Laplace transform

About: Laplace transform is a research topic. Over the lifetime, 17104 publications have been published within this topic receiving 284357 citations.

On some applications of the space-time fractional derivative

Abstract: In this work, we study and extend the one-dimensional fractional derivative to the multidimensional space-time fractional derivative, determine the exact solution via the Laplace transform, and develop mathematical foundations of the respective operators. The multidimensional space-time fractional derivative is developed to augment the equations. The results show that this method is effective, convenient, and easily executable. The main advantage of the proposed approach is that it is an accurate analytical method (the Laplace transform method) that can be implemented for both space and time discretizations of the fractional derivatives and allows us to present new solutions to problems by certain applications for solving space-time fractional derivatives.

The Buffon–Laplace needle problem in three dimensions

Abstract: We generalize the Buffon?Laplace needle problem, one of the earliest statistical calculations, to three dimensions, account for finite widths, and find the probability for particles of arbitrary length l to make contact with a 2D sieve-like mesh. We assume a frictionless interaction with the mesh to account for particle rotations that occur after contact, and find that the probability for a stable intersection goes to 1 for long particle lengths as P(l) = 1?(0.5 ? 0.02)l?1.00 ? 0.01. This approach represents the first step toward modeling the efficacy of a square-grid mesh for filtering rod-like granular particles from solution, assuming that the volume fraction is kept low, so particles cannot cooperatively jam. We argue that our approach results in an upper bound on the filtration probability and confirm this with comparison to a simple experiment in which cylindrical particles are filtered with a square-hole mesh.

Several representations of the euler-cauchy equation with respect to integral transforms

Abstract: The subjects on the solution of differential equations with variable coefficients have aroused the interest of many researchers. Existing many in- tegral transforms are playing a role to get the solution of it. In this sense, we have researched about several forms of Euler-Cauchy equation, using Laplace transform. Related to the topic, the proposed idea can be also applied to other transforms (Sumudu/Elzaki, etc.).

Generation of internal and surface waves by seafloor movement in a two layer fluid system

Abstract: Internal and surface waves generated by the deformations of the solid bed in a two layer fluid system of infinite lateral extent and uniform depth are investigated. An integral solution is developed for an arbitrary bed displacement on the basis of a linear approximation of the complete description of wave motion using a transform method (Laplace in time and Fourier in space) analogous to that used to study the generation of tsunamis by many researchers. The theoretical solutions are presented for three interesting specific deformations of the seafloor; the spatial variation of each seafloor displacement consists of a block section of the seafloor moving vertically either up or down while the time-displacement history of the block section is varied. The generation process and the profiles of the internal and surface waves for the case of the exponential bed movement are numerically illustrated, and the effects of the deformation parameters, densities and depths of the two layers on the solutions are discussed. As expected, the solutions derived from the present work include as special cases that obtained by Kervella et al. [Theor Comput Fluid Dyn 21:245-269, 2007] for tsunamis cased by an instantaneous seabed deformation and those presented by Hammack [J Fluid Mech 60:769-799, 1973] for the exponential and the half-sine bed displacements when the density of the upper fluid is taken as zero.