Numerical Inverse Laplace Transformation Research Articles

Numerical inverse transformation of double sided Laplace transform with parameter optimization

There are efficient numerical inverse Laplace transformation (NILT) procedures for double sided Laplace transforms (DSLT), which are based on some hyperparameters. If those hyperparameters are set properly, the computed inverse is accurate, while using an incorrect hyperparameter results in a completely incorrect inverse.To the best of the authors knowledge, there is no NILT procedure for DSLT, which automatically sets the hyperparameters based on the DSLT and the point where the inverse needs to be computed. In this paper we propose a procedure which automatically optimizes the hyperparameters of the NILT method and this way it eliminates the unforeseen dependency of the computed inverse on the inherent hyperparameters.

An Analytic RLC Model for Coupled Interconnects Which Uses a Numerical Inverse Laplace Transform

An Analytic <i>RLC</i> Model for Coupled Interconnects Which Uses a Numerical Inverse Laplace Transform

Accurate Modeling and Wave Propagation Analysis of Cracked Slender Structural Members by the Spectral Element Method

The analysis of elastic wave propagation in cracked structures is very useful in the crack detection by the ultrasonic guided wave method. This study presents an accurate spectral element modeling method for cracked slender structural members by using refined waveguide models and a more realistic crack model. Firstly, a spatial spectral beam element model is established for uncracked slender structural member based on the Love rod theory, the modified Timoshenko beam theory, and the Saint-Venant’s torsion theory. Then, the complete local additional flexibility matrix for crack in the structural member with rectangular cross section is derived from the theory of elastic fracture mechanics, and a two-node condensed spectral element model considering the stiffness coupling effect caused by the crack is established for cracked slender structural member. The wave response in cracked structures is solved by the numerical inverse Laplace transformation method. A thorough comparison of the wave responses in cracked structural member evaluated by the presented spectral element model and the 3D solid finite element model is given in the numerical example, which verifies the accuracy and high efficiency of the presented method.

Memory response on a piezoelectric rod under three-phase-lag model

ABSTRACT In this article, the concept of the memory-dependent derivative has been utilized to build up the heat equation to investigate the piezoelectric effect on a finite rod. The governing equations are written under the generalized theory of thermoelasticity based on the three-phase-lag model and solved in the Laplace transform domain. The numerical inverse Laplace transformation technique has been adapted to obtain the solution in the space–time domain. The effect of the different choices of kernel functions, time delay and comparison with other models are discussed and represented graphically. It is noted that the field variables attain the maximum amplitudes for the highest values of the time delay for a fixed kernel.

Fast Computation Method of Energization Overvoltage in Cable Lines Based on Numerical Inverse Laplace Transform

Currently, the research on the energization overvoltage in cable lines is mainly realized through simulation calculation. However, the operation process is complex and the computation speed is slow when using the simulation software to calculate the energization overvoltage in the whole line. This paper presents a numerical calculation method of energization overvoltage in the entire cable line, which can greatly improve the calculation speed, simplify the operation process and have accurate calculation results. The method is based on transmission line theory, modal theory, transmission line equation’s complex frequency domain solution and Numerical Inverse Laplace Transform (NILT). This method first performs phase-to-mode conversion on the cable line and converts the mutually coupled cable conductors into mutually independent propagation modes. Furthermore, each independent propagation mode is conducted by Laplace Transform to obtain the complex frequency domain of its voltage. Finally, after the NILT is performed on the voltage complex frequency domain solution of each propagation mode and the time domain solution is obtained, the energization overvoltage of the full cable is obtained through the inverse phase-to-mode conversion. This method is applied to a 330kV cable line, and the calculation speed is significantly improved. The overvoltage calculation results have a small error compared to the obtained results using the simulation software, indicating that the method has high accuracy.

On the Approximation of Fractal-Fractional Differential Equations Using Numerical Inverse Laplace Transform Methods

Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects. Using the Laplace transform for solving differential equations, however, sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analytical means. Thus, we need numerical inversion methods to convert the obtained solution from Laplace domain to a real domain. In this paper, we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with order . Our proposed numerical scheme is based on three main steps. First, we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense, and then into Caputo sense. Secondly, we transform the fractional differential equation in Caputo sense to an equivalent equation in Laplace space. Then the solution of the transformed equation is obtained in Laplace domain. Finally, the solution is converted into the real domain using numerical inversion of Laplace transform. Three inversion methods are evaluated in this paper, and their convergence is also discussed. Three test problems are used to validate the inversion methods. We demonstrate our results with the help of tables and figures. The obtained results show that Euler’s and Talbot’s methods performed better than Stehfest’s method.

Transient Electrophoresis of a Cylindrical Colloidal Particle

We develop the theory of transient electrophoresis of a weakly charged, infinitely long cylindrical colloidal particle under an application of a transverse or tangential step electric field. Transient electrophoretic mobility approaches steady electrophoretic mobility with time. We derive closed-form expressions for the transient electrophoretic mobility of a cylinder without involving numerical inverse Laplace transformations and the corresponding time-dependent transient Henry functions. The transient electrophoretic mobility of an arbitrarily oriented cylinder is also derived. It is shown that in contrast to the case of steady electrophoresis, the transient Henry function of an arbitrarily oriented cylinder at a finite time is significantly smaller than that of a sphere with the same radius and mass density as the cylinder so that a cylinder requires a much longer time to reach its steady mobility than the corresponding sphere.

Parameter estimation of Markov modulated fluid arrival processes

Markov modulated discrete arrival processes have a wide literature, including parameter estimation methods based on expectation–maximization (EM). In this paper, we investigate the adaptation of these EM based methods to Markov modulated fluid arrival processes (MMFAP), and conclude that only the generator matrix of the modulating Markov chain of MMFAPs can be approximated by EM based method. For the rest of the parameters, the fluid rates and the fluid variances, we investigate the efficiency of numerical likelihood maximization.To reduce the computational complexity of the likelihood computation, we accelerate the numerical inverse Laplace transformation step of the procedure with function fitting.

Optimized numerical inverse Laplace transformation

Among the numerical inverse Laplace transformation (NILT) methods, those that belong to the Abate-Whitt framework (AWF) are considered to be the most efficient ones currently. It is a characteristic feature of the AWF NILT procedures that they are independent of the transform function and the time point of interest. In this work we propose an NILT procedure that goes beyond this limitation and optimize the accuracy of the NILT utilizing also the transform function and the time point of interest.

Approximate Analytic Expression for the Time-Dependent Transient Electrophoretic Mobility of a Spherical Colloidal Particle.

The general expression is derived for the Laplace transform of the time-dependent transient electrophoretic mobility (with respect to time) of a spherical colloidal particle when a step electric field is applied. The transient electrophoretic mobility can be obtained by the numerical inverse Laplace transformation method. The obtained expression is applicable for arbitrary particle zeta potential and arbitrary thickness of the electrical double layer around the particle. For the low potential case, this expression gives the result obtained by Huang and Keh. On the basis of the obtained general expression for the Laplace transform of the transient electrophoretic mobility, we present an approximation method to avoid the numerical inverse Laplace transformation and derive a simple approximate analytic mobility expression for a weakly charged particle without involving numerical inverse Laplace transformations. The transient electrophoretic mobility can be obtained directly from this approximate mobility expression without recourse to the numerical inverse Laplace transformation. The results are found to be in excellent agreement with the exact numerical results obtained by Huang and Keh.

Numerical inverse Laplace transformation beyond the Abate–Whitt framework

Among the numerical inverse Laplace transformation (NILT) methods, those that belong to the Abate–Whitt framework (AWF) are considered to be the most efficient ones currently. It is a characteristic feature of the AWF NILT procedures that they are independent of the transform function and the time point of interest.In this work we propose an NILT procedure that goes beyond this limitation and optimize the accuracy of the NILT utilizing also the transform function and the time point of interest.

Algorithm for fast calculating the energization overvoltages along a power cable based on modal theory and numerical inverse Laplace transform

Transient overvoltage caused by energization may damage the insulation in a power cable transmission system. Therefore, overvoltage along the cable should be calculated and evaluated carefully. Calculation based on traditional commercial software is time-consuming, therefore, there is a need to develop a new algorithm to improve the computational efficiency. First, coupled cable conductors are transformed into independent modes through phase-mode transformation. Then, the voltage formula of each independent mode in the complex frequency domain is obtained through Laplace transform. Finally, energization overvoltages along the cable are calculated using numerical inverse Laplace transform (NILT) and mode-phase transformation. The proposed algorithm is applied to a 330 kV cable line with a length of 9.6 km to certify the accuracy and computational efficiency. Compared with frequency dependent phase model (FDPM) in PSCAD/EMTDC, relative errors of the obtained results are less than 1%, and the maximum CPU time consumed is only about 8%. The proposed algorithm can improve the computational speed of cable energization overvoltages greatly on the premise of ensuring accuracy and may be contributing to actual applications.

The CME method

Numerical inverse Laplace transformation (NILT) is an important tool in the field of system modelling and performance analysis. The recently introduced CME method has many important advantages over the alternative numerical inverse Laplace transformation (NILT) methods. It avoids Gibbs oscillation (i.e., does not generate overshoot and undershoot), preserves the monotonicity of functions, its accuracy is gradually improving with the order, and it is numerically more stable than the alternative methods. In this paper we demonstrate these advantages and introduce our tool which implements the CME method and other popular NILT methods.

Transient analysis of Markov modulated processes with Erlangization, ME-fication and inverse Laplace transformation

The paper investigates the relation of random clock based and numerical inverse Laplace transformation based transient analysis of Continuous time Markov chains (CTMCs) and Markov fluid models (MFMs) and proves that these methods are identical. This identity leads to new intuitive understanding about the analysis approaches.

Simplified analysis of MHD flow in a porous surrounding bounded by an oscillating vertical cylindrical surface

This study developed the relatively simple mathematical model by adding a radiation effect and transverse magnetic field to the physical model, taking into account heat and mass transfer of a permeable, saturated porous medium, infinite oscillating cylindrical plate. In comparison, magnetohydrodynamics reality flow pattern in most earlier works was ignored in porous media. We have performed a semi-analytical method for numerical inverse Laplace transformation into PDEs of radiation on Newtonian fluids’ heat and mass transfer. The numerical stability method is used to analyze the influence of parameter variation for different values of M and N. We took into account both the local flow velocity distribution and concentration profile with the effect of Sc and time t and the skin friction against time t. There are several variations examined between the heat transfer under the cylinder boundary conditions and the plane boundary conditions. The findings given by empirical and numerical approaches are in good agreement. As seen in the literature, the numerical results on the local flow velocity agree closely with the solutions available. Our observations, meanwhile, show that the rate of heat transfer can be boosted using a magnetic field. The curved surface of the boundary configuration will restrain velocity and the temperature rise, while it has little effect on the mass transfer rate.

Two temperature generalized thermoelasticity involving memory-dependent derivative under fuzzy environment

A one-dimensional two-temperature model of generalized thermoelasticity theory in half-space has been studied in this present discussion with memory-dependent derivative. The fuzzy approach has been applied to construct the governing equations. The fuzzy variables such as stress, strain and so on are represented in r-cut. The coupled partial differential equations' analytical solutions are obtained in Laplace transform domain subject to a stress-free boundary with a time-dependent imprecise thermal shock. A suitable numerical inverse Laplace transformation technique is applied to get the space time-domain results for different time delay parameter values and several kernel functions. Two examples of numerical results in the fuzzy environment are performed and are compared with the crisp results graphically. The concluding remarks are made based on numerical results and graphs.

Concentrated matrix exponential distributions with real eigenvalues

AbstractConcentrated random variables are frequently used in representing deterministic delays in stochastic models. The squared coefficient of variation ($\mathrm {SCV}$) of the most concentrated phase-type distribution of order $N$ is $1/N$. To further reduce the $\mathrm {SCV}$, concentrated matrix exponential (CME) distributions with complex eigenvalues were investigated recently. It was obtained that the $\mathrm {SCV}$ of an order $N$ CME distribution can be less than $n^{-2.1}$ for odd $N=2n+1$ orders, and the matrix exponential distribution, which exhibits such a low $\mathrm {SCV}$ has complex eigenvalues. In this paper, we consider CME distributions with real eigenvalues (CME-R). We present efficient numerical methods for identifying a CME-R distribution with smallest SCV for a given order $n$. Our investigations show that the $\mathrm {SCV}$ of the most concentrated CME-R of order $N=2n+1$ is less than $n^{-1.85}$. We also discuss how CME-R can be used for numerical inverse Laplace transformation, which is beneficial when the Laplace transform function is impossible to evaluate at complex points.

Semi‐analytical solution for one‐dimensional consolidation of a two‐layered soil system with unsaturated and saturated conditions

AbstractSemi‐analytical solutions to one‐dimensional (1D) consolidation of the two‐layered soils are developed in this paper that can consider a two‐layered soil system with unsaturated upper soils and saturated lower soils. In this case, the upper soil layer remains the water‐air two‐phase flows, while the lower soil layer only involves the flow of the water phase. Based on the assumptions made in the 1D consolidation framework for saturated and unsaturated soils, the analytical solutions in the transformation space for the consolidation model are derived using a decoupling technique and the Laplace transformation. Subsequently, following the Crump algorithm, the numerical inverse Laplace transformation is carried for the related semi‐analytical results that can analyze the consolidation characteristics of the two‐layered soils in the time domain. Furthermore, two typical cases for saturated soils in the literature are shown to examine the methodology and final solutions. Finally, results of a case study in this paper indicate that the excess pore pressures within the unsaturated soil layer near the interface are significantly affected by the water flow in the saturated soil layer, and that the final primary consolidation settlement of soils would be overestimated if the unsaturated condition of upper soils is ignored.

Convolutive modeling of cyclic voltammetry, AC-voltammetry, sine wave voltammetry and impedance spectroscopy with interfacial CPE behaviour and uncompensated ohmic resistances: A Unified Theory

The constant phase element (CPE) is a well-established circuit component for the frequency domain simulation of electrochemical reactions at disperse and heterogeneous electrodes. Computing the time-dependent current response of voltammetric experiments with interfacial CPE behaviour is, however, not a straightforward task, owing to the lack of analytical time domain solutions of the convoluted capacitive current. In this paper, a universal framework for the convolutive treatment of cyclic voltammetry (CV), alternating current cyclic voltammetry (ACCV), (large) sine wave voltammetry (LSWV) and electrochemical impedance spectroscopy (EIS) in presence of interfacial CPEs and uncompensated Ohmic resistances is presented. A combination of numerical and analytical inverse Laplace transformation techniques allows the accurate assessment of the non-ideal capacitive behaviour, which can be used in turn for classical convolution algorithms. The convolutive approach bears the advantage that different electrode geometries with any kind of spatial boundary conditions as well as first order homogeneous chemical reactions, coupled to the charge transfer step, can be implemented readily. This offers an exceptionally large degree of flexibility in the simulations for the electrochemical system at hand.

A localized collocation scheme with fundamental solutions for long-time anomalous heat conduction analysis in functionally graded materials

This paper makes the first attempt to develop a novel localized collocation scheme based on fundamental solutions for long-time anomalous heat conduction analysis in functionally graded materials. In the present numerical framework, the Laplace transformation and Fixed Talbot numerical inverse Laplace transformation are employed to deal with the time fractional derivative in anomalous time fractional heat conduction model. The localized collocation scheme based on fundamental solutions is applied to solve Laplace-transformed time-independent partial differential equations (PDEs), and the generalized reciprocity method (GRM) coupled with the derived problem-dependent generalized fundamental solution is proposed to deal with the inhomogeneous term in Laplace-transformed time-independent PDEs. Numerical tests demonstrate that the developed computational strategy can be considered as one of the competitive alternatives in the simulation of anomalous heat conduction behavior of functionally graded materials. Furthermore, the effect of time fractional derivative order on the behavior of anomalous heat conduction is numerically investigated.