Inverse Laplace Transform Research Articles

Poroelastic response of a functionally graded hollow cylinder under an asymmetric loading condition

This paper aims to develop semi-analytical results of the poromechanical fields in a hollow cylinder subjected to an asymmetric loading condition. The poroelastic properties of the hollow cylinder vary in the radial direction. The heterogeneous hollow cylinder is approximated by a multilayer structure in which each cylindrical layer is assumed to be homogeneous. The analytical results of the poromechanical field are first established in the transformed Laplace space, and then the ones in time space are obtained by the inverse Laplace transformation. They are perfectly in coherence with existent results in the literature that were developed for the homogeneous and the heterogeneous cases at the steady state regime. Sensitivity analysis shows a significant effect of the radial variations of the poroelastic properties on the response of the hollow cylinder.

Prediction of the stability and compatibility of Colombian heavy crude oils by 1D low field nuclear magnetic resonance relaxometry and chemometric methods

Stability and compatibility can be described by the insolubility number (IN), the solubility blending number (SBN) and the ratio (R) between them, which has become a critical index in daily refineries operation. In this work, a new methodology for predicting the stability and compatibility of heavy crude oils by 1D low field nuclear magnetic resonance (LF-NMR) relaxometry has been developed. Various chemometric models were developed to predict the IN,SBN and R properties of Colombian heavy crude oils (°API from 6 to 27), by integrating intervals of either the 1D free induction transverse magnetization decay curves (T2 FID) obtained with a low-field (LF) nuclear magnetic resonance (NMR) spectrometer, or their T2 relaxation distributions obtained via the inverse Laplace transform (ILT). Correlations between the relaxometry curves and said properties were evaluated using the principal component regression (PCR) and the partial least squares regression (PLSR) methodologies. The obtained prediction models yielded coefficients of determination (R2) above 95% with as few as 6 components and values of cross validated root mean squared error of prediction (RMSEP) as low as 0.71 for the IN, 2.37 for the SBN and of 0.03 for the R property, respectively. The best prediction models were attained using PLSR and employing the FIDs as the predictors. The use of chemometric predictive models from LF-NMR measurements represent a low-cost and faster alternative to estimate critical properties of the independent crude oils to be blended, and thus, will help in predicting the stability and compatibility of crude oil blends, which in turn may help accelerate the decision-making process in daily refineries tasks.

Deconvolution of the segmental and chain modes in amorphous polymers: Do the short-chain modes affect the bulk relaxation?

A new method for calculating the retardation spectra for linear viscoelastic materials is presented. The method uses the analytical expressions among the viscoelastic functions through the numerical evaluation of the Inverse Laplace Transform (ILT) with the Gaver–Wynn–Rho (GWR) algorithm. It is applied to study three polymers' relaxation mechanisms: a star Polystyrene and two Polycyanurate networks with two different crosslink densities. The shear spectrum is de-convoluted to separate the segmental and the chain modes responsible for the softening dispersion. The retardation mechanisms extracted from the shear response are then compared with the volumetric bulk response. The results show that the shear segmental contribution is narrower than the volumetric bulk response and finishes about two decades before the shoulder's appearance in the shear retardation spectrum. Thus, the volumetric bulk response, mainly exerted by the segmental mode, is also affected at long times by the short time chain mechanism, usually claimed to be Sub-Rouse modes.

Simultaneous Flow of n-Immiscible Fractional Maxwell Fluids with Generalized Thermal Flux and Robin Boundary Conditions

In a rectangular region, the multilayered laminar unsteady flow and temperature distribution of the immiscible Maxwell fractional fluids by two parallel moving walls are studied. The flow of the fluid occurs in the presence of Robin’s boundaries and linear fluid-fluid interface conditions due to the motion of the parallel walls on its planes and the time-dependent pressure gradient. The problem is defined as a mathematical model which focuses on the fluid memory, which is represented by a constituent equation with the Caputo time-fractional derivative. The integral transformations approach (the Laplace transform and the finite sine-Fourier transform) is used to determine analytical solutions for velocity, shear stress, and the temperature fields with fluid interface, initial, and boundary conditions. For semianalytical solutions, the algorithms of Talbot are used to calculate the Laplace inverse transformation. We used the Mathcad software for graphical illustration and numerical computation. It has been observed that the memory effect is significant on both fluid motion and temperature flow.

Enhanced Resolution Analysis for Water Molecules in MCM-41 and SBA-15 in Low-Field T2 Relaxometric Spectra.

Mesoporous silica materials are the subjects for relaxometric NMR studies in which we obtain information on the properties of molecules in confined geometries. The signal analysis in such investigations is generally carried out with the help of the Inverse Laplace Transform (ILT), which is accompanied by a regularization procedure. The appropriate selection of the regularization method may positively affect the resolution of the spectrum and the essence of the final conclusions. In this work, we examined the MCM-41 and SBA-15 model systems in various saturation states, using L-Curve regularization for relaxation spectra based on our own version of the fast fast ILT implementation. In a single relaxometric spectrum, the water contributions from the internal volume in the pores and between the silica particles were identified, which allowed us to trace the dynamics of the corresponding drying trends during the removal of water from the sample as a function of total water saturation.

Revisiting the 63Cu NMR Signature of Charge Order in La1.875Ba0.125CuO4

We use single crystal $^{63}$Cu NMR techniques to revisit the early $^{63}$Cu NQR signature of charge order observed for La$_{1.875}$Ba$_{0.125}$CuO$_{4}$ ($T_{\text{c}} =4$~K) [A. W. Hunt et al., Phys. Rev. Lett. {\bf 82}, 4300 (1999)]. We show that the growth of spin correlations is accelerated below $\sim 80$~K, where the inverse Laplace transform (ILT) T$_{1}$ analysis of the $^{139}$La NMR spin-lattice relaxation curve recently uncovered emergence of the slow components in the lattice and/or charge fluctuations [P. M. Singer et al., {\bf 101}, 174508 (2020)]. From the accurate measurements of the $^{63}$Cu NMR signal intensity, spin echo decay $M(2\tau)$, spin-lattice relaxation rate $^{63}1/T_1$, and its density distribution function $P(^{63}1/T_{1})$, we also demonstrate that charge order at $T_{\text{charge}}\simeq 54$~K turns on strong enhancement of spin fluctuations {\it within charge ordered domains}, thereby making the CuO$_2$ planes extremely inhomogeneous. The charge ordered domains grow quickly below $T_{\text{charge}}$, and the volume fraction $F_{\text{CA}}$ of the canonical domains unaffected by charge order gradually diminishes by $\sim 35$~K. This finding agrees with our independent estimations of $F_{\text{CA}}$ based entirely on the $^{139}$La ILTT$_{1}$ analyses, but is in a stark contrast with much slower growth of charge ordered domains observed for La$_{1.885}$Sr$_{0.115}$CuO$_{4}$ from its $T_{\text{charge}}\simeq 80$~K to $T_{\text{c}}\simeq 30$~K.

The Fractional Derivative of the Dirac Delta Function and Additional Results on the Inverse Laplace Transform of Irrational Functions

Motivated from studies on anomalous relaxation and diffusion, we show that the memory function M(t) of complex materials, that their creep compliance follows a power law, J(t)∼tq with q∈R+, is proportional to the fractional derivative of the Dirac delta function, dqδ(t−0)dtq with q∈R+. This leads to the finding that the inverse Laplace transform of sq for any q∈R+ is the fractional derivative of the Dirac delta function, dqδ(t−0)dtq. This result, in association with the convolution theorem, makes possible the calculation of the inverse Laplace transform of sqsα∓λ where α<q∈R+, which is the fractional derivative of order q of the Rabotnov function εα−1(±λ,t)=tα−1Eα,α(±λtα). The fractional derivative of order q∈R+ of the Rabotnov function, εα−1(±λ,t) produces singularities that are extracted with a finite number of fractional derivatives of the Dirac delta function depending on the strength of q in association with the recurrence formula of the two-parameter Mittag–Leffler function.

Application of the Efros Theorem to the Function Represented by the Inverse Laplace Transform of s−μ exp(−sν)

Using a special case of the Efros theorem which was derived by Wlodarski, and operational calculus, it was possible to derive many infinite integrals, finite integrals and integral identities for the function represented by the inverse Laplace transform. The integral identities are mainly in terms of convolution integrals with the Mittag–Leffler and Volterra functions. The integrands of determined integrals include elementary functions (power, exponential, logarithmic, trigonometric and hyperbolic functions) and the error functions, the Mittag–Leffler functions and the Volterra functions. Some properties of the inverse Laplace transform of s−μexp(−sν) with μ≥0 and 0<ν<1 are presented.

TRANSFORMADA INVERSA DE LAPLACE PARA ANÁLISE DE SINAIS DE RESSONÂNCIA MAGNÉTICA NUCLEAR DE BAIXO CAMPO

INVERSE LAPLACE TRANSFORM FOR SIGNAL ANALYSIS OF LOW FIELD NUCLEAR MAGNETIC RESONANCE. The objective of this work is to describe the mathematical procedure denominated ‘Inverse Laplace Transform’ used to process Low Field Nuclear Magnetic Resonance signals of experiments as CPMG, Inversion Recovery and Saturation Recovery. Also, we present a developed script, “ILT Origin script” v.1, in the environment of the software Origin Lab 9 to execute it. The presented ILT is executed with an algorithm of non-negative last square, with Tikhonov regularization and singular values decomposition. The main features of the code were tested using simulated and experimental signals, and in order to validate the results the same sets of data were analyzed using similar ILT softwares. The developed scripts for Origin Lab, Matlab and other languages are available for free for scientific and teaching uses.

APPROXIMATE ANALYTICAL SOLUTIONS TO NONLINEAR DAMPED OSCILLATORY SYSTEMS USING A MODIFIED ALGEBRAIC METHOD

In the current paper, a modified algebraic method (MAGM) is proposed as an effective semi-analytical technique for solving nonlinear damped oscillatory systems. A polynomial is supposed as the trial solution, and its unknown coefficients are easily determined through the algebraic method (AGM). In order to improve the solution, the Laplace transformation is applied to the series solution, and then the Pade approximants of the resultant equation are constructed. Finally, the inverse Laplace transformation is adopted to obtain a periodic solution for the nonlinear problem under consideration. The proposed method is then applied for obtaining approximate analytical solutions of a damped rotatory oscillator as well as nonlinear vibrations of a flexible beam excited by an axial force. The results are compared with those obtained by the fourth-order Runge–Kutta method, and good agreement is observed.

Supercooled nano-droplets of water confined in hydrophobic rubber.

Hydrophobic elastomers are capable of absorbing a small amount of water that forms droplets around hydrophilic sites. These systems allow the study of confinement effects by a hydrophobic environment on the dynamics and thermodynamic behaviour of water molecules. The freezing-melting properties and the dynamics of water inside nano-droplets in butyl rubber are affected, as revealed by differential scanning calorimetry (DSC) and deuterium nuclear magnetic resonance (2H-NMR). Upon cooling down, all water crystalizes with a bimodal droplet population (da = 3.4 nm and db = 4.4 nm) in a temperature range associated with the droplet size distribution. However, the melting temperature is not shifted according to the Gibbs-Thomson equation. The relative decrease of the 2H-NMR longitudinal magnetization is not a single exponential and, by inverse Laplace transformation, it was deduced to be bimodal in agreement with the DSC measurements (T1,a ∼ 10 ms and T1,b ∼ 200 ms). The deduced correlation time of molecular reorientation is longer than that of bulk water and the behaviour with temperature follows the Vogel-Fulcher-Tammann (VFT) equations with a changing fragility as the droplet size is reduced when reducing hydration.

MHD Oldroyd-B Fluid with Slip Condition in view of Local and Nonlocal Kernels

We examine the velocity field of an incompressible Oldroyd-B fluid over a horizontal plate of continual length in a permeable medium with magnetohydrodynamics effect. Firstly, the results for the dimensionless classical model (governing equation) have been studied analytically then the study is extended for different fractional operators. The relations to determine the velocity fields of this problem are found by Laplace transformation and different numerical inversion algorithms. The impact of physical parameters on velocity profiles is analyzed graphically for integer and non-integer models. Non-integer operators are used to analyzing the impact of fractional parameters on the fluid curves of the fluid.

On the approximate inverse Laplace transform of the transfer function with a single fractional order

The history of fractional calculus dates back to 1600s and it is almost as old as classical mathematics. Although many studies have been published on fractional-order control systems in recent years, there is still a lack of analytical solutions. The focus of this study is to obtain analytical solutions for fractional order transfer functions with a single fractional element and unity coefficient. Approximate inverse Laplace transformation, that is, time response of the basic transfer function, is obtained analytically for the fractional order transfer functions with single-fractional-element by curve fitting method. Obtained analytical equations are tabulated for some fractional orders of [Formula: see text]. Moreover, a single function depending on fractional order alpha has been introduced for the first time using a table for [Formula: see text]. By using this table, approximate inverse Laplace transform function is obtained in terms of any fractional order of [Formula: see text] for [Formula: see text]. Obtained analytic equations offer accurate results in computing inverse Laplace transforms. The accuracy of the method is supported by numerical examples in this study. Also, the study sets the basis for the higher fractional-order systems that can be decomposed into a single (simpler) fractional order systems.

Three infinite series and integral formulations for the voltammetric responses of reversible electrochemical reactions at a rotating disk electrode. Discrimination using the method of lines, and generalization to the integral equation method with finite kinetics, Ohmic potential drop and (pseudo-) capacitive interfacial behaviour

Given the closed-form expression of the mass transfer function pertaining to one-dimensional mass transport conditions in the electrolyte, the voltammetric response of one-step reversible electrochemical (E r ) reactions to a linear potential ramp can be derived first as an infinite series well suited to formal analysis, and then as an integral formulation well suited to numerical calculation. This approach applies in this article to the E r reaction studied on a uniformly accessible Rotating Disk Electrode (RDE). Using three closed-form approximations of the mass transfer function, three infinite series and three integral formulations can be derived for the Faradaic current response versus the electrode potential. Discrimination of the approximating formulations of the voltammetric response can be performed by comparison to benchmark voltammetric data obtained using the numerical method of lines to solve the relevant partial differential equation for diffusion-convection at RDE. Once the best approximating model is selected, based on its voltammetric features, the appropriate mass transfer function can be used together with the inverse Laplace transformation to derive the kernel function involved in the Integral Equation method. Application to the E reaction investigated by cyclic voltammetry on RDE is an illustration example in this article, first disregarding and then taking into account Ohmic drop and double-layer charging effects with capacitive or pseudo-capacitive (constant phase element) interfacial behaviour. • Infinite series formulations of the voltammetric response for the E r reaction on RDE . • Integral formulations of the voltammetric response for the E r reaction on RDE . • Derivation of a new kernel function for the dissolved species reacting on RDE. • Application to the Integral Equation method for cyclic voltammetry on RDE. • Voltammetric modelling of Ohmic drop and double-layer (capacitance/CPE) effects.

An Application of Laplace Transform, Inverse Laplace Transform and Pade’s Approximant in the Periodic Solution of Duffing Equation of Motion

An Application of Laplace Transform, Inverse Laplace Transform and Pade’s Approximant in the Periodic Solution of Duffing Equation of Motion

Study of transversely isotropic nonlocal thermoelastic thin nano-beam resonators with multi-dual-phase-lag theory

This study deals with a novel model of forced vibrational analysis of nonlocal transversely isotropic thermoelastic nanobeam resonators due to ramp-type heating and due to time varying exponentially decaying load with multi-dual-phase-lag theory of thermoelasticity. The mathematical model is prepared for the nanobeam in a closed form with the application of Euler–Bernoulli (E–B) beam theory using nonlocal generalized thermoelasticity with multi-dual phase lags. The nonlocal nanobeam theory has a nonlocal parameter to depict small-scale effect. The Laplace Transform technique has been used to find the expressions for lateral deflection, conductive temperature, thermal moment, nonlocal axial stress and thermodynamic temperature for (i) clamped–clamped, (ii) simply supported–simply supported, (iii) clamped–simply supported, (iv) clamped–free, and (v) free–free nanobeam in the transformed domain. The general algorithm of the inverse Laplace Transform is developed to compute the results numerically in physical domain. The results exhibit that the amplitude of deflection and thermal moment is attenuated and depends upon the schematic design of the nanobeam being considered. Also, it can be found from both the numerical calculations and the analytic results that nonlocal multi-dual-phase-lag theory of thermoelasticity with two temperatures due to time varying exponentially decaying load has significant effect on deflection and thermal moment. The effect of different theories of nonlocal thermoelasticity, due to time varying exponentially decaying load, has been depicted on the various quantities. Some particular cases have also been deduced.

Implementing PRED Subroutine of NONMEM for Versatile Pharmacokinetic Analysis Using Fast Inversion of Laplace Transform (FILT).

In pharmacokinetic (PK) analysis, conventional models are described by ordinary differential equations (ODE) that are generally solved in their Laplace transformed forms. The solution in the Laplace transformed forms is inverse Laplace transformed to derive an analytical solution. However, inverse Laplace transform is often mathematically difficult. Consequently, numerical inverse Laplace transform methods have been developed. In this study, we focus on extending the modeling functions of Nonlinear Mixed Effect Model (NONMEM), a standard software for PK and population pharmacokinetic (PPK) analyses, by adding the Fast Inversion of Laplace Transform (FILT) method, one of the representative numerical inverse Laplace transform methods. We implemented PREDFILT, a specialized PRED subroutine, which functions as an internal model unit in NONMEM to enable versatile FILT analysis with second-order precision. The calculation results of the compartment models and a dispersion model are in good agreement with the ordinary analytical solutions and theoretical values. Therefore, PREDFILT ensures enhanced flexibility in PK or PPK analyses under NONMEM environments.

Transient analysis of piecewise homogeneous QBD process

The article presents a numerical analysis approach for the transient solution of a piecewise homogeneous quasi-birth-death process. The proposed approach computes the transient probabilities based on the linear combination of matrix geometric series in Laplace transform domain, and builds on the availability of an efficient numerical inverse Laplace transformation method.

Optimized Portable Unilateral Magnetic Resonance Sensor for Assessing the Aging Status of Silicon Rubber Insulators

Silicone rubber insulators (SRIs) play an important role in high-voltage power transmission lines. Given that SRIs critically threaten the safe operation of the power grid, quantitatively assessing the aging status of SRIs in situ is crucial. In this article, a portable unilateral magnetic resonance (UMR) sensor with a U-shaped magnet structure and its measuring circuit were designed for the nondestructive detection of the aging status of SRIs in situ. First, the optimization efficiency of the magnet structure was improved through a uniform design method. Then, the planar radio frequency (RF) coil was designed by the finite-element method. Furthermore, the demagnetization effect of eddy currents induced in metal materials on the RF magnetic field was analyzed. We measured the SRI sheds at different service times and obtained the corresponding 1H transverse relaxation decay curves by importing the Carr- Purcell-Meiboom-Gill (CPMG) sequence. Finally, the transverse relaxation decay curves were postprocessed using inverse Laplace transformation, and the 1-D spectrum distributions of transverse relaxation time (T2) were obtained. The results reveal that T2 of the surface of the SRI shed decreases as the service time of the SRI increases. In addition, the difference in T2 between the surface and interior of the SRI shed increases. The comparison experiments demonstrate that the U-shaped sensor designed in this article is superior to the curved sensor previously designed by our team. The effect of temperature variation on SRIs aging assessment was considered at last.

A Semi-Analytical Fractal-Fractional Mathematical Model for Multi-Fractured Horizontal Wells in Coalbed Methane Reservoirs

Abstract Coalbed methane (CBM) which is clean energy has received great emphasis recently, and the multi-fracturing technology is widely applied in the exploitation of CBM. Due to the complexity, the randomness, and the anisotropism of the porous medium and the anomalous diffusion process, the fractal theory and fractional calculus are utilized to establish a semi-analytical fractal-fractional mathematical model considering the stress sensitivity of the cleat system for multi-fractured horizontal wells in CBM reservoirs. Through line-sink theory, Pedrosa transformation, perturbation theory, Laplace transformation, element discretization, superposition principle, and Stehfest numerical inversion, the pressure-transient analysis curves are plotted in the double logarithmic coordinates. By comparing with the existing model, the validation of the proposed model is illustrated. Also, nine flowing stages are identified according to different characteristics. Then, sensitivity analysis is conducted and influence laws are summarized. At last, a field application is introduced to furtherly verify the reliability of the proposed model. The relevant results analysis can provide some new significant guidance for interpreting the field data more precisely.