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1

Positive solutions for fractional differential equation at resonance under integral boundary conditions

Wang Youyu, Huang Yue, Li Xianfei

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

238--253

EN

By using the theory of fixed point index and spectral theory of linear operators, we study the existence of positive solutions for Riemann-Liouville fractional differential equations at resonance. Our approach will provide some new ideas for the study of this kind of problem.

2

A new iteration method for the solution of third-order BVP via Green's function

Akgun Fatma Aydın, Rasulov Zaur

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

425--435

EN

In this study, a new iterative method for third-order boundary value problems based on embedding Green’s function is introduced. The existence and uniqueness theorems are established, and necessary conditions are derived for convergence. The accuracy, efficiency and applicability of the results are demonstrated by comparing with the exact results and existing methods. The results of this paper extend and generalize the corresponding results in the literature.

3

Temperature-time profiles of a tubular bus in shorting conditions

Gołębiowski Jerzy, Zaręba Marek

Przegląd Elektrotechniczny

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2020

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R. 96, nr 3

146--149

EN

In the paper temperature-time profiles generated in a tubular bus under short-circuit conditions were determined. The initial-boundary parabolic problem was the mathematical model. Adiabatic boundaries of the system and field uniformity at the initial moment of shorting were assumed and motivated. The two models of the bus-bar electric resistivity were considered: the temperature dependent (variable) and averaged one (constant). The problem was solved by means of Green’s function. The one-second shorting current and its multi-second equivalents were determined based on the above. The analytical results were verified numerically by the finite element method.

PL

W artykule wyznaczono czasowe przebiegi temperatury generowane w szynoprzewodzie rurowym podczas zwarcia. Matematycznym modelem jest początkowo-brzegowe zagadnienie paraboliczne. Przyjęto i uzasadniono założenie o adiabatycznych brzegach układu oraz o równomierności pola w początkowej chwili zwarcia. Rozpatrywano dwa modele elektrycznej rezystywności szynoprzewodu: uzależnionej termicznie (zmiennej) i uśrednionej (stałej). Zagadnienie rozwiązano za pomocą funkcji Greena. Na tej podstawie wyznaczono jednosekundowy prąd zwarcia i jego wielosekundowe równoważniki. Wyniki analityczne zweryfikowano numerycznie metodą elementu skończonego.

4

New bounds for Shannon, Relative and Mandelbrot entropies via Hermite interpolating polynomial

Mehmood N., Butt S. I., Pečarić D., Pečarić J.

Demonstratio Mathematica

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2018

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Vol. 51, nr 1

112--130

EN

To procure inequalities for divergences between probability distributions, Jensen’s inequality is the key to success. Shannon, Relative and Zipf-Mandelbrot entropies have many applications in many applied sciences, such as, in information theory, biology and economics, etc. We consider discrete and continuous cyclic refinements of Jensen’s inequality and extend them from convex function to higher order convex function by means of different new Green functions by employing Hermite interpolating polynomial whose error term is approximated by Peano’s kernal. As an application of our obtained results, we give new bounds for Shannon, Relative and Zipf-Mandelbrot entropies.

5

Comparison of heating curves of a rectangular busbar in different conditions of heat abstraction during the short-circuit heating

Zaręba M.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2018

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Vol. 66, nr 1

79--85

EN

The paper compares heating curves for a rectangular busbar in the conditions of convective and adiabatic heat transfer during short-circuit heating. Different coefficients of heat transfer from the external surface of the busbar and different busbar cross-sections have been assumed. This has allowed determining the error value occurring when the adiabatic rather than convective boundary condition is presumed at short circuit. The analysis takes into account a change of resistivity in the temperature function. The respective boundary-initial problems have been solved with analytical methods using Green’s function. The calculated results show that no considerable errors occur for long-lasting short circuits with an adiabatic rather than convective boundary condition.

6

Fizyczne podstawy metody interferometrii sejsmicznej

Czarny R.

Zeszyty Naukowe Instytutu Gospodarki Surowcami Mineralnymi i Energią PAN

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2017

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nr 101

195--202

PL

Interferometria sejsmiczna jest dynamicznie rozwijającą się metodą, której pierwsze zastosowania sięgają początków obecnego stulecia. Aktualnie znajduje coraz szersze zastosowanie w zagadnieniach m.in. obrazowania głębokich struktur ziemi oraz utworów przypowierzchniowych, monitorowania procesów wulkanicznych oraz analizowania wpływu silnych trzęsień ziemi na obiekty budowlane. Metoda ta pozwala na odtworzenie odpowiedzi impulsowej tzw. funkcji Greena ośrodka pomiędzy parą odbiorników na podstawie zarejestrowanych w tym samym czasie sejsmicznych pól falowych na tych odbiornikach. W wyniku odpowiednich operacji matematycznych metoda ta zamienia zarejestrowane na odbiornikach koherentne fale sejsmiczne o nieznanym czasie oraz miejscu ich wzbudzenia na układ tzw. wirtualnych źródeł emitujących sejsmiczne pole falowe z dowolnego odbiornika. W artykule przedstawiono fizyczne uzasadnienie wyników eksperymentu akustyki odwróconego czasu (ang. time-reversed acoustics) według Derode i in. (2003), które jest zarazem wytłumaczeniem metody interferometrii sejsmicznej. Eksperyment laboratoryjny w pierwszym etapie polegał na rejestracji akustycznego pola falowego wyemitowanego na brzegu naczynia wypełnionego cieczą i stalowymi prętami. Następnie rejestracje zostały odwrócone w czasie i wysłane powtórnie do wewnątrz naczynia i odebrany po przeciwnej stronie. Zarejestrowany na końcu sygnał okazał się zbliżony do sygnału wyemitowanego, pomimo przejścia przez ośrodek wielokrotnie rozpraszający. Doświadczenie to uzasadniono wykorzystując technikę korelacji wzajemnej (ang. cross-correlation), zasadę superpozycji pola falowego oraz zasadę wzajemności Rayleigha.

EN

Seismic interferometry is a geophysical method which has been developing very rapidly over the last decade. It has been applied to image deep structures of the Earth as well as near-surface, monitor volcanic processes, geothermal reservoirs within exploitation, rock mass deformation induced by mining, landslides, ground water storage, ice sheet or the impact of strong earthquakes to buildings. The vast majority of these applications use ambient seismic noise as a seismic source. This method involves reconstructing thte impulse response, the socalled Green’s function, between pair of receivers based on the wave field registered by them. Using seismic interferometry with various data processing flows the registered coherent seismic waves by the receivers can be changed to virtual sources which are placed in the receiver locations. In the article, the physical derivation of the time-reversed acoustics experiment which was introduced by Derode et. al. (2003) is presented. This derivation also explains the seismic interferometry method. The laboratory experiment contained two phases. First, an acoustics signal was emitted into the medium with hundreds of scatterers (cube with liquid and rods) and registered on the opposite side of the medium. Then, registrations were reversed and emitted back. Finally, the wave field refocused exactly in the point of initial excitation. Derode et. al. explain these results using the cross-correlation technique, superposition and Rayleigh’s reciprocity principles.

7

Analiza wpływu zmiennej rezystywności elektrycznej na pole termiczne szynoprzewodu cylindrycznego metodą funkcji Greena

Gołębiowski J., Zaręba M.

Przegląd Elektrotechniczny

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2017

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R. 93, nr 6

111--114

PL

W pracy zbadano wpływ zmiennej rezystywności elektrycznej na nieustalone pole termiczne szynoprzewodu cylindrycznego. Analizę przeprowadzono metodą funkcji Greena. Na tej podstawie wyznaczono długotrwałe prądy dopuszczalne, krzywe rozgrzewu oraz termiczne stałe czasowe modeli układu ze zmiennym i stałym oporem właściwym. Wyniki pozytywnie zweryfikowano metodą elementów skończonych za pomocą modułu matematycznego modelowania programu Comsol Multiphysics.

EN

The influence of the variable electric resistivity on the transient thermal field of a cylindrical bus bar has been analysed in this paper. The Green’s function was used for this purpose. On this basis, the steady state current rating, the heating curves and the thermal time constants of the model of the system with variable and state resistivity were determined. Results were verified with the finite elements method. A mathematical modeling module of the Comsol Multiphysics software was applied for this purpose.

8

Assessment of continental hydrosphere loading using GNSS measurements

Zygmunt M., Rajner M., Liwosz T.

Reports on Geodesy and Geoinformatics

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2016

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Vol. 101

36--53

EN

Presented paper is dedicated to problems of deformation of the Earth's crust as a response to the surface loading caused by continental waters. The aim of this study was to specify areas particularly vulnerable to studied deformation and to compare calculated and observed displacements. Information of the continental water volume was taken from the WaterGAP Global Hydrological Model. Calculated values of the deformations were verified with the results obtained with programs SPOTL and grat. Vertical deformations were almost 10 times higher than the deformation in the horizontal plane, for which reason later part of the paper focuses on the former. In order to check agreement of the calculated and observed deformation 23 stations of International GNSS Service (IGS) were selected and divided into three groups (inland, near the shoreline and islands). Before comparison outliers and discontinuities were removed from GNSS observations. Modelled and observed signals were centred. The analysed time series of the vertical displacements showed that only for the inland stations it is possible to effectively remove displacements caused by mass transfer in the hydrosphere. For stations located in the coastal regions or islands, it is necessary to consider additional movement effects resulting from indirect ocean tidal loading or atmosphere loading.

9

Vibration analysis of composite circular and annular membranes

Siedlecka U., Zamorska I., Kukla S.

Journal of Applied Mathematics and Computational Mechanics

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2016

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Vol. 15, nr 1

149--159

EN

In this paper a solution to the free vibration problem of composite circular and annular membranes is presented. The vibrations of membranes whose material densities and/or thicknesses varied step-wise with the radial co-ordinate are considered. This approach is applied to approximate the solution to the vibration problem of a membrane with continuously varying density and/or thickness with the radial co-ordinate. The obtained analytical solutions are used in numerical investigations into the effect of parameters characterizing the composite membranes on their eigenfrequencies.

10

Damage identification of a beam with a variable cross-sectional area

Cekus D., Miara M., Zamorska I.

Journal of Applied Mathematics and Computational Mechanics

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2016

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Vol. 15, nr 4

23--32

EN

In the paper, the process of identification of crack parameters occurring in the cantilever beam with the variable cross-sectional area has been presented. For identification, the non-destructive vibration method has been applied. The analytical solution of the free vibration problem of the beam described according to the Bernoulli-Euler theory has been obtained with the help of Green’s functions.

11

Green’s function approach to frequency analysis of thin circular plates

Żur K. K.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2016

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Vol. 64, nr 1

181--188

EN

The free vibration analysis of homogeneous and isotropic circular thin plates by using the Green’s functions is considered. The formulae for construction of the influence function for all nodal diameters are presented in a closed form. The limited independent solutions of differential Euler equations were expanded in the Neumann power series using the method of successive approximation. This approach allows to obtain the analytical frequency equations as power series rapidly convergent to exact eigenvalues for different number of nodal diameters. The first ten dimensionless frequencies for eight different natural modes of circular plates are calculated. A part of obtained results have not been presented yet in open literature for thin circular plates. The results of investigation are in good agreement with selected results obtained by other methods presented in literature.

12

Green’s function in free axisymmetric vibration analysis of annular thin plates with different boundary conditions

Żur K. K.

International Journal of Applied Mechanics and Engineering

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2015

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Vol. 20, no. 4

939--951

EN

Free vibration analysis of homogeneous and isotropic annular thin plates by using Green’s functions is considered. The formula of the influence function for uniform thin circular and annular plates is presented in closed-form. The limited independent solutions of differential Euler equation were expanded in the Neumann power series based on properties of integral equations. The analytical frequency equations as power series were obtained using the method of successive approximations. The natural axisymmetric frequencies for singularities when the core radius approaches zero are calculated. The results are compared with selected results presented in the literature.

13

Przegląd zastosowania metody interferometrii sejsmicznej

Czarny R.

Przegląd Górniczy

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2014

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T. 70, nr 7

84--91

PL

Artykuł w sposób przeglądowy przedstawia szerokie możliwości zastosowania interferometrii sejsmicznej. Metoda jest stosunkowo nowa, mając na uwadze pierwsze eksperymenty wykonane w 2001 roku, i aktualnie znajduje się w fazie dynamicznego rozwoju. W pracy omówiono jej zastosowanie w wielu zagadnieniach związanych z odwzorowaniem budowy wnętrza Ziemi oraz warstw przypowierzchniowych, a także zagadnień geoinżynierskich z np. monitorowania czynnych osuwisk. Przedstawione przykłady badań zostały opublikowane w uznanych czasopismach naukowych.

EN

The aim of this paper is to review the wide range of seismic interferometry applications. Despite the fact that only 10 years have passed since the first application was implemented in seismology, seismic interferometry is currently used in many issues beginning with analysis of the Earth´s deep interior and ending with the time-laps velocity changes at active landslides. The presented research results were published in leading scientific journals.

14

Możliwości wykorzystania interferometrii sejsmicznej w górnictwie

Marcak H., Pilecki Z., Isakow Z., Czarny R.

Przegląd Górniczy

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2014

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T. 70, nr 7

74--83

PL

Metoda interferometrii sejsmicznej znalazła zastosowanie w zagadnieniach rozpoznawania budowy geologicznej głębokiego podłoża i w sejsmice poszukiwawczej do odwzorowania budowy ośrodka. Istnieje również możliwość wykorzystania tej metody dla potrzeb górniczych. W części wstępnej przedstawiono podstawy matematyczne i fizyczne metody interferometrii sejsmicznej. Następnie omówiono możliwości wykorzystania tej metody w rozwiązywaniu problemów geologiczno-górniczych. Scharakteryzowano rodzaje zjawisk sejsmicznych, występujących na terenach górniczych, możliwych do wykorzystania w interferometrii sejsmicznej. Omówiono przykłady sytuacji geologiczno-górniczych, w których metoda interferometrii sejsmicznej może dostarczyć wartościowej informacji.

XX

Seismic interferometry method has been used to identify the deep geological structure of the ground and seismic exploration for mapping the structure medium. It is also possible to use this method for mining purposes. In the introductory part we present the mathematical foundations and physical methods of seismic interferometry. It then discusses the possibility of using this method in solving geological-mining tasks. Seismic phenomena occurring in mining areas and suitable for application in seismic interferometry have been described. Examples of geological and mining situation in which seismic interferometry method can provide valuable information have been presented.

15

Radial heat conduction in a multilayered sphere

Siedlecka U.

Journal of Applied Mathematics and Computational Mechanics

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2014

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Vol. 13, nr 4

109--116

EN

In this paper, an exact analytical solution of the radial heat conduction problem in a hollow multilayered sphere is presented. The boundary conditions of the third kind and perfect contact at the interfaces are assumed. The temperatures are obtained by using the Green’s function method. The Green’s functions are expressed in the form of the series of appropriate eigenfunctions.

16

Green's function for heat conduction problems in a multi-layered hollow cylinder

Kukla S., Siedlecka U.

Journal of Applied Mathematics and Computational Mechanics

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2014

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Vol. 13, nr 3

115--122

EN

In this paper, derivation of the Green’s function for the heat conduction problems in a finite multi-layered hollow cylinder is presented. Formulation and solution of the problem includes an arbitrary number of the cylinder layers characterized by various thermal properties. At the interfaces perfect thermal contact was assumed. The Green’s function for the three-dimensional heat conduction problems in the cylindrical coordinate has been presented in the form of a product of two other Green’s functions.

17

Frequency analysis of a double-walled nanotubes system

Ciekot A., Kukla S.

Journal of Applied Mathematics and Computational Mechanics

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2014

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Vol. 13, nr 4

27--34

EN

In the present paper, a problem of free vibration of a double-walled nanotubes system is considered. The nanotubes of the system are coupled by an arbitrary number of elastic rings. The Green’s functions method was used to obtain a solution to the problem. Numerical results are presented graphically.

18

Frequency analysis of a double-nanobeam system

Ciekot A., Kukla S.

Journal of Applied Mathematics and Computational Mechanics

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2014

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Vol. 13, nr 1

23--31

EN

In this paper, a problem of transverse free vibration of a double-nanobeam-system is considered. The nanobeams of the system are coupled by an arbitrary number of translational springs. The solution of the problem by using the Green’s functions properties is obtained. A numerical example is presented.

19

Application of a Green’s function method to heat conduction problems in multi-layered cylinders

Siedlecka U., Kukla S.

Journal of Applied Mathematics and Computational Mechanics

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2013

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Vol. 12, nr 3

105--113

EN

In this paper, the Green’s functions which are used in solving the heat conduction problem in a composite cylinder are derived. The functions are obtained by the solution of one dimensional eigenproblems and are presented in the form of eigenfunction series. The temperature in the cylinder as a function of time and space coordinates are expressed by the Green’s functions. A numerical example is presented.

20

Green's functions for interior and exterior Helmholtz problems

Kukla S., Siedlecka U., Zamorska I.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2012

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Vol. 11, nr 1

53-62

EN

In the paper, the derivation of Green's functions for Helmholtz equation in circular, annular and exterior circular domains is presented. The Green's functions are assumed of the form a cosine series. An example of application of the Green's functions in frequency analysis of a composite membrane is presented.