Laplace Transform Inversion Algorithm Research Articles

Solving the Nonlinear Two-Dimension Wave Equation Using Dual Reciprocity Boundary Element Method

The boundary element method (BEM) is a very effective numerical tool which has been widely applied in engineering problems. Wave equation is a very important equation in applied mathematics with many applications such as wave propagation analysis, acoustics, dynamics, health monitoring and etc. This paper presents to solve the nonlinear 2-D wave equation defined over a rectangular spatial domain with appropriate initial and boundary conditions. Numerical solutions of the governing equations are obtained by using the dual reciprocity boundary element method (DRBEM). Two-dimension wave equation is a time-domain problem, with three independent variables . At the first the Laplace transform is used to reduce by one the number of independent variables (in the present work ), then Salzer's method which is an effective numerical Laplace transform inversion algorithm is used to recover the solution of the original equation at time domain. The present method has been successfully applied to 2-D wave equation with satisfactory accuracy.

Vibro-acoustic analysis of a coach platform under random excitation

Vibro-acoustic analysis of a rail vehicle cabin is presented in this paper. Vehicle is modeled by an air cavity coupled to a flexible floor panel. Analytical procedure is employed to predict the structural-borne noise in the vehicle model generated by random excitation of the panel. In this study, natural frequencies of the coupled system are obtained and then the sound pressure field inside the cavity is analytically determined. In order to find dynamic responses of the coupled system in the time domain, Durbin's numerical Laplace transform inversion algorithm is employed. Convergence of the algorithm is verified and eventually a parametric study is carried out to investigate the effects of rail irregularity and vehicle speed on the time and frequency responses.

Transient acoustic radiation from an eccentric sphere

A vigorous semi-analytical model is offered to describe the 3D coupled nonaxisymmetric non-steady acousto-elastodynamic behavior of a submerged solid sphere with an off-center fluid-filled spherical cavity, acted upon by general distributed time-varying mechanical loads at its internal and/or external boundaries. The solid medium is designated by the 3D Navier’s linear elasticity model, while the internal/external ideal compressible fluids are assumed to obey the classical linear acoustic theory. Laplace transformation is applied to the time variable, and the separation of variables technique together with the pertinent fluid/solid interface conditions, the classical orthogonality properties of spherical harmonics, and a modified form of the translational addition theorem for spherical vector wave functions, are utilized to attain the final matrix equations in terms of unknown modal coefficients. Durbin’s Laplace transform inversion algorithm is subsequently applied to compute the pressure/displacement time response histories of water-submerged eccentric and coaxial metallic spheres excited by a couple of external concentrated Ricker-pulse radial excitations. Also, some key characteristics of the transient structure/liquid interaction phenomena with respect to cavity eccentricity are noted based on selected two-dimensional visualizations of the internal/external sound fields. Lastly, the accuracy of numerical simulations is verified by employing a standard FEM software.

The erlangization method for Markovian fluid flows

For applications of stochastic fluid models, such as those related to wildfire spread and containment, one wants a fast method to compute time dependent probabilities. Erlangization is an approximation method that replaces various distributions at a time t by the corresponding ones at a random time with Erlang distribution having mean t. Here, we develop an efficient version of that algorithm for various first passage time distributions of a fluid flow, exploiting recent results on fluid flows, probabilistic underpinnings, and some special structures. Some connections with a familiar Laplace transform inversion algorithm due to Jagerman are also noted up front.

NUMERICAL TRANSFORM INVERSION USING GAUSSIAN QUADRATURE

Numerical inversion of Laplace transforms is a powerful tool in computational probability. It greatly enhances the applicability of stochastic models in many fields. In this article we present a simple Laplace transform inversion algorithm that can compute the desired function values for a much larger class of Laplace transforms than the ones that can be inverted with the known methods in the literature. The algorithm can invert Laplace transforms of functions with discontinuities and singularities, even if we do not know the location of these discontinuities and singularities a priori. The algorithm only needs numerical values of the Laplace transform, is extremely fast, and the results are of almost machine precision. We also present a two-dimensional variant of the Laplace transform inversion algorithm. We illustrate the accuracy and robustness of the algorithms with various numerical examples.

Transient Analysis of Rewarded Continuous Time Markov Models by Regenerative Randomization with Laplace Transform Inversion

In this paper we develop a variant, regenerative randomization with Laplace transform inversion, of a previously proposed method (the regenerative randomization method) for the transient analysis of rewarded continuous time Markov models. Those models find applications in dependability and performability analysis of computer and telecommunication systems. The variant differs from regenerative randomization in that the truncated transformed model obtained in that method is solved using a Laplace transform inversion algorithm instead of standard randomization. As regenerative randomization, the variant requires the selection of a regenerative state on which the performance of the method depends. For a class of models, class C’, including typical failure/repair models, a natural selection for the regenerative state exists and, with that selection, theoretical results are available assessing the performance of the method in terms of “visible” characteristics. Using dependability class C’ models of moderate size of a RAID 5 architecture we compare the performance of the variant with those of regenerative randomization and randomization with steady-state detection for irreducible models, and with those of regenerative randomization and standard randomization for models with absorbing states. For irreducible models, the new variant seems to be about as fast as randomization with steady-state detection for models which are not too small when the initial probability distribution is concentrated in the regenerative state, and significantly faster than regenerative randomization when the model is stiff and not very large. For stiff models with absorbing states, the new variant is much faster than standard randomization and significantly faster than regenerative randomization when the model is not very large. In addition, the variant seems to be able to achieve stringent accuracy levels safely.

Inventory Control and Regenerative Processes: Computations

This article is a continuation of the paper Inventory Control and Regenerative processes: Theory (Bazsa et al.; 1998) and presents closed form expressions for Laplace transforms associated with the cost functions of the classical single item inventory models with indivisible items, a fixed lead time and backlogging. For given instances of these inventory control models these Laplace transforms will be used as input functions in a newly developed Laplace transform inversion algorithm (Iseger; A new method for inverting Laplace transforms; 1998) which yields highly accurate computational results of the associated cost functions.

The solution of the time-independent neutron slowing down equation using a numerical laplace transform inversion

Abstract The transport equation describing steady state neutron slowing down in an infinite medium with constant cross sections is solved numerically using a recently developed numerical Laplace transform inversion algorithm. The basis for the algorithm (described below) is an evaluation of the Bromwich integral without analytical continuation. This method of evaluation avoids the requirement of determining the singularities of the transform in detail and readily allows consideration of scattering from multiple species when constant cross sections are assumed.

Evaluation of the electric field generated by longitudinal plasma waves

The stabilizing effects of the addition of short‐range collisions to a partially ionized plasma are investigated through a recently developed numerical Laplace transform inversion algorithm. The technique is demonstrated using a two‐stream equilibrium distribution and double Maxwellian equilibrium distributions. The regions of plasma stability obtained in previous studies are verified.

The numerical solution of linear time-dependent partial differential equations by the Laplace transform and finite difference approximations

A feasible method is presented for the numerical solution of a large class of linear partial differential equations which may have source terms and boundary conditions which are time-varying. The Laplace transform is used to eliminate the time-dependency and to produce a subsidiary equation which is then solved in complex arithmetic by finite difference methods. An effective numerical Laplace transform inversion algorithm gives the final solution at each spatial mesh point for any specified set of values of t. The single-step property of the method obviates the need to evaluate the solution at a large number of unwanted intermediate time points. The method has been successfully applied to a variety of test problems and, with two alternative numerical Laplace transform inversion algorithms, has been found to give results of good to excellent accuracy. It is as accurate as other established finite difference methods using the same spatial grid. The algorithm is easily programmed and the same program handles equations of parabolic and hyperbolic type.

The numerical solution of linear time-dependent partial differential equations by the laplace and fast fourier transforms

The method proposed may be used to solve a large class of linear parabolic and hyperbolic equations involving as independent variables time and two further spatial variables on a rectangular domain. The Laplace transform is used to reduce by one the number of independent variables, the resulting subsidiary equation having then to be one of the class which may be solved numerically by the FFT method. An effective numerical Laplace transform inversion algorithm is used to recover the solution of the original equation. The method has been successfully applied to a variety of test problems and found to be as accurate as established ADI methods. It is easily programmed and handles a wide range of time-varying boundary conditions and source functions without complication. For such problems it can be very competitive with regard to computation times. The method possesses the single-step property in that the solution at a particular time point may be evaluated without requiring the solution at intermediate time points, in contrast with the requirement for most finite difference methods. Both parabolic and hyperbolic equations are solved by precisely the same algorithm.