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1
Content available Application of second order inhomogeneous linear recurrences to solving a tridiagonal system
Borowska J., Łacińska L.
Journal of Applied Mathematics and Computational Mechanics
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2016
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Vol. 15, nr 2
5--10
EN
In this paper we consider a linear system of algebraic equations of a tridiagonal type. We show that the solution of such a system can be represented by a corresponding second order inhomogeneous linear recurrence equation. This approach enables us to represent the solution to the tridiagonal Toeplitz linear system of equations in a closed form.
2
Content available Eigenvalues of 2-tridiagonal Toeplitz matrix
Borowska J., Łacińska L.
Journal of Applied Mathematics and Computational Mechanics
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2015
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Vol. 14, nr 4
11--17
EN
In this article an explicit formula for eigenvalues of a 2-tridiagonal Toeplitz matrix can be derived on the basis of a certain relation between the determinant of this matrix and the determinant of a pertinent tridiagonal matrix. It can be pointed out that the problem is investigated without imposing any conditions on the elements of matrix.
3
Content available Relation between determinants of tridiagonal and certain pentadiagonal matrices
Borowska J., Łacińska L.
Journal of Applied Mathematics and Computational Mechanics
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2014
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Vol. 13, nr 3
21--28
EN
In this paper we consider a pentadiagonal matrix which consists of only three non-zero bands. We prove that the determinant of such a matrix can be represented by a product of two determinants of corresponding tridiagonal matrices. It is shown that such an approach gives greatly shorter time of computer calculations.
4
Content available Recurrence form for determinant of a heptadiagonal symmetric Toeplitz matrix
Borowska J., Łacińska L.
Journal of Applied Mathematics and Computational Mechanics
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2014
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Vol. 13, nr 4
19--26
EN
In this paper we show that the determinant of heptadiagonal symmetric Toeplitz matrix can be represented by a particular solution of the system of three homogeneous linear recurrence equations. The general considerations are illustrated by certain numerical example implementated in the Maple system.
5
Content available A system of linear recurrence equations for determinant of pentadiagonal matrix
Borowska J., Łacińska L., Rychlewska J.
Journal of Applied Mathematics and Computational Mechanics
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2014
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Vol. 13, nr 2
5--12
EN
In this paper we present an application of the system of two homogeneous linear recurrence equations to evaluate the determinant of pentadiagonal matrix. The general considerations are illustrated by two examples. It is shown that the proposed approach is suited for implementation using computer algebra systems.
6
Content available On determinant of certain pentadiagonal matrix
Borowska J., Łacińska L., Rychlewska J.
Journal of Applied Mathematics and Computational Mechanics
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2013
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Vol. 12, nr 3
21--26
EN
In this paper, using the LU factorization, the relation between the determinant of a certain pentadiagonal matrix and the determinant of a corresponding tridiagonal matrix will be derived. Moreover, it will be shown that determinant of this special pentadiagonal matrix can be calculated by applying the fourth order homogeneous linear difference equation.
7
Content available remote Application of difference equations to certain tridiagonal matrices
Borowska J., Łacińska L., Rychlewska J.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2012
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Vol. 11, nr 3
15-20
EN
In this paper we present an application of second order homogeneous linear difference equations with constant coefficients to evaluate the determinant of tridiagonal matrices. Comparing the obtained results with a certain alternative approach [1] some formulae for the finite sum are derived.
8
Content available remote On the tolerance wave-type solutions to the hyperbolic heat conduction in microperiodic composites
Łacińska L., Wierzbicki E.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2008
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Vol. 7, nr 1
93-97
EN
In the note wave-type solutions to the hyperbolic heat conduction in the microperiodic composites are investigated. The considerations are related to the tolerance averaged model of the hyperbolic heat transfer problems in microperiodic composites. The Lapunov exponent notation is applied. The open form of these solutions are formulated in one shape function case.
9
On the boundary effect in elastodynamics of laminated solids
Wierzbicki E., Łacińska L.
Zeszyty Naukowe Politechniki Świętokrzyskiej. Nauki Techniczne
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2007
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Z. 6
45-54
EN
The object of the contribution is the analysis of a certain aspects of the boundary effect behaviour in elastodynamics of FGM-type multiphase laminated composites. The aim of the paper is to answer the question what kind of assumptions are enough to be taken into account in the framework of the simplified tolerance model for the elastodynamics of the considered laminates the boundary effect will be observed. The answer to this question is obtained for a certain class of two-phased FGM-type laminates and for tolerance description based on the exclusively one arbitrary shape function.
PL
Przedmiotem rozważań jest analiza zjawisk typu warstwy brzegowej w elastodynamice wielofazowych laminatów z gradacją własności efektywnych. Celem pracy jest odpowiedź na pytanie, przy jakich założeniach przyjętych w ramach uproszczonego tolerancyjnego modelu elastodynamiki takich laminatów efekt brzegowy jest obserwowany. Odpowiedź na to pytanie uzyskano dla pewnej klasy dwufazowych laminatów typu FGM i tolerancyjnego modelu z jedną funkcją kształtu, której postać nie musi być precyzowana dla realizacji celu pracy.
10
Content available remote On tolerance averaged model of the propagation of boundary displacement fluctuations in FGM-type laminated composite
Łacińska L., Wierzbicki E.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2007
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Vol. 6, nr 1
133-140
EN
The object of the contribution is the analysis of a certain initial-boundary problems in elastodynamics of FGM-type multiphased laminated composites. The aim of the paper is to formulate an answer to the question what kind of assumptions are enough to be taken into account in the framework of the simplified tolerance model for the elastodynamics of the considered laminates the boundary effect will be observed.
11
Content available remote On the probabilistic analogy between homogenized and tolerance models of two-constituent microperiodic isotropic laminated composites
Łacińska L., Wierzbicki E.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2006
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Vol. 5, nr 1
71-82
EN
In the note an interrelation between tolerance and homogenized effective moduli for two-constituent elastic microperiodic composite are examined. It is shown that, in the case in which constituents are isotropic, effective modulae are identical. The proof is based on the probabilistic analogy formulated for both kinds of modulae.
12
Content available remote Boundary and initial fluctuation effect on dynamic behaviour of a laminated solid
Łacińska L., Wierzbicki E.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2005
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Vol. 4, nr 1
110-122
EN
In the paper an approximate description of the evolution of boundary traction fluctuations in the two-component linear-elastic laminates is discussed. In the paper [1] it has been shown that in the framework of the tolerance averaged model of linear elastodynamics of the aforementioned laminates, cf. [2], in the first approximation such evolution is independent of the mean (averaged) displacement. The aim of this contribution is to discuss solutions to the boundary problem formulated in [1].
13
Content available remote On a certain boundary value problem for a laminated half-space
Łacińska L.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2004
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Vol. 3, nr 1
99--104
EN
In this contribution we are to show that on the boundaries of a laminated medium, which are perpendicular to the laminae interfaces, we observe some near-boundary phenomena related to the oscillatory character of the boundary tractions. The aim of this note is to describe and discuss those phenomena in dynamic problems. The analysis is carried out in the framework of the tolerance averaging technique. General considerations are illustrated by a certain special problem describing decay of the traction oscillations.
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