Laplace Transform Research Articles

Applying the Laplace Transform Procedure, Testing Exponentiality against the NBRUmgf Class

This paper addresses a hypothesis testing problem for comparing exponentially distributed data against a new class termed “New Better than Renewal Used in Moment Generating Function” (NBRUmgf). A measure of departure from exponentiality is constructed using the Laplace transform, followed by the development of a U-statistic-based test for the hypothesis. Additionally, a test based on the goodness of fit approach is examined as a special case. The asymptotic normality of the proposed statistic is introduced, and Pitman’s asymptotic efficiency of the two tests is computed and compared with other tests. Percentiles of the test statistics are computed for certain sample sizes in the case of complete data, and the powers of the tests are computed for popular reliability distributions. Finally, practical applications of the proposed tests are demonstrated in multiple cases.

From micro to macro: Creep behaviour and closed-form expression of viscoelastic parameters for a rheological particle assembly

This study is devoted to analytically establish macro–micro quantitative relationships for creep characteristics of a rheological particle assembly, contributing to a more comprehensive insight into the interplay between different scales. To represent the various rheological behaviour in microscopic for a hexagonal close-packed (HCP) circular particle assembly, the contact responses are assumed as viscoelastic models. By extending the traditional homogenization method to the Laplace domain, the macroscopic one/two-dimensional creep equations of the rheological particle assembly, expressed with the microscopic contact parameters in a closed-form manner, are innovatively identified. It is found that the macroscopic creep behaviour is not always consistent with the microscopic one and in general exhibits more complex creep characteristics. The parametric analysis presents the quantitative influence of contact parameters on macroscopic creep behaviour. Extending the one-dimensional case to three dimensions (3D), the relaxation moduli as well as creep compliance, expressed in closed-form by contact parameters, are obtained in 3D viscoelastic constitutive equations. Distinct viscoelastic behaviours are observed for macroscopic shear and bulk responses. The proposed macro–micro quantitative relations of creep characteristics in this study benefit to deeply understand the macro–micro relations of viscoelastic properties, providing an alternative approach to calibrate microscopic parameters when involving the DEM simulations.

DEVELOPMENT AND ASSESSMENT OF MANUAL AND AUTOMATED PID CONTROLLERS FOR THE OPTIMUM PRODUCTION OF ETHYLENE GLYCOL IN A CSTR

Industrially, one of the techniques used to enhance the optimum production of petrochemicals is the configuration of process control units such as gas chromatography (G.C. Analyzer) for concentration and thermocouple for temperature in the production system. This research focused on the application of the principles of mass and energy conservation in the development of dynamic or unsteady state models for predicting the behaviour of the system or process involved in the production of ethylene glycol from a reaction involving oxidation of ethylene-to-ethylene oxide which further undergoes hydrolysis to produce ethylene glycol in a continuous stirred tank reactor (CSTR). The high mathematical complexities of the mass and energy balance models of the Process as a result of the reaction kinetic scheme of the non-elementary reaction of the process which integrated both linear and non-linear terms into the models were resolved by the application of the principles of Taylor’s series expansion, Laplace Transform, partial fraction and matrix in the development of the dynamic models. Process simulation tool (Simulink) was utilized in the configuration of closed-loop systems with thermocouple and G. C. Analyzer, at the feed point of the reactor for process variables (temperature and concentration) control during the process using proportional, integral and derivative (PID) as controller parameters. The results of the process behaviour showed that manual tuning of different controller parameters experienced a high level of fluctuations and would not attain its stability or steady state even at a processing time above 16 seconds. Whereas, automatic tuning of desired controller gains of 1.361, 1.056 and 0.4109 for K P, K I and K D for temperature and pressure with a tuning time of 143.9 Seconds, requires a maximum time of 8 seconds for the system to attain stability. This study showed that automated tuning of the controller resulted in better performance characteristics for optimum production of ethylene glycol in a non-isotheral continuous stirred tank reactor within the shortest possible time.

Transient Pressure Behavior of CBM Wells during the Injection Fall-Off Test Considering the Quadratic Pressure Gradient.

Conventional coalbed methane (CBM) reservoir models for injection fall-off testing often disregard the quadratic pressure gradient's impact. This omission leads to discrepancies in simulating the transient behavior of formation fluids and extracting critical reservoir properties. Accurate determination of permeability, storability, and other properties is crucial for effective reservoir characterization and production forecasting. Inaccurate estimations can lead to suboptimal well placement, ineffective production strategies, and ultimately, missed economic opportunities. To address this shortcoming, we present a novel analytical model that explicitly incorporates the complexities of the quadratic pressure gradient and dual-permeability flow mechanisms, prevalent in many CBM formations where nanopores are rich, presenting a kind of natural nanomaterial. This model offers significant advantages over traditional approaches. By leveraging variable substitution, it facilitates the derivation of analytical solutions in the Laplace domain, subsequently converted to real-space solutions for practical application. These solutions empower reservoir engineers to generate novel type curves, a valuable tool for analyzing wellbore pressure responses during injection fall-off tests. By identifying distinct flow regimes within the reservoir based on these type curves, engineers gain valuable insights into the dynamic behavior of formation fluids. This model goes beyond traditional approaches by investigating the influence of the quadratic pressure gradient coefficient, inter-porosity flow coefficient, and storability ratio on the pressure response. A quantitative comparison with traditional models further elucidates the key discrepancies caused by neglecting the quadratic pressure gradient. The results demonstrate the proposed model's ability to accurately depict the non-linear flow behavior observed in CBM wells. This translates to more reliable pressure and pressure derivative curves that account for the impact of the quadratic pressure gradient.

The Generalized Fox–Wright Function: The Laplace Transform, the Erdélyi–Kober Fractional Integral and Its Role in Fractional Calculus

In this paper, we consider and study in detail the generalized Fox–Wright function Ψ˜qp introduced in our recent work as an extension of the Fox–Wright function Ψqp. This special function can be seen as an important case of the so-called I-functions of Rathie and H¯-functions of Inayat-Hussain, that in turn extend the Fox H-functions and appear to include some Feynman integrals in statistical physics, in polylogarithms, in Riemann Zeta-type functions and in other important mathematical functions. Depending on the parameters, Ψ˜qp is an entire function or is analytic in an open disc with a final radius. We derive its basic properties, such as its order and type, and its images under the Laplace transform and under classical fractional-order integrals. Particular cases of Ψ˜qp are specified, including the Mittag-Leffler and Le Roy-type functions and their multi-index analogues and many other special functions of Fractional Calculus. The corresponding results are illustrated. Finally, we emphasize the role of these new generalized hypergeometric functions as eigenfunctions of operators of new Fractional Calculus with specific I-functions as singular kernels. This paper can be considered as a natural supplement to our previous surveys “Going Next after ‘A Guide to Special Functions in Fractional Calculus’: A Discussion Survey”, and “A Guide to Special Functions of Fractional Calculus”, published recently in this journal.

Semi-Analytical Solution for a Single-Phase Thermal Conduction Problem of a Hollow Cylinder with a Growing or Receding Inner Radius

Abstract A semi-analytical solution for the thermal conduction of a single phase, homogeneous, circular, hollow-cylinder with a growing or receding inner radius at a constant rate under unit-loading was derived using conformal mapping, Laplace transformation, and a Zakian series representation of the inverse Laplace transform. All solutions allow for convection on the fixed outer radius. Predictions were compared to finite-element simulations with excellent agreement observed. Given the changing thickness, thermal transients could not reach true steady-state equilibrium, especially for faster growth or recession rates. Indeed, the temperature states become somewhat linear with respect to time, reflecting the constant velocity of growth or recession. In practice, the resulting solutions can be used to determine temperatures during machining, wear, erosion, corrosion, and/or additive manufacturing, especially for lower temperature solid-state methods such as cold-spray.

Characterizing the Oligomers Distribution along the Aggregation Pathway of Amyloid Aβ1-40 by NMR.

This study delves into the early aggregation process of the Aβ1-40 amyloid peptide, elucidating the associated oligomers distribution. Motivated by the acknowledged role of small oligomers in the neurotoxic damage linked to Alzheimer's disease, we present an experimental protocol for preparing 26-O-acyl isoAβ1-40, a modified Aβ1-40 peptide facilitating rapid isomerization to the native amide form at neutral pH. This ensures seed-free solutions, minimizing experimental variability. Additionally, we demonstrate the efficacy of coupling NMR diffusion ordered spectroscopy (DOSY) with the Inverse Laplace Transform (ILT) reconstruction method, for effective characterization of early aggregation processes. This innovative approach efficiently maps oligomers distributions across a wide spectrum of initial peptide concentrations offering unique insights into the evolution of oligomers relative populations. As a proof of concept, we demonstrate the efficacy of our approach assessing the impact of Epigallocathechin gallate, a known remodeling agent of amyloid fibrils, on the oligomeric distributions of aggregated Aβ1-40. The DOSY-ILT proposed approach stands as a robust and discriminating asset, providing a powerful strategy for rapidly gaining insight into potential inhibitors' impact on the aggregation process.

Laplace-domain crosstalk-free source-encoded elastic full-waveform inversion using time-domain solvers

Crosstalk-free source-encoded elastic full-waveform inversion (FWI) using time-domain solvers demonstrates skill and efficiency at conducting seismic inversions involving multiple sources and receivers with limited computational resources. A drawback of common formulations of the procedure is that, by sweeping through the frequency domain randomly at a rate of one or a few sparsely sampled frequencies per shot, it is difficult to simultaneously incorporate time-selective data windows, as necessary for the targeting of arrivals or wave packets during the various stages of the inversion. Here, we solve this problem by using the Laplace transform of the data. Using complex-valued frequencies allows for damping the records with flexible decay rates and temporal offsets that target specific traveltimes. We present the theory of crosstalk-free source-encoded FWI in the Laplace domain, develop the details of its implementation, and illustrate the procedure with numerical examples relevant to exploration-scale scenarios.

Interferometric phase diagram generation method based on dot trace texture information and semantic segmentation of correlation coefficients

This paper proposes a method for generating interferometric phase diagrams, which includes interferometric image registration based on Dot Trace Texture Information (DTTI) and interferometric phase filtering based on semantic segmentation. First, the InSAR images are sharpened according to the different backscattering characteristics of various pixel values, and DTTI is extracted to generate the Dot Trace Texture Map (DTTM). Second, SIFT feature points are extracted from two InSAR coherent complex images. They are then screened by comparison with DTTI. These selected points are used to calculate the registration relationship, ensuring precise registration of the two images. Third, the interferometric phase diagram is segmented by semantics based on correlation coefficients. For high-quality interferometric phase regions, we use two-dimensional frequency domain rotational symmetric Gaussian Laplace and wavelet domain filters successively. For low-quality regions, we use an average filter to smooth out noise. The superiority of the proposed algorithm was verified through comparative experiments.

Adaptive consensus control of uncertain nonlinear multi-agent systems with communication delays and relative output information

This paper explores the challenges of distributed adaptive consensus control problems in nonlinear multi-agent systems with unknown parameters and communication delays. To address these issues, a novel backstepping-based adaptive state feedback control protocol is proposed by utilizing a reference dynamic system. Furthermore, an output-feedback consensus control algorithm incorporating a state observer is introduced. Through the application of Laplace transformation, matrix properties, and graph theory, it is demonstrated that the closed-loop multi-agent system ensures the boundedness of all signals and achieves output consensus, even in the presence of nonuniform communication delays. Numerical simulations on a group of inverted pendulums illustrate the effectiveness of these algorithms.

Dam control system based on DDPG algorithm

Abstract Dams, regarded as a form of hydraulic engineering, are purposed to intercept river or canal flow to either elevate water levels or regulate discharge rates. The discharge of a dam is subject to influence by various factors, including precipitation, temperature, and regulatory policies. Nevertheless, in real-world scenarios, the precise value of the expected dam flow rate is challenging to ascertain through simple arithmetic operations. In this study, a novel dam flow model has been formulated. Through the utilization of the DDPG algorithm and deep learning techniques applied to data from the current month, the parameters of the dam control differential equations have been determined, and the transfer function after Laplace transformation has been established. Consequently, only the current month’s fluctuations in the water level of Lake Huron need to be determined to derive the corresponding dam flow rate for that month.

Laplace Transform Based Modeling of Drug Delivery with Reversible Drug Binding in a Multilayer Tissue.

Drug delivery from a drug-loaded device into an adjacent tissue is a complicated process involving drug transport through diffusion and advection, coupled with drug binding kinetics responsible for drug uptake in the tissue. This work presents a theoretical model to predict drug delivery from a device into a multilayer tissue, assuming linear reversible drug binding in the tissue layers. The governing mass conservation equations based on diffusion, advection and drug binding in a multilayer cylindrical geometry are written, and solved using Laplace transformation. The model is used to understand the impact of various non-dimensional parameters on the amounts of bound and unbound drug concentrations as functions of time. Good agreement for special cases considered in past work is demonstrated. The effect of forward and reverse binding reaction rates on the multilayer drug binding process is studied in detail. The effect of the nature of the external boundary condition on drug binding and drug loss is also studied. For typical parameter values, results indicate that only a small fraction of drug delivered binds in the tissue. Additionally, the amount of bound drug rises rapidly with time due to early dominance of the forward reaction, reaches a maxima and then decays due to the reverse reaction. The general model presented here can account for other possible effects such as drug absorption within the device. Besides generalizing past work on drug delivery modeling, this work also offers analytical tools to understand and optimize practical drug delivery devices.

Efficient computational hybrid method for the solution of 2D multi-term fractional order advection-diffusion equation

Abstract Multi-term time-fractional advection diffusion equations are vital for simulating a wide range of physical phenomena, including fluid dynamics and environmental transport processes. However, due to their natural complexity, these equations pose challenges for conventional numerical approaches. In this article, we develop a high order accurate method to solve the multi-term time-fractional advection diffusion equations. We combine the Laplace transform (LT) to integrate the considered equations in time, with Chebyshev spectral method (CSM) for spatial terms The proposed method produces highly accurate solutions with remarkably low computational cost as compared to finite difference method. The propose numerical scheme first employs the LT which reduces the considered problem into a finite set of elliptic equations which may be solved in parallel. Then, the CSM is employed for the disctrezation of spatial operators, which makes it possibly to accurately represent the solution chebyshev grid. Finally, numerical inversion of LT is used to convert the obtain solution from the Laplace domain into the real domain. This work utilizes the modified Talbot’s method and Stehfest’s method for numerical inversion of the LT. To measure the performance, efficiency, and accuracy of the suggested approach, numerical approximations of three models are acquired and verified against the exact solution. The outcomes presented in tables and figures demonstrate that the modified Talbot’s method performed better as compared to Stehfest’s method.

Thermal effect on the transient behavior of a piezomagnetic half-space subjected to dynamic anti-plane load

Considering the importance of understanding the propagation of transient waves in the piezomagnetic solids, the thermal effect on the transient behavior of a piezomagnetic half-space subjected to dynamic anti-plane load is investigated analytically in this paper. Using one-sided, two-sided Laplace transformation and Cagniard–de Hoop (CH) technique, an efficient and accurate analytical derivation for the solution of the anti-plane displacement, shear stress, magnetic potential, and induction in Laplace domain is presented. The study shows that the thermal stresses developed in x-axis and y-axis directions have significant influence on the transient response of the half-space. The magnetic induction B x increases obviously when the thermal stress is applied in x-axis direction, while it decreases when the thermal stress is applied in y-axis direction. Approaching time of magnetic induction B x and B y will become longer with higher thermal stress in x-axis direction. With the growth of the thermal stress in x-direction, contribution from the electromagnetic–elastic head (EH) wave increases, while the contribution from the shear elastic (SE) wave and the static value of shear stress decrease.

A Novel Approach to Solve Nonlinear Higher Order VFIDE Using the Laplace Transform and Adomian Decomposition Method

A Novel Approach to Solve Nonlinear Higher Order VFIDE Using the Laplace Transform and Adomian Decomposition Method

Decoupling of the structure functions in momentum space based on the Laplace transformation

Using Laplace transform techniques, we describe the determination of the longitudinal structure function FL(x,Q2), at the leading-order approximation in momentum space, from the structure function F2(x,Q2) and its derivative with respect to lnQ2 in a kinematical region of low values of the Bjorken variable x. Since the x dependence of F2(x,Q2) and its evolution with Q2 are determined much better by the data than FL(x,Q2), this method provides both a direct check on FL(x,Q2) where measured, and a way of extending FL(x,Q2) into regions of x and Q2 where there are currently no data. In our calculations, we ultilize the Block-Durand-Ha parametrization for the structure function F2(x,Q2) [M. M. Block , .]. We find that the Laplace transform method in momentum space provides correct behaviors of the extracted longitudinal structure function FL(x,Q2) and that our obtained results are in line with data from the H1 Collaboration and other results for FL(x,Q2) obtained using Mellin transform method. Published by the American Physical Society 2024

Analysis of a hollow cylinder with variable thermal conductivity and diffusivity by fractional thermoelastic diffusion theory

Based on fractional thermoelastic diffusion theory, an infinite hollow cylinder with variable thermal conductivity and diffusivity is studied. The inner surface of the hollow cylinder is free and subjected to thermal and chemical shocks, while the outer surface is fixed and adiabatic. The closed forms of dimensionless displacement, temperature, chemical potential, stress, and concentration in the Laplace transform domain are given by eigenvalue method. These physical quantities are then numerically obtained through the inverse Laplace transformation. The effects of fractional order parameters, variable thermal conductivity and diffusivity on these quantities are discussed. The present model may be helpful for better understanding the thermoelastic diffusion problems in actual engineering.

Dynamic Thermal Response of Multiple Interface Cracks between a Half-Plane and a Coating Layer under General Transient Temperature Loading.

This paper explores the thermal behavior of multiple interface cracks situated between a half-plane and a thermal coating layer when subjected to transient thermal loading. The temperature distribution is analyzed using the hyperbolic heat conduction theory. In this model, cracks are represented as arrays of thermal dislocations, with densities calculated via Fourier and Laplace transformations. The methodology involves determining the temperature gradient within the uncracked region, and these calculations contribute to formulating a singular integral equation specific to the crack problem. This equation is subsequently utilized to ascertain the dislocation densities at the crack surface, which facilitates the estimation of temperature gradient intensity factors for the interface cracks experiencing transient thermal loading. This paper further explores how the relaxation time, loading parameters, and crack dimensions impact the temperature gradient intensity factors. The results can be used in fracture analysis of structures operating at high temperatures and can also assist in the selection and design of coating materials for specific applications, to minimize the damage caused by temperature loading.

Discharge variations of qanat near an ephemeral stream

Ephemeral streams are the most important sources of groundwater recharge in arid and semi-arid regions of the world. To extract this replenishable water, most of the time, qanats are constructed parallel to the ephemeral streams. Analytical modeling of a stream-qanat system can provide an effective tool for managing such a system in a timely fashion and, thus, is commonly applied in the hydrological communities. However, the available analytical solutions of discharge variation of qanat only consider the distributed recharge from rainfall and cannot be used to investigate the effects of geometric parameters of stream and hydraulic parameters of aquifer on the discharge of qanat near an ephemeral stream. Thus, the goal of this work is to obtain a semi-analytical solution of qanat discharge due to recharge from an ephemeral stream of an arbitrary shape in a three-dimensional anisotropic homogeneous unconfined aquifer. The solution is obtained via Laplace and Fourier transformations along with the principle of superposition. The results of this study are presented in the form of qanat discharge-time curves. The influences of the qanat length, shape, and recharge distribution along the ephemeral stream are explored in great detail. The effects of the hydraulic parameters of the aquifer on the discharge of qanat are also investigated. The results of this study can be used to predict the discharge of qanat due to recharge from an ephemeral stream, to calculate the optimum length of a qanat for maximum extraction of the replenished water, to estimate the adequate recharge from the ephemeral stream and the hydraulic parameters of aquifer using the discharge data of qanat, among other applications.

Annular unidirectional slip Couette flow with heat transfer and variable viscosity by Laplace transform and homotopy perturbation method

This article investigates the thermal, variable viscosity axial Couette flow between two con- centric circular cylinders, taking into account both the Navier and the dynamic slip boundary conditions. In order to put in evidence the effect of dynamic slip on the behavior of the solution, we have kept the flowing fluid to a simple Newtonian viscous fluid. The nonlinear governing equations for momentum and energy balance are solved under startup condition using the Laplace transform (LT) technique, in conjunction with the Homotopy Perturbation Method (HPM). The first two orders of approximation for temperature and velocity are obtained, as well as the entropy generation in the space occupied by the flow and the volumetric flux. Numerical results and graphical representations of the solution are provided and discussed to depict the effects of the slip parameters on the velocity and the temperature profiles, on the entropy generation rate and on the volumetric flux.