The basic three-dimensonal boundary-contact dynamic problems are considered for a piecewise-homogeneous isotropic elastic medium bounded by several closed surfaces. Using the Fourier method, the considered problems are proved to be solvable under much weaker restrictions on the initial data of the problems as compared with other methods. 1. Two well-known methods – the Laplace transform and the...

In a rectangular domain, boundary-value problem is considered for mixed-type equation with regularized Caputo-like counterpart of hyper-Bessel differential operator and the bi-ordinal Hilfer's fractional derivative. Using method separation variables, Laplace transform, unique solvability has been established. Moreover, we have found explicit solution initial problems derivative non-zero startin...

In this dissertation, we first focus on the generalized Laplace transform on time scales. We prove several properties of the generalized exponential function which will allow us to explore some of the fundamental properties of the Laplace transform. We then give a description of the region in the complex plane for which the improper integral in the definition of the Laplace transform converges,...

We study stability of subsonic phase boundary solutions in the Suliciu model for phase transitions under tri-linear structural relation. With the help of Laplace transform, the evolution of perturbation is described by a linear dynamical system, and explicit solution is obtained in terms of inverse Laplace transform. Stability is established through energy estimates. The relaxed system is also ...

The Laplace transform theory of Banach space valued functions and the field of evolution equations presently offer many research problems which are of general interest in mathematical analysis. Although there is a longstanding relationship between the two fields, it was not until recently that the basic transform principles for Banach space valued functions could be formulated in a satisfactory...

In this study, shear and peel stress distributions in the viscoelastic adhesive layer of a single-lap joint (SLJ) with functionally graded (FG) adherends are investigated. The study focuses on the effect of different adherend profiles and material composition on the time-dependent stress concentration/distribution in balanced and unbalanced SLJs. For this purpose, the Reddy model is applied to ...

The generalized Stieltjes transform (GST) is an integral transform that depends on a parameter ρ > 0. In previous work a convenient form of the inverse transformation was derived for the case ρ = 3/2. This paper generalizes that result to all ρ > 0. It is a well-known fact that the GST can be formulated as an iterated Laplace transform, and that therefore its inverse can be expressed as an iter...

The roller bearing characteristic frequencies contain very little energy, and are usually overwhelmed by noise and higher levels of structural vibrations. Therefore, envelope spectrum analysis is widely used to detection bearing localized fault. In order to overcome the shortcomings in the traditional envelope analysis in which manually specifying a resonant frequency band is required, a new ap...

The purpose of this document is to summarize main points of the paper, “Numerical Inversion of Laplace Transforms of Probability Distributions”, and provide R code for the Euler method that is described in the paper. The code is used to invert the Laplace transform of some popular functions. Context Laplace transform is a useful mathematical tool that one must be familiar with, while doing appl...

This note presents the explicit form of the Laplace transform of the state transition matrix in the case of a system matrix being a companion matrix. Starting from this special case, we show how to compute the Laplace transform of the state transition matrix for any system matrix.1