Roman Wituła

On computing the determinants and inverses of some special type of tridiagonal and constant-diagonals matrices

The determinants and inverses of some tridiagonal and constant-diagonals matrices are discussed in the paper. Linear recurrence equations of the second and higher orders which satisfy these determinants are presented here. In connection with these determinants new families of polynomials are defined. The relationships between these polynomials and normed Chebyshev polynomials of the first and second kind are investigated. Incidentally, a number of new identities for Chebyshev polynomials and some linear modifications of these polynomials are shown, as well as many new trigonometric identities and identities for Fibonacci and Lucas numbers.

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.

Partial fractions decompositions of some rational functions

In this paper there are presented three new methods of partial fractions decomposition of some rational functions. These methods can be applied to generate the general numerical algorithms and to calculations by hand.

Some phenomenon of the powers of certain tridiagonal and asymmetric matrices

In this paper, a new method of calculation of real (and even complex) powers of some asymmetric matrices obeying constance tridiagonals is presented.

Reconstruction of the Thermal Conductivity Coefficient in the Time Fractional Diffusion Equation

This paper describes reconstruction of the thermal conductivity coefficient in the time fractional diffusion equation. Additional information for the considered inverse problem was given by the temperature measurements at selected points of the domain. The direct problem was solved by using the finite difference method. To minimize functional defining the error of approximate solution the Fibonacci search algorithm was used.