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1

Continuous-state branching processes with migration

Vidmar Matija

Probability and Mathematical Statistics

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2022

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Vol. 42, Fasc. 2

227--249

EN

Continuous-state branching processes (CSBPs) with immigration (CBIs), stopped on hitting zero, are generalized by allowing the process governing immigration to be any Lévy process without negative jumps. Unlike CBIs, these newly introduced processes do not appear to satisfy any natural affine property on the level of the Laplace transforms of the semigroups. Basic properties of these processes are described. Explicit formulae (on neighborhoods of infinity) for the Laplace transforms of the first passage times downwards and of the explosion time are derived.

2

Exact solution for heat conduction inside a sphere with heat absorption using the regularized Hilfer-Prabhakar derivative

Elhadedy Hager, Latif Mohamed S. Abdel, Nour Hamed M., Kader Abbas H. Abdel

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 2

27--37

EN

In this article, we utilize the finite Sine-Fourier transform and the Laplace transform for solving fractional partial differential equations with regularized Hilfer-Prabhakar derivative. These transforms are used to get analytical solutions for the time fractional heat conduction equation (TFHCE) with the regularized Hilfer-Prabhakar derivative associated with heat absorption in spherical coordinates. Two cases of Dirichlet boundary conditions are considered by obtaining an analytical solution in each case. The effect of the parameters of the regularized Hilfer-Prabhakar derivative on the heat transfer inside the sphere is discussed using some figures.

3

Approximate analytical solution to the Cattaneo heat conduction model with various laser sources

Nuruddeen Rahmatullah Ibrahim

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 1

67--78

EN

The present manuscript investigates the role being played by various laser short heating sources in a conduction process of a metallic substrate. The Cattaneo heat conduction model is considered in favour of its finiteness of conduction speed. The analytical solutions for the temperature fields are determined via the application of the Laplace integral transform. Finally, we sought a numerical Laplace inversion scheme where the analytical inversion failed and graphically examined the significance of the heating parameters on the temperature fields.

4

Analysis of deflection in visco-thermoelastic beam resonators subjected to harmonic loading

Chopra Deepti, Singh Prince

International Journal of Applied Mechanics and Engineering

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2022

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Vol. 27, no. 2

35--52

EN

This paper analyses the transverse deflection in a homogeneous, isotropic, visco-thermoelastic beam when subjected to harmonic load. The ends of the beam are considered at different boundary conditions (both axial ends clamped, both axial ends simply supported and left end clamped and right end free). The deflection has been studied by using the Laplace transform. Numerical computation of analytical expression of deflection obtained after Inverse Laplace transform has been done using MATLAB software. The graphical observations have been discussed under various boundary conditions for different values of time and length. The above work has applications in design of resonators.

5

Modified Laplace based variational iteration method for the mechanical vibrations and its applications

Rehman Shahida, Hussain Akhtar, Rahman Jamshaid Ul, Anjum Naveed, Munir Taj

Acta Mechanica et Automatica

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2022

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Vol. 16, no. 2

98--102

EN

In this paper, we are putting forward the periodic solution of non-linear oscillators by means of variational iterative method (VIM) using Laplace transform. Here, we present a comparative study of the new technique based on Laplace transform and the previous tech-niques of maximum minimum approach (MMA) and amplitude frequency formulation (AFF) for the analytical results. For the non-linear oscillators, MMA, AFF and VIM by Laplace transform give the same analytical results. Comparison of analytical results of VIM by Laplace transform with numerical results by fourth-order Runge–Kutta (RK) method conforms the soundness of the method for solving non-linear oscillators as well as for the time and boundary conditions of the non-linear oscillators.

6

On Mixtures of Gamma Distributions, Distributions with Hyperbolically Monotone Densities and Generalized Gamma Convolutions (GGC)

Sjödin Tord

Probability and Mathematical Statistics

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2021

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Vol. 41, Fasc. 1

1--7

EN

Let Y be a standard Gamma(k) distributed random variable (rv), k > 0, and let X be an independent positive rv. If X has a hyperbolically monotone density of order k (HMk), then Y · X and Y/X are generalized gamma convolutions (GGC). This extends work by Roynette et al. and Behme and Bondesson. The same conclusion holds with Y replaced by a finite sum of independent gamma variables with sum of shape parameters at most k. Both results are applied to subclasses of GGC.

7

Asymptotic expansions for the first hitting times of bessel processes

Hamana Yuji, Kaikura Ryo, Shinozaki Kosuke

Opuscula Mathematica

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2021

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Vol. 41, no. 4

509--537

EN

We study a precise asymptotic behavior of the tail probability of the first hitting time of the Bessel process. We deduce the order of the third term and decide the explicit form of its coefficient.

8

Fourier, Laguerre, Laplace Transforms with applications

Aghili Arman

Journal of Mathematics and Applications

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2021

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Vol. 44

5--17

EN

In this article, the author considered certain time fractional equations using joint integral transforms. Transform method is a powerful tool for solving singular integral equations, integral equation with retarded argument, evaluation of certain integrals and solution of partial fractional differential equations. The obtained results reveal that the transform method is very convenient and effective. Illustrative examples are also provided.

9

Analysis of solutions of the 1D fractional Cattaneo heat transfer equation

Siedlecka Urszula, Ciesielski Mariusz

Journal of Applied Mathematics and Computational Mechanics

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2021

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Vol. 20, nr 4

87--98

EN

In this paper, a solution of the single-phase lag heat conduction problem is presented. The research concerns the generalized 1D Cattaneo equation in a whole-space domain, where a second order time derivative is replaced by the fractional Caputo derivative. The Fourier-Laplace transform technique is used to determine a solution of the considered problem. The numerical inversion of the Laplace transforms is applied. The effect of the order of the fractional derivative on the temperature distribution is investigated.

10

A method for Soft Fault Diagnosis of Linear Analog Circuits Using the Laplace Transform Technique

Tadeusiewicz Michał, Ossowski Marek, Korzybski Marek

International Journal of Electronics and Telecommunications

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2021

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Vol. 67, No. 3

531--536

EN

This paper is focused on multiple soft fault diagnosis of linear time-invariant analog circuits and brings a method that achieves all objectives of the fault diagnosis: detection, location, and identification. The method is based on a diagnostic test arranged in the transient state, which requires one node accessible for excitation and two nodes accessible for measurement. The circuit is specified by two transmittances which express the Laplace transform of the output voltages in terms of the Laplace transform of the input voltage. Each of these relationships is used to create an overdetermined system of nonlinear algebraic equations with the circuit parameters as the unknown variables. An iterative method is developed to solve these equations. Some virtual solutions can be eliminated comparing the results obtained using both transmittances. Three examples are provided where laboratory or numerical experiments reveal effectiveness of the proposed method.

11

Queueing systems with random volume customers and a sectorized unlimited memory buffer

Tikhonenko Oleg, Ziółkowski Marcin, Kempa Wojciech M.

International Journal of Applied Mathematics and Computer Science

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2021

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Vol. 31, no. 3

471--486

EN

In the present paper, we concentrate on basic concepts connected with the theory of queueing systems with random volume customers and a sectorized unlimited memory buffer. In such systems, the arriving customers are additionally characterized by a non-negative random volume vector. The vector’s indications can be understood as the sizes of portions of information of a different type that are located in the sectors of memory space of the system during customers’ sojourn in it. This information does not change while a customer is present in the system. After service termination, information immediately leaves the buffer, releasing its resources. In analyzed models, the service time of a customer is assumed to be dependent on his volume vector characteristics, which has influence on the total volume vector distribution. We investigate three types of such queueing systems: the Erlang queueing system, the single-server queueing system with unlimited queue and the egalitarian processor sharing system. For these models, we obtain a joint distribution function of the total volume vector in terms of Laplace (or Laplace-Stieltjes) transforms and formulae for steady-state initial mixed moments of the analyzed random vector, in the case when the memory buffer is composed of two sectors. We also calculate these characteristics for some practical case in which the service time of a customer is proportional to the customer’s length (understood as the sum of the volume vector’s indications). Moreover, we present some numerical computations illustrating theoretical results.

12

Generalized thermoelasticity model of nonlocal strain gradient Timoshenko nanobeams

Yue Xuejie, Yue Xuezheng, Borjalilou Vahid

Archives of Civil and Mechanical Engineering

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2021

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Vol. 21, no. 3

655--674

EN

The aim of this study is to establish a thorough model for appraisal of size-dependent thermoelastic vibrations of Timoshenko nanobeams by capturing small-scale effect on both structural and thermal fields. With the intention of incorporating size effect within motion and heat conduction equations, nonlocal strain gradient theory (NSGT) as well as nonclassical heat conduction model of Guyer and Krumhansl (GK model) are exploited. For the sake of generalization and clarifying the impact of nonclassical scale parameters on results, by introducing some nondimensional quantities, the size-dependent coupled thermoelastic equations are written in dimensionless form. By applying the Laplace transform to this system of differential equations, thermoelastic responses of a simply supported Timoshenko nanobeam under dynamic load are extracted in closed forms. In order to highlight the influence of scale parameters on thermoelastic behavior of Timoshenko nanobeams, a variety of numerical results is provided. The discrepancy between classical and nonclassical outcomes betokens the salient role of structural and thermal scale parameters in accurate analysis of nanobeams. In addition, findings reveal that utilization of NSGT gives the means to capture both stiffness softening and stiffness enhancement characteristic of small-sized structures, so that according to the relative values of two scale parameters of NSGT, the nonclassical model of Timoshenko nanobeam can exhibit either softening or hardening behavior in comparison with the classical one.

13

Laplace-Carson integral transform for exact solutions of non-integer order initial value problems with Caputo operator

Kumar Prem, Qureshi Sania

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 1

57--66

EN

Finding the exact solution to dynamical systems in the field of mathematical modeling is extremely important and to achieve this goal, various integral transforms have been developed. In this research analysis, non-integer order ordinary differential equations are analytically solved via the Laplace-Carson integral transform technique, which is a technique that has not been previously employed to test the non-integer order differential systems. Firstly, it has proved that the Laplace-Carson transform for n-times repeated classical integrals can be computed by dividing the Laplace-Carson transform of the underlying function by n-th power of a real number p which later helped us to present a new result for getting the Laplace-Carson transform for d-derivative of a function under the Caputo operator. Some initial value problems based upon Caputo type fractional operator have been precisely solved using the results obtained thereof.

14

Theoretical studies of dynamic soil compaction by wheeled forestry machines

Grigorev Igor, Kunickaya Ol'ga, Burgonutdinov Albert, Ivanov Viktor, Shuvalova Svetlana, Shvetsova Viktoria, Stepanishcheva Marina, Tikhonov Evgeniy

Diagnostyka

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2020

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Vol. 21, No. 4

3--13

EN

The impact of forest skidding machine tires on the soil differs depending on topography, soil properties, and type of the wheel system. The development of a mathematical model describing the entire dynamic process is a challenging but relevant task to assess the level of impact. The work aims mathematical modeling of the impact caused by the skidding system on the forest soil employing Kelvin-Voigt theory with additional elastic element and Laplace transform equations. A dynamic model represents the "traktor-timber bundle– soil" system. According to the results of mathematical modeling, it was found that studying the vertical vibrations and vibrations of sprung mass in longitudinal and transverse planes is sufficient for examining dynamic soil compaction. Developed methods of statistical dynamics with the presentation of the track surface microroughness and the theory of linear elastic and viscous soil deformation showed that each pass of the skidding system is ac-companied by additional dynamic soil compaction. Its maximum value depends on the properties of the soil and skidding system, as well as on the presence of resonant zones in the frequency spectrum. The results of these studies provide an opportunity to predict the exposure level of skidders and establish new solutions to minimize negative consequences for the environment and productivity of the forest industry.

15

Acoustic propagation in inhomogeneous fluids: regularization via the introduction of fine particles

Jordan P. M.

Archives of Mechanics

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2020

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Vol. 72, nr 1

59--73

EN

It is shown, using analytical methodologies, that the velocity field blow-up suffered by vertically ascending acoustic waves in an isothermal atmosphere can be eliminated via the introduction of fine particles. Assuming the inhomogeneous generalization of the particle-laden flow model known as the (linearized) Marble–Thompson model-1, it is established that bounded, exponentially decreasing, shock amplitudes can be obtained provided the mass fraction of particles exceeds a critical value, for which an exact expression is derived. Lastly, supporting numerical results are presented, special cases are discussed, and possible follow-on studies are noted.

16

Resonance of nanoscale beam due to various sources in modified couple stress thermoelastic diffusion with phase lags

Kumar Rajneesh, Devi Shaloo, Sharma Veena

Mechanics and Mechanical Engineering

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2019

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Vol. 23, nr 1

36--49

EN

This paper deals with the study of thermoelastic thin beam in a modified couple stress with three-phaselag thermoelastic diffusion model subjected to thermal and chemical potential sources. The governing equations are derived by using the Euler-Bernoulli beam assumption and eigenvalue approach. The Laplace transform technique is employed to obtain the expressions for displacements, lateral deflection, temperature change, axial stress and chemical potential. A particular type of instantaneous and distributed sources is taken to show the utility of the approach. The general algorithm of the inverse Laplace transform is developed to compute the results numerically. The numerical results are depicted graphically to show the effects of phase lags, with and without energy dissipation on the resulting quantities. Some special cases are given.

17

Using Shehu integral transform to solve fractional order Caputo type initial value problems

Qureshi Sania, Kumar Prem

Journal of Applied Mathematics and Computational Mechanics

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2019

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Vol. 18, nr 2

75--83

EN

In the present research analysis, linear fractional order ordinary differential equations with some defined condition (s) have been solved under the Caputo differential operator having order α > 0 via the Shehu integral transform technique. In this regard, we have presented the proof of finding the Shehu transform for a classical nth order integral of a piecewise continuous with an exponential nt h order function which leads towards devising a theorem to yield exact analytical solutions of the problems under investigation. Varying fractional types of problems are solved whose exact solutions can be compared with solutions obtained through existing transformation techniques including Laplace and Natural transforms.

18

Phase-lag effects in skin tissue during transient heating

Kumar R., Vashishth A. K., Ghangas S.

International Journal of Applied Mechanics and Engineering

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2019

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Vol. 24, no. 3

603--623

EN

A three-phase-lag (TPL) model is proposed to describe heat transfer in a finite domain skin tissue with temperature dependent metabolic heat generation. The Laplace transform method is applied to solve the problem. Three special types of heat flux are applied to the boundary of skin tissue for thermal therapeutic applications. The depth of tissue is influenced by the different oscillation heat flux. The comparison between the TPL and dual-phase-lag (DPL) models is analyzed and the effects of phase lag parameters […] and material constant […] on the tissue temperature distribution are presented graphically.

19

Metoda równań różnicowych dla opisu stanu nieustalonego obwodu szeregowego RLC załączonego do napięcia sinusoidalnego z użyciem programu MathCAD

Frączak Piotr Stanisław, Czajkowski Andrzej Antoni

Problemy Nauk Stosowanych

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2018

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T. 9

33--40

PL

Wstęp i cel: W pracy przedstawiono opis i symulacje stanu nieustalonego w obwodzie elektrycznym szeregowym RLC. Pokazano zastosowanie metody równań różnicowych do rozwiązywania równania różniczkowego drugiego rzędu w programie MathCAD. Materiał i metody: W wyniku zastosowania metody równań różnicowych wskazano na możliwość przejścia od równań różniczkowych liniowych drugiego rzędu o stałych współczynnikach. Zastosowano metodę analityczno-numeryczną. W analizie numerycznej użyto program MathCAD. Wyniki: Otrzymano jednakowy kształt przebiegu krzywej prądu nieustalonego przy wyznaczaniu metodą równań różnicowych drugiego rzędu i równaniem różniczkowo-całkowym z wykorzystaniem przekształcenia odwrotnego Laplace’a. Ponadto otrzymane kształty prądów nieustalonych w rozpatrywanym obwodzie elektrycznym zweryfikowano w programie numerycznym PSpice Wniosek: Stosując zarówno metodę równań różnicowych i metodę przekształceń Laplace’a otrzymuje się jednakowe przebiegi prądu nieustalonego w funkcji czasu.

EN

Introduction and aim: Some description and simulation of the transient in RLC circuit have been presented in this paper. Also has been shown the application of the Laplace transform to solve the differential equation. Material and methods: By using the Laplace transformation to the option of the transition from linear differential equations of the second order with constant coefficients to the algebraic equations. The analytical and numerical methods have been used in the considerations. In numerical analysis, a reversed Laplace transform was applied by using the MathCAD program. Results: It has been obtained the same curve shape of the transient current at the determination by the second-order differential equation (classical solution) and the different-integral equation by using the inverse Laplace transform. In addition, the obtained shapes of transients in the considered electrical circuit were verified in the numerical program PSpice Conclusion: By applying both the Laplace transform method and the analytical method, the same transient currents are obtained as a function of time.

20

Certain Laplace transforms of convolution type integrals involving product of two special pFp functions

Milovanović G. V., Parmar R. K., Rathie A. K.

Demonstratio Mathematica

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2018

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Vol. 51, nr 1

264--276

EN

Recently the authors obtained several Laplace transforms of convolution type integrals involving Kummer’s function 1F1 [Appl. Anal. Discrete Math., 2018, 12(1), 257–272]. In this paper, the authors aim at presenting several new and interesting Laplace transforms of convolution type integrals involving product of two special generalized hypergeometric functions pFp by employing classical summation theorems for the series 2F1, 3F2, 4F3 and 5F4 available in the literature.