The Laplace transform induced by the deformed exponential function of two variables

2018 ◽  
Vol 21 (3) ◽  
pp. 775-785
Author(s):  
Predrag M. Rajković ◽  
Miomir S. Stanković ◽  
Sladjana D. Marinković
Keyword(s):  
Laplace Transform ◽  
Integral Equations ◽  
Kernel Function ◽  
Exponential Function ◽  
Liouville Type ◽  
Type Integral ◽  
The Laplace Transform ◽  
Function Of Two Variables ◽  
Differential And Integral Equations

Abstract Based on the easy computation of the direct transform and its inversion, the Laplace transform was used as an effective method for solving differential and integral equations. Its various generalizations appeared in order to be used for treating some new problems. They were based on the generalizations and deformations of the kernel function and of the notion of integral. Here, we expose our generalization of the Laplace transform based on the so-called deformed exponential function of two variables. We point out on some of its properties which hold on in the same or similar manner as in the case of the classical Laplace transform. Relations to a generalized Mittag-Leffler function and to a kind of fractional Riemann-Liouville type integral and derivative are exhibited.

scholarly journals Analytical solutions of some general classes of differential and integral equations by using the Laplace and Fourier transforms

Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 2869-2876
Author(s):  
H.M. Srivastava ◽  
Mohammad Masjed-Jamei ◽  
Rabia Aktaş
Keyword(s):  
Fourier Transform ◽  
Laplace Transform ◽  
Integral Equations ◽  
Analytical Solutions ◽  
General Class ◽  
Fourier Transforms ◽  
Laplace And Fourier Transforms ◽  
The Laplace Transform ◽  
The Fourier Transform ◽  
Differential And Integral Equations

This article deals with a general class of differential equations and two general classes of integral equations. By using the Laplace transform and the Fourier transform, analytical solutions are derived for each of these classes of differential and integral equations. Some illustrative examples and particular cases are also considered. The various analytical solutions presented in this article are potentially useful in solving the corresponding simpler differential and integral equations.

scholarly journals The Solutions of Some Riemann–Liouville Fractional Integral Equations

2021 ◽  
Vol 5 (4) ◽  
pp. 154
Author(s):  
Karuna Kaewnimit ◽  
Fongchan Wannalookkhee ◽  
Kamsing Nonlaopon ◽  
Somsak Orankitjaroen
Keyword(s):  
Laplace Transform ◽  
Integral Equations ◽  
Exponential Function ◽  
Fractional Integral ◽  
Laplace Transform Technique ◽  
The Laplace Transform

In this paper, we propose the solutions of nonhomogeneous fractional integral equations of the form I0+3σy(t)+a·I0+2σy(t)+b·I0+σy(t)+c·y(t)=f(t), where I0+σ is the Riemann–Liouville fractional integral of order σ=1/3,1,f(t)=tn,tnet,n∈N∪{0},t∈R+, and a,b,c are constants, by using the Laplace transform technique. We obtain solutions in the form of Mellin–Ross function and of exponential function. To illustrate our findings, some examples are exhibited.

Asymptotic solution of the Vlasov and Poisson equations for an inhomogeneous plasma

1991 ◽  
Vol 45 (1) ◽  
pp. 59-70 ◽  
Author(s):  
Riccardo Croci
Keyword(s):  
Magnetic Field ◽  
Laplace Transform ◽  
Integral Equations ◽  
Initial Conditions ◽  
Inhomogeneous Plasma ◽  
Algebraic Equations ◽  
Poisson Equations ◽  
The Laplace Transform ◽  
Eigenvalue Parameter ◽  
The Fourier Transform

The purpose of this paper is to derive the asymptotic solutions to a class of inhomogeneous integral equations that reduce to algebraic equations when a parameter ε goes to zero (the kernel becoming proportional to a Dirac δ function). This class includes the integral equations obtained from the system of Vlasov and Poisson equations for the Fourier transform in space and the Laplace transform in time of the electrostatic potential, when the equilibrium magnetic field is uniform and the equilibrium plasma density depends on εx, with the co-ordinate z being the direction of the magnetic field. In this case the inhomogeneous term is given by the initial conditions and possibly by sources, and the Laplace-transform variable ω is the eigenvalue parameter.

Diffraction by a wave-guide of finite length

1952 ◽  
Vol 48 (1) ◽  
pp. 118-134 ◽  
Author(s):  
D. S. Jones
Keyword(s):  
Electromagnetic Wave ◽  
Approximate Solution ◽  
Laplace Transform ◽  
Integral Equations ◽  
Finite Length ◽  
Wave Guide ◽  
The Laplace Transform ◽  
The Difference

AbstractWhen the electric intensities on two parallel planes, of which the two perfectly conducting sides of a wave-guide of finite length and infinite width are portions, are taken as unknowns, the problem of the diffraction of a plane harmonic electromagnetic wave polarized parallel to the edges of the guide leads to two integral equations. By means of the Laplace transform these equations are converted into others suitable for solution by successive substitutions. The series thus obtained is too complex for practical purposes, and so an approximate solution is found for the case when the length of the guide is large compared with the wavelength. Finally, there is a brief discussion of the difference between the distant fields when l is large and when l is infinite.

scholarly journals A generalized Meijer transformation

1985 ◽  
Vol 8 (2) ◽  
pp. 359-365 ◽  
Author(s):  
G. L. N. Rao ◽  
L. Debnath
Keyword(s):  
Laplace Transform ◽  
Kernel Function ◽  
Distributional Sense ◽  
The Laplace Transform

In a series of papers [1-6], Kratzel studies a generalized version of the classical Meijer transformation with the Kernel function(st)νη(q,ν+1; (st)q). This transformation is referred to as GM transformation which reduces to the classical Meijer transform whenq=1. He also discussed a second generalization of the Meijer transform involving the Kernel functionλν(n)(x)which reduces to the Meijer function whenn=2and the Laplace transform whenn=1. This is called the Meijer-Laplace (or ML) transformation. This paper is concerned with a study of both GM and ML transforms in the distributional sense. Several properties of these transformations including inversion, uniqueness, and analyticity are discussed in some detail.

scholarly journals The Study of Generalized Synchronization between Two Identical Neurons Based on the Laplace Transform Method

2021 ◽  
Vol 11 (24) ◽  
pp. 11774
Author(s):  
Bin Zhen ◽  
Ran Liu
Keyword(s):  
Laplace Transform ◽  
Integral Equations ◽  
Time Difference ◽  
Coupling Function ◽  
Successive Approximation Method ◽  
Generalized Synchronization ◽  
Laplace Transform Method ◽  
Transform Method ◽  
The Laplace Transform ◽  
Analytical Criterion

In this paper, a new method is proposed based on the auxiliary system approach to investigate generalized synchronization between two identical neurons with unidirectional coupling. Different from other studies, the synchronization error system between the response and auxiliary systems is converted into a set of Volterra integral equations according to the Laplace transform method and convolution theorem. By using the successive approximation method in the theory of integral equations, an analytical criterion for the detection of generalized synchronization between two identical neurons is obtained. It is found that there is a time difference between two signals of neurons when the generalized synchronization between them is achieved. Furthermore, the value of the time difference has no relation to the generalized synchronization condition but depends on the coupling function between two neurons. The study in this paper shows that one can construct a coupling function between two identical neurons using the current signal of the drive system to predict its future signal or make its past signal reappear.

scholarly journals Alenezi Transform–A New Transform to Solve Mathematical Problems

2021 ◽  
pp. 1-8
Author(s):  
Ahmad M. Alenezi
Keyword(s):  
Laplace Transform ◽  
Integral Equations ◽  
Integral Transform ◽  
The Other ◽  
The Laplace Transform ◽  
Mathematical Problems ◽  
Sumudu Transforms

In this paper, we present a new integral transform called Alenezi-transform in the category of Laplace transform. We investigate the characteristic of Alenezi-transform. We discuss this transform with the other transforms like J, Laplace, Elzaki and Sumudu transforms. We can demonstrate that Alenezi transforms are near to the condition of the Laplace transform. We can explain the new Properties of transforms using Alenezi transform. Alenezi transform can be applied to solve differential, Partial and integral equations.

scholarly journals Determination of stress concentration near the holes under dynamic loadings

2021 ◽  
pp. 19-24
Author(s):  
O Maksymovych ◽  
T Solyar ◽  
A Sudakov ◽  
I Nazar ◽  
M Polishchuk
Keyword(s):  
Stress Concentration ◽  
Laplace Transform ◽  
Integral Equations ◽  
Dynamic Loads ◽  
Initial Time ◽  
Structural Elements ◽  
Dynamic Problems ◽  
Dynamic Stresses ◽  
Method Of Integral Equations ◽  
The Laplace Transform

Purpose. To develop an approach for determining the stress state of plate structural elements with holes under dynamic loads with controlled accuracy. Methodology. The study was carried out on the basis of the Laplace transform and the method of integral equations. Findings. An approach to determining the dynamic stresses at the holes in the plates is proposed, which includes: the Laplace transform in the time coordinate; a numerical method for determining transformants of displacements and stresses based on the method of integral equations; finding originals on the basis of Prudnikovs formula adapted to dynamic problems of elasticity theory. The problem of determining the Laplace images for displacements is reduced to solving singular integral equations. Integral equations were solved numerically based on the approaches developed in the boundary element method. To find displacements and stresses, the Laplace transform inversion formulas proposed by Prudnikov are adapted to dynamic problems. The study on dynamic stresses at holes of various shapes was carried out. Originality. A new approach to the regularization of the Prudnikov formula for inverting the Laplace transform as applied to dynamic problems of the theory of elasticity has been developed. For its implementation: convergence of Fourier series based on pre-set stresses at the initial time is improved; the remainder is taken into account in the conversion formula. Practical value. A method has been developed for calculating the stress concentration at holes of arbitrary shape in lamellar structural elements under dynamic loads. The proposed approach makes it possible to determine stresses with controlled accuracy. The studies performed at circular and polygonal holes with rounded tops can be used in strength calculations for dynamically loaded plates. The influence of Poissons ratio on the concentration of dynamic stresses is analyzed.

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