Mariusz Pleszczyński

Comparing the Adomian decomposition method and the Runge–Kutta method for solutions of the Stefan problem

The solution of the one-phase Stefan problem is presented. A Stefans task is first approximated with a system of ordinary differential equations. A comparison between the Adomian decomposition method and the fourth-order Runge–Kutta method for solving this system is then presented.

A Certain Analytical Method Used for Solving the Stefan Problem

The paper presents an analytic method applied for finding the approximate solution of Stefan problem reduced to the one-phase solidification problem of a plate with the a priori unknown, and varying in time, boundary of the region in which the solution is sought. Proposed method is based on the known formalism of initial extension of a sought function describing the temperature field into the power series, some coefficients of which can be determined with the aid of boundary conditions, and also based on the approximation of a function defining the freezing front location with the broken line, parameters of which can be obtained by solving the appropriate differential equations. Results received by applying the proposed procedure are compared with the results obtained using a classical numerical method for solving the Stefan problem.

Inverse Continuous Casting Problem Solved by Applying the Artificial Bee Colony Algorithm

The paper presents an application of the Artificial Bee Colony algorithm in solving the inverse continuous casting problem consisted in reconstruction of selected parameters characterizing the cooling conditions in crystallizer and in secondary cooling zone. In presented approach we propose to use the bee algorithm for minimization of appropriate functional representing the crucial part of the method.

Kaprekar's transformations. Part I - theoretical discussion

The paper is devoted to discussion of the minimal cycles of the so called Kaprekars transformations and some of its generalizations. The considered transformations are the self-maps of the sets of natural numbers possessing n digits in their decimal expansions. In the paper there are introduced several new characteristics of such maps, among others, the ones connected with the Sharkovskys theorem and with the Erdös-Szekeres theorem concerning the monotonic subsequences. Because of the size the study is divided into two parts. Part I includes the considerations of strictly theoretical nature resulting from the definition of Kaprekars transformations. We find here all the minimal orbits of Kaprekars transformations Tn, for n = 3,..., 7. Moreover, we define many different generalizations of the Kaprekars transformations and we discuss their minimal orbits for the selected cases. In Part II (ibidem), which is a continuation of the current paper, the theoretical discussion will be supported by the numerical observations. For example, we notice there that each fixed point, familiar to us, of any Kaprekars transformation generates an infinite sequence of fixed points of the other Kaprekars transformations. The observed facts concern also several generalizations of the Kaprekars transformations defined in Part I.

Application of the Taylor Transformation to the Systems of Ordinary Differential Equations

In the paper the Taylor transformation is applied to systems of ordinary differential equations, including nonlinear differential equations. Apart from the description of the method, its computational effectiveness is demonstrated on example. Efficiency of the proposed method is confirmed it with the selected classical methods devoted to problems of considered kind. The present paper is an introduction to some further research in this area, which is very important for a wide range of problems described by means of the systems of ordinary differential equations.

Harmonic Numbers of Any Order and the Wolstenholme’s-Type Relations for Harmonic Numbers

The concept of harmonic numbers has appeared permanently in the mathematical science since the very early days of differential and integral calculus. Firsts significant identities concerning the harmonic numbers have been developed by Euler (see Basu, Ramanujan J, 16:7–24, 2008, [1], Borwein and Bradley, Int J Number Theory, 2:65–103, 2006, [2], Sofo, Computational techniques for the summation of series, 2003, [3], Sofo and Cvijovic, Appl Anal Discrete Math, 6:317–328, 2012, [4]), Goldbach, and next by the whole gallery of the greatest XIX and XX century mathematicians, like Gauss, Cauchy and Riemann. The research subject matter dealing with the harmonic numbers is constantly up-to-date, mostly because of the still unsolved Riemann hypothesis – let us recall that, thanks to J. Lagarias, the Riemann hypothesis is equivalent to some “elementary” inequality for the harmonic numbers (see Lagarias, Amer Math Monthly, 109(6):534–543, 2002, [5]). In paper (Sofo and Cvijovic, Appl Anal Discrete Math, 6:317–328, 2012, [4]) the following relation for the generalized harmonic numbers is introduced

Kaprekar's transformations. Part II-numerical results and intriguing corollaries.

\begin{aligned} H_n^{(r)} := \sum _{k=1}^n \frac{1}{k^r} = \frac{(-1)^{r-1}}{(r-1)!} \left( \psi ^{(r-1)}(n+1) - \psi ^{(r-1)}(1) \right) , \end{aligned}

Problem of the Moving Boundary in Continuous Casting Solved by The Analytic-Numerical Method

(1) for positive integers n, r. The main goal of our paper is to derive the generalization of this formula for every \(r \in \mathbb {R}\), \(r>1\). It was important to us to get this generalization in possibly natural way. Thus, we have chosen the approach based on the discussion of the Weyl integral. In the second part of the paper we present the survey of results concerning the Wolstenholme’s style congruence for the harmonic numbers. We have to admit that we tried to define the equivalent of the universal divisor (the polynomial, some kind of the special function) for the defined here generalized harmonic numbers. Did we succeed? We continue our efforts in this matter, we believe that such universal divisors can be found.