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1
Content available Numerical solution of two-dimensional Fredholm integro-differential equations by Chebyshev integral operational matrix method
Ishola Christie Yemisi, Taiwo Omotayo Adebayo, Adebisi Ajimot Folasade, Peter Olumuyiwa James
Journal of Applied Mathematics and Computational Mechanics
|
2022
|
Vol. 21, nr 1
29--40
EN
This paper presents the Chebyshev Integral Operational Matrix Method (CIOMM) for the numerical solution of two-dimensional Fredholm Integro-Differential Equations (2D-FIDEs). The process of the method is obtaining the operational matrix of integration by evaluating a 2D integral of 2D Chebyshev polynomial basis functions and assuming approximate solutions of the 2D-FIDEs as a truncated 2D Chebyshev series. This leads to a system of linear algebraic equations which are solved to obtain the values of the unknown constants using Maple 18. Some numerical problems are solved to illustrate the practicability of the method.
2
Content available Irregularity of post mining deformations as indicator revealing effects of processes of unknown origin in area of Bochnia
Szczerbowski Zbigniew
Geoinformatica Polonica
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2020
|
T. 19
81--93
EN
The presented work deals with the problem of terrain surface and rock mass deformation in the area of the Bochnia Salt Mine. The deformations are related to natural causes (mainly the tectonic stress of the Carpathian orogen) as well as anthropogenic ones related to the past mining activity conducted directly under the buildings of the town of Bochnia. The discussed characteristics of land surface deformation are important from the point of view of threats to surface features and contribute to spatial development. Particularly anomalous zones of observed subsidence basins are examined as places of second order deformation effects. The author presents a method of determinations of these anomalous areas and he discusses their origins.
PL
Przestawiona praca dotyczy problematyki deformacji powierzchni i górotworu w rejonie Kopalni Soli Bochnia. Deformacje te związane są zarówno z przyczynami naturalnymi (głównie nacisk tektoniczny orogenu karpackiego), jak i antropogenicznymi związanymi z minioną aktywnością górniczą prowadzoną bezpośrednio pod zabudową miejską Bochni. Omówione charakterystyki deformacji powierzchni terenu są istotne z punktu widzenia zagrożeń dla obiektów powierzchni oraz są elementem planowania przestrzennego. Szczególnie przeanalizowane zostały strefy anomalne niecki obniżeniowej jako efekty drugiego rzędu deformacji. Autor przedstawił sposób ich wyznaczania oraz omówił genezę ich powstania.
3
Content available remote 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
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2019
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Vol. 20
41--45
EN
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.
4
Content available On some conjectures regarding tridiagonal matrices
Fonseca C. M.
Journal of Applied Mathematics and Computational Mechanics
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2018
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Vol. 17, nr 4
13--17
EN
We discuss several conjectures proposed recently by A.Z. Küçük and M. Düz on the permanent of certain type of tridiagonal matrices. We recall some less known results on tridiagonal matrices and, at the same time, bring other results together to a common framework.
5
Content available Non-linear unsteady inverse boundary problem for heat conduction equation
Joachimiak M., Ciałkowski M.
Archives of Thermodynamics
|
2017
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Vol. 38, no. 2
81--100
EN
Direct and inverse problems for unsteady heat conduction equation for a cylinder were solved in this paper. Changes of heat conduction coefficient and specific heat depending on the temperature were taken into consideration. To solve the non-linear problem, the Kirchhoff’s substitution was applied. Solution was written as a linear combination of Chebyshev polynomials. Sensitivity of the solution to the inverse problem with respect to the error in temperature measurement and thermocouple installation error was analysed. Temperature distribution on the boundary of the cylinder, being the numerical example presented in the paper, is similar to that obtained during heating in the nitrification process.
6
Content available A geometric property of the roots of Chebyshev polynomials
Wituła R., Hetmaniok E., Słota D.
Journal of Applied Mathematics and Computational Mechanics
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2014
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Vol. 13, nr 1
143--148
EN
In this text a new property of geometric nature of the Chebyshev polynomials is given. It is proven that the lengths of diagonals of a regular n-gon with the side of length equal to one are the sums of positive roots of the respective renormalized Chebyshev polynomials of one from among four types. Some new special decompositions of differences of values of the Chebyshev polynomials are also presented.
7
Content available Aproksymacja quasijednostajna
Purczyński J.
Poznan University of Technology Academic Journals. Electrical Engineering
|
2013
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No. 73
43--50
PL
W pracy wykorzystano metodę aproksymacji średniokwadratowej wielomianowej, przy czym jako kryterium przyjęto wielkość błędu maksymalnego uzyskanego przybliżenia -stąd proponowana nazwa metody „aproksymacja quasijednostajna”. Wartość tego błędu zależy od liczby punktów aproksymacji w przedziale. Zmieniając liczbę punktów aproksymacji stwierdza się, że wartość błędu maksymalnego posiada minimum dla określonej wartości L liczby uwzględnionych punktów. Dla wielomianu stopnia N wyznacza się optymalną liczbę równoodległych punktów aproksymacji L oraz maksymalny błąd aproksymacji. Proponowana metoda została porównana z metodą aproksymacji jednostajnej, jaką są wielomiany Czebyszewa. Na rozpatrzonych przykładach wykazano, że metoda „aproksymacji quasijednostajnej” prowadzi do mniejszych wartości błędu maksymalnego, niż wielomiany Czebyszewa.
EN
In the paper the polynomial mean-square approximation method was applied, where the applied criterion was the value of the maximum error of the obtained approximation - hence the proposed name for this method - ‘quasi-uniform approximation’. The value of this error depends on the number of approximation points within the range. By changing the number of points within the range, it can be noticed that the value of the maximum error has the minimum value for a particular value of L number of considered points. For a polynomial of N degree, the optimum number of equidistant points of approximation L and the maximum error of approximation are determined. The proposed method was compared with a uniform approximation method, namely the Chebyshev polynomial. The examples included in the paper show that the ‘quasi-uniform approximation’ method yields smaller values of the maximum error than Chebyshev polynomials.
8
Content available remote On the q-Hermite polynomials and their relationship with some other families of orthogonal polynomials
Szabłowski P. J.
Demonstratio Mathematica
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2013
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Vol. 46, nr 4
679--708
EN
We review properties of the q-Hermite polynomials and indicate their links with the Chebyshev, Rogers–Szegö, Al-Salam–Chihara, continuous q-utraspherical polynomials. In particular, we recall the connection coefficients between these families of polynomials. We also present some useful and important finite and infinite expansions involving polynomials of these families including symmetric and non-symmetric kernels. In the paper, we collect scattered throughout literature useful but not widely known facts concerning these polynomials. It is based on 43 positions of predominantly recent literature.
9
Approximation of entire function solutions of the Helmholtz equation having slow growth
Kumar D.
Journal of Applied Analysis
|
2012
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Vol. 18, nr 2
179-196
EN
In this paper, we study the Chebyshev polynomial approximation of entire solutions of Helmholtz equations in R2 in Banach spaces (B(p, q, m) space, Hardy space and Bergman space). Some bounds on generalized order of entire solutions of Helmholtz equations of slow growth have been obtained in terms of the coefficients and approximation errors using function theoretic methods.
10
Content available remote Metoda kolokacji rozwiązywania równań różniczkowych typu parabolicznego
Povstenko Y.
Studia i Materiały / Europejska Uczelnia Informatyczno-Ekonomiczna w Warszawie
|
2011
|
Nr 1(1)
43--50
PL
W artykule omówiono metodę spektralną rozwiązywania równań typu parabolicznego. Opisano procedurę numeryczną i przedstawiono wyniki eksperymentu numerycznego.
EN
In this paper the spectral method for solving equations of parabolic type is discussed. Numerical procedure is described, and results of numerical experiment are presented.
11
Content available remote High-speed modular structures for parallel computing in the space of orthogonal projections
Selianinau M.
Prace Naukowe Akademii im. Jana Długosza w Częstochowie. Edukacja Techniczna i Informatyczna
|
2010
|
T. 5
89-98
EN
The numerical-analytical method of digital signal processing based on the modular model of space of orthogonal projections are presented in the article. This gives new possibilities for the high-performance processing of discrete signals on numerical-analytical level at the realization not only arithmetic but also more complicated operations such as convolution, correlation, algorithms of spectral analysis and others.
PL
W artykule przedstawiono liczbowo-analityczną metodę cyfrowego przetwarzania sygnałów, która oparta jest na wykorzystaniu modułowego modelu przestrzeni rzutów ortogonalnych. Daje ona nowe możliwości dla wysokowydajnego przetwarzania sygnałów dyskretnych na liczbowo-analitycznym poziomie przy realizacji nie tylko arytmetycznych, ale również bardziej skomplikowanych operacji takich, w szczególności, jak splot, korelacja, algorytmy widmowej analizy i inne.
12
Content available remote Aproksymacja jednostajna odwzorowań kartograficznych
Pędzich P.
Roczniki Geomatyki
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2009
|
T. 7, z. 4
67-76
PL
W artykule zaprezentowane zostaną własności wielomianów Czebyszewa oraz ich zastosowanie do aproksymacji odwzorowań kartograficznych. Ponadto przedstawione zostanie porównanie wyników aproksymacji jednostajnej oraz aproksymacji średniokwadratowej odwzorowań kartograficznych.
EN
Usually the least square method is used for approximation of map projection. Determination of polynomial coefficients requires solution of a complicated system of equations. It is possible to avoid such a problem using orthogonal Chebyshev polynomials. This is a completely different method of approximation, where the maximum difference between the value of the function and the value calculated from polynomial is minimized. In the paper, properties of Chebyshev polynomials approximations are presented as wall as their application to map projection approximation and comparison with other methods of map projection.
13
Content available remote Parametric excitation of pipes through fluid flow
Szabó Z.
Computer Assisted Mechanics and Engineering Sciences
|
1999
|
Vol. 6, No. 3-4
487-494
EN
In this paper the dynamic behaviour of a continuum inextensible pipe containing fluid flow is investigated. The fluid is considered to be incompressible, frictionless and its velocity relative to the pipe has the same but time-periodic magnitude along the pipe at a certain time instant. The equations of motion are derived via Lagrangian equations and Hamilton's principle. The system is non-conservative, and the amount of energy carried in and out by the flow appears in the model. It is well-known, that intricate stability problems arise when the flow pulsates and the corresponding mathematical model, a system of ordinary or partial differential equations, becomes time-periodic. The method which constructs the state transition matrix used in Floquet theory in terms of Chebyshev polynomials is especially effective for stability analysis of systems with multi-degree-of-freedom. The stability charts are created w.r.t. the forcing frequency omega, the perturbation amplitude nu and the average flow velocity U.
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