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Symbolic description of the polynomial roots and their numerical implementation - better than in Mathematica software?

Hetmaniok Edyta, Słota Damian, Pleszczyński Mariusz, Wituła Roman, Różański Michał, Szczygieł Marcin

Annals of Computer Science and Information Systems

2019

Vol. 20

41--45

This paper is a continuation of the discussion undertaken in one of our previous papers. We present in the current paper the corrected, and also given in a slightly changed form, Vandermonde formulae for the roots of some quintic polynomials considered in J.P. Tignol's monograph. The proofs of selected trigonometric identities from our previous paper are given and some new identities have been generated by the occasion, which also can be used for testing our Langrange algorithm for the case of cubic polynomials. Moreover, we present here the decomposition of polynomials belonging to some two-parameter family of polynomials related to the Chebyshev polynomials of the first kind.

Application of the immune algorithm IRM for solving the inverse problem of metal alloy solidification including the shrinkage phenomenon

Zielonka Adam, Hetmaniok Edyta, Słota Damian

Computer Methods in Materials Science

2018

Vol. 18, No. 1

1--10

In the paper the mathematical model of the inverse one-dimensional problem of binary alloy solidification, with the material shrinkage phenomenon taken into account, is defined. The process is described by using the model of solidification in the temperature interval, whereas the shrinkage of material is modeled basing on the mass balance equation. The inverse problem consists in reconstruction of the heat transfer coefficient on the boundary of the casting mould separating the cast from the environment. Lack of this data is compensated by the measurements of temperature in the control point located inside the mould. The method of solving the investigated problem is based on two procedures: the implicit scheme of finite difference method supplemented by the procedure of correcting the field of temperature in the vicinity of liquidus and solidus curves and the immune optimization algorithm IRM.