results.

A stronger version of the second mean value theorem for integrals

Abstract We prove a stronger version of the classic second mean value theorem for integrals.

A study of the convergence of and error estimation for the homotopy perturbation method for the Volterra–Fredholm integral equations

Abstract In this work the solution of the Volterra–Fredholm integral equations of the second kind is presented. The proposed method is based on the homotopy perturbation method, which consists in constructing the series whose sum is the solution of the problem considered. The problem of the convergence of the series constructed is formulated and a proof of the formulation is given in the work. Additionally, the estimation of the approximate solution errors obtained by taking the partial sums of the series is elaborated on.

Using the swarm intelligence algorithms in solution of the two-dimensional inverse Stefan problem

In the paper a procedure for solving the two-dimensional inverse Stefan problem is presented. In considered problem the heat transfer coefficient is identified with the aid of known measurements of temperature in selected points of the region as the additional information. Direct Stefan problem is solved by using the alternating phase truncation method. Goal of the paper is to compare two swarm intelligence algorithms-the Ant Colony Optimization algorithm and the Artificial Bee Colony algorithm-applied for minimizing a functional expressing the error of approximate solution.

Using the Artificial Bee Colony Algorithm for Determining the Heat Transfer Coefficient

In this paper we present an application of the Artificial Bee Colony (ABC) algorithm for solving the inverse heat conduction problem, consisting in determining the state function and some of the boundary conditions. The ABC algorithm belongs to the group of swarm intelligence algorithms and is inspired by the technique of searching for the nectar around the hive by the colony of bees. We propose to use this algorithm for minimizing the proper functional, which allows to reconstruct the value of heat transfer coefficient in the successive cooling zones.