Laplace Transform Method Research Articles

Numerical inverse Laplace transformation using concentrated matrix exponential distributions

This paper investigates the performance of the numerical inverse Laplace transformation (ILT) method based on concentrated matrix exponential (CME) distributions, referred to as the CME method.The CME method does not generate overshoot and undershoot (i.e., avoids Gibbs oscillation), preserves monotonicity of functions, its accuracy is gradually improving with the order, and it is numerically stable even for order 1000 when using machine precision arithmetic, while other methods get unstable already for order 100 using the same arithmetic.For ILT based tail approximation, the paper recommends an abscissa shifting approach which improves the accuracy of most ILT methods and proposes a heuristic procedure to approximate the numerical accuracy of some ILT methods.

The application of the improved Talbot’s inverse Laplace transformation method in solving the flow problem of porous materials

In this paper, we extend the fixed Talbot’s method to the complex-valued function in order to get the general solutions for the Biot’s consolidation in the physical domain. We derive a solution for the unsteady flow field of layered porous media with anisotropic permeability under a point fluid source. By a Laplace and two-dimensional(2D) Fourier transform, the continuity equation of the fluid can be solved, and the flow field can be expressed in an analytical form in the transformed domain. Using the boundary and interface condition, the flow field for general layered porous media can be solved in the transform domain. The actual solutions in the physical domain can be obtained by inverting the Laplace-2D Fourier transform. Numerical examples are given to demonstrate the validity of the extended fixed Talbot’s inverse Laplace transformation method, and its application in solving dymanic problems of porous materials.

Computation of State Vector of Networked Control System using Combination of Laplace Transform and Variational Iteration Method

Networked Control System (NCS) is a control system in which system components such as sensor, actuator and controller communicate over a network. Networked control system model is represented by , where the state vector and the control input . In this paper the state vector of the networked control system is computed by obtaining the solution of first order delay differential equation using the combination of Laplace and Variational Iteration method.

Unsteady water functionalized oxide and non-oxide nanofluids flow over an infinite accelerated plate

The purpose of this communication is to examine the collective influence of heat and mass transfer on magnetohydrodyanmics (MHD) water functionalized oxide or non-oxide nanofluid flows through porous medium together with thermal radiation and chemical reaction effects. The Newtonian heating on the accelerated wall in the presence of different nanoparticles is considered first time and not explored in earlier investigations. To calculate the temperature and velocity fields, the Maxwell model for effective thermal conductivity with relatively large particles is used. Three types of base fluids namely water, kerosene and engine oil containing different types of oxide and non-oxide nanoparticles are taken for this study. Analytic solutions of the governing equations are obtained with the help of Laplace transform method (LTM). Some important cases such as free convection and motion of plate with variably accelerated, constant accelerated, uniform velocity as well as in the absence of free convection are obtained. Graphs are plotted to analyse the characteristics of different involved parameters on velocity, temperature and concentration.

New approximate and asymptotic closed-form expressions for the outage probability and the average BER of MIMO-FSO system with MRC diversity technique over Gamma–Gamma fading channels with generalized pointing errors

This paper studies the outage probability (OP) and the average bit-error rate (BER) of multiple-input multiple-output (MIMO) free-space optical (FSO) system with maximal ratio combining (MRC) diversity technique over Gamma–Gamma (GG) fading channels with generalized pointing errors modeled by the Beckmann distribution. A new approximate closed-form probability density function (PDF) at high signal-to-noise ratio (SNR) for MRC scheme is derived by inverse Laplace transformation method. Based on the developed PDF, new approximate and asymptotic closed-form expressions for the OP and the average BER are derived, respectively. Then, the OP and the average BER performance of the MIMO-FSO system with MRC scheme at high SNR are analyzed with different parameters, including number of transmitters and receivers, receiver aperture sizes, nonzero boresight, and different jitters. Finally, numerical results and Monte Carlo simulations corroborate the accuracy of the derived approximate and asymptotic expressions over a wide range of SNR.

Influence of drying methods on the thermodynamic parameters, effective moisture diffusion and drying rate of wastewater sewage sludge

In this study, thin layer drying experiments of direct and indirect natural convection solar drying of wastewater sewage sludge in Algeria were conducted. The purpose of these was to analyze the effect of two methods of drying on different parameters, namely: drying rate, efficient diffusion of moisture, and thermodynamic parameters. Three new universal drying kinetic models were proposed and compared with four other models existing in the literature to model the drying process. The effective diffusivity of moisture of sludge (Deff) changed from 1.91 10−8 to 9.12 10−8 m2 s−1. It was obtained by using Fick's second law, based on the resolution by Laplace's transformation method and the Fourier series. Thermodynamic parameters, to know, activation energy, free energy change, enthalpy and entropy change, were also determined using a new methodology build on the evolution of effective diffusion coefficient in terms of temperature by adopting the Eyring equation.

A photothermal excitation for a semiconductor medium due to pulse heat flux and volumetric source of heat with thermal memory

In the present work, the interaction between the generalized thermoelasticity theory and photothermal theory, under the effect of volumetric source of heat with thermal memory, is investigated. The governing equations for this model are taken of a semiconductor medium in one dimension with pulse heat flux. The main goal is to study the overlap between the waves as plasma, thermal and elastic waves. The analytical solutions are obtained for some physical quantities as temperature, displacement, radial stress and carrier intensity in thin circular plate by using the Laplace transformation method. To obtain the complete solutions, the inversion numerical method is used. The semiconductor silicon material is used to obtain numerical results. The obtained physical quantities are discussed and illustrated graphically.

Time-Dependent Nonlinear Convective Flow and Radiative Heat Transfer of Cu-Al2O3-H2O Hybrid Nanoliquid with Polar Particles Suspension: a Statistical and Exact Analysis

The statistical and exact analysis of heat transfer rate and skin friction coefficient of a nonlinear convective flow of Cu − Al2O3 − H2O hybrid nanofluid with polar particle suspension is performed. The heat transport phenomenon includes radiative heat effect. A micropolar fluid model is accounted. Exact solutions to the governing problem are found via Laplace transform method (LTM). The heat transfer rate and skin friction are analysed critically via statistical methods like probable error and regression models. The slope of linear regression of data points for skin friction and Nusselt number is estimated to quantify the increase/decrease. The Nusselt number and thermophysical properties for twenty-four different hybrid nanofluids are presented. A novel idea of a nonlinear convective flow of Cu − Al2O3 − H2O hybrid nanofluid with polar particle suspension is investigated for the first time. Opposite behaviour of velocity and microrotation profile are established when the physical parameters are varied.

Effect of Heat Source/Sink on Free Connective MHD Flow past an Exponentially Accelerated Infinite Plate with Mass Diffusion and Chemical Reaction

The effects of heat source/sink and chemical reaction with mass diffusion on free convective incompressible viscous fluid flow past an accelerated vertical plate with magnetic field has been investigated. Laplace transformation method has been applied to solve the system of linear partial differential equations. The result is presented in form of complementary error function and exponential function. The effect of non dimensional parameters such as Schmidt number (Sc), Accelerated parameter (a), Chemical reaction parameter (K), Prandtl number (Pr), Magnetic field parameter (M), Mass Grashof number (Gm), Heat source/sink parameter (H), Thermal Grashof number (Gr) on temperature, concentration, velocity has been discussed with graphs.

Time Domain Electromagnetic Transient Analysis of Aerial Nonuniform Transmission Lines Excited by an Incident Electromagnetic Field

In this paper a model for Nonuniform Transmission Lines for electromagnetic transient analysis that incorporates frequency dependency of electrical parameters, variation of line electrical parameters with respect to distance, and distributed excitations due to incident electromagnetic fields is presented. The model is developed using the Method of Characteristics in the actual physical domain instead of the Modal Domain; this simplifies the mathematical development and the final equations. Moreover, the equations of the resultant model are valid either for two-conductor lines or for multiconductor lines without any change; this can be an advantage when the computer programming language considers a scalar as a 1x1 matrix. The proposed numerical model is developed under the hypothesis that the dielectric surrounding the conductors is homogenous. It is shown that in this case the characteristic curves of a Nonuniform Transmission Line become straight lines. Finally, the model is validated by comparison with results obtained using the Numerical Laplace Transform method.

Sensor fault diagnosis and fault‐tolerant control for stochastic distribution time‐delayed control systems

SummaryIn this paper, a new fault diagnosis and fault‐tolerant control method based on the model equivalent transformation is proposed for the stochastic distribution time‐delayed control system, in which the random delay between the controller and the actuator and the external disturbance is considered. The system is modeled by using a linear B‐spline to approximate the probability density function (PDF) of system output. The original system is transformed into an equivalent system without random delay based on the Laplace transformation method. Then, the equivalent system that is converted to the augmentation system with a new state variable is introduced. The observer is designed to estimate the fault information based on the augmentation system. Observer gain matrices and controller parameters are obtained by solving the linear matrix inequality. The PI control algorithm is used to make the PDF of the system output track the desired distribution. Finally, the validity of the proposed method is verified by computer simulation results.

DOUBLE LAPLACE TRANSFORM METHOD FOR SOLVING TELEGRAPH EQUATION

Telegraph equation is a partial differential equation which includes the one dimensional wave equation. This equation can be used to solve several problems that physically regulate voltage and current in the electricity transmission line in distance and time. In this study, telegraph equation and partial integrodifferential equation were analyzed and resolved by double Laplace transform. This equation was accompanied by initial values and boundary values solved by Laplace transform. Next, there are some examples of equations to discover a solution. The solution of these examples showed that double Laplace transform is one method that can be used to solve telegraph equation. Then the solutions are simulated in graphical form which shows the waveform of the equation.

S-domain stability analysis of a turning tool with process damping

This work involved S-domain stability analysis of a turning tool with process damping. Process damping is a phenomenon of dry friction between the tool flank face and workpiece which induces high stability and smoothness at low turning speed. A pair of valid equations for stability analysis of turning with process damping was derived in this work using Laplace transformation method. This work for the first time introduced non-linear feed term in the existing process damping model leading to process damping force of form Fpd(vτ)α where Fpd processed damping force model in related literature. It is seen that in light of experiments of the related literature, the new proposal can only be valid for low values of α in the neighbourhood of 0.01. MATLAB delay differential equation solver called dde23 was used to simulate the vibration response of turning processes at selected points on the stability diagram of turning with and without process damping. This helped in confirming that stable points in the stability diagram of turning with process damping became unstable points in the stability diagram of turning without process damping. The stability was much higher at low speed of 200rpm than at speed of 4000rpm. Process damping coefficient Cpd of the experiment was estimated at Cpd of 61,000 N/m by comparing experimental depths of cut with theoretical depths of cut of known process damping coefficient.

Q-homotopy analysis transform method applied to fractional Kundu–Eckhaus equation and fractional massive Thirring model arising in quantum field theory

In this paper, [Formula: see text]-homotopy analysis transform method ([Formula: see text]-HATM) is used to solve fractional Kundu–Eckhaus equation and fractional massive Thirring model. This method is a mixture of Homotopy analysis method (HAM) and Laplace transform method (LTM) when [Formula: see text]. The approximate solutions obtained by [Formula: see text]-HATM are compared with the exact solutions. Numerical results are known through different graphs and tables. The results light the power, efficiency, simplicity, and reliability of the proposed method.

Analytical solution for three-dimensional radial heat transfer in a cold-region tunnel

The study of the temperature field of cold-region tunnels is fundamental for preventing frost heave. At present, considerable knowledge has been obtained by numerical simulation and physical experimentation. In this paper a computational model for three-dimensional radial heat transfer was established according to some reasonable assumptions, and the complex boundary conditions inside the tunnel surface including axial coordinate z and time t were obtained based on the analysis of the field temperature monitoring data. The z and t terms in the boundary were split to simplify the problem. Laplace and Fourier integral transformation methods were combined to solve this problem, and the Den Iseger method was adopted to solve the Laplace inverse transformation. Yuximolegai tunnel was introduced to study its temperature distribution. The freezing length in z direction is reduced from 143 m to 70 m and the maximum radial freezing depth at the entrance is also shortened from 4.7 m to 2.4 m due to 5 cm insulation layer, which reveals that the effect of 5 cm insulation layer was remarkable. Considering the efficiency of insulation layer, it needs to be combined with other measures to prevent frost damage of surrounding rock. The analysis of convective heat transfer coefficient, formation temperature, and thickness of the insulation layer was performed to obtain their influence on the lining temperature. The results show that the influence of the convection heat transfer coefficient is small. The effect of the insulation layer improves with increasing formation temperature and worsens as the thickness of the insulation layer increases. These achievements can enrich the research of temperature fields of tunnels in cold regions and are significant for guiding tunnel design in cold regions.

Analysis of wavepacket tunneling with the method of Laplace transformation

We use the method of Laplace transformation to determine the dynamics of a wavepacket that passes a barrier by tunneling. We investigate the transmitted wavepacket and find that it can be resolved into a sequence of subsequent wave packages. This result sheds new light on the Hartman effect for the tunneling time and gives a possible explanation for an experimental result obtained by Spielmann et al.

Analytical solution of a heat transfer model for a tubular co-current diluted moving bed reactor with indirect heating and intraparticle gradients

The understanding of the multiphase flow and the thermal behavior in moving beds is important for design and process optimization. Thus, in the present work, the heat transfer in a tubular moving bed reactor indirect heated by a flue gas is modeled, considering the thermal interactions: “conveyor fluid-flue gas”, “conveyor fluid-particle”, as well as the intraparticle heat transfer. The model is solved analytically by the powerful Self-Adjoint Operators Method (SAO), which allows the maximum maintenance of the physical characteristics of the problem. The consistency of this novel solution is shown through limiting solutions and comparison with simpler models: a lumped capacity model (LP), whose solution is independently obtained through the Laplace Transform Method, and an existing isothermal wall moving bed heat exchanger (MBHE) solution. For low particle Biot numbers systems, the LP and SAO solutions showed a very good agreement. The results also showed good validation of the solutions by comparing with experimental data. A numerical hybrid solution, through a combination of the Finite Difference Method (FD) and the Finite Analytical Method (FA), is also given and the comparison with the analytical solution shows excellent agreement. Finally, a scale analysis is proposed and performed. The scale analysis made it possible to assign a physical meaning to the model's parameters, to infer about the thermal behavior of the system, to evaluate the use of simpler models and to present semi-theoretical expressions for the maximum temperature difference into the particle and between particle surface and conveyor fluid, for systems with low Bip.

Generalized thermoviscoelastic analysis with fractional order strain in a thick viscoelastic plate of infinite extent

Viscoelastic materials (e.g. rubber, resin, concrete skeleton, etc.) serve as one of the most promising candidate for new multifunctional materials due to their excellent rheological properties, as they are widely used in biology, petroleum industry, chemical and civil engineering, and other areas. The interaction of high-heat-flux loads (e.g. ultrafast pulsed lasers) with viscoelastic materials is a hot topic of numerous theoretical and experimental studies. Nevertheless, classical and extended generalized thermoviscoelasticity models available in literatures cannot characterize the strain relaxation behavior for viscoelastic materials. In this work, the time-based fractional thermoviscoelasticity model is further extended with a new consideration of fractional order strain. The governing equations involving strain and thermal relaxation times as well as fractional order parameters of strain and heat flux are formulated and are also applied to investigate thermal shock problem of a transient heated thick viscoelastic plate of infinite extent. With the aids of Laplace transformation methods, the transient solutions are also obtained. Finally, the effects of fractional order parameters of strain and heat flux on the transient thermoviscoelastic responses are studied and discussed.

Exact and statistical computations of radiated flow of nano and Casson fluids under heat and mass flux conditions

Abstract The statistical and exact analysis of the unsteady radiative flow of Nano and Casson fluids past a vertical plate with Dufour effect is carried out. The heat transport phenomenon is studied under uniform heat flux (UHF) and uniform wall temperature (UWT) conditions. The exact solution to the problem is found using the Laplace transform method (LTM). The effects of various parameters on velocity, temperature and concentration profiles are examined via graphs. The heat transfer rate and skin friction are analyzed through statistical tools like probable error and regression. The Dufour effect enhances the velocity and temperature profiles. It is also observed that the velocity profile is slightly greater in the case of UWT than the UHF case for both nanofluid and Casson fluid. From the regression analysis, it is established that the Dufour number and nanoparticle volume fraction have a negative impact whereas the radiative heat parameter has a positive impact on the rate of heat transfer. Highlights The statistical analysis of the unsteady radiated flow on a vertical plate with Dufour effect is performed. The uniform heat flux (UHF) and uniform wall temperature (UWT) conditions are accounted. The Nusselt number and friction factor are analysed through statistical tools. The velocity profile is greater for UWT case than the UHF case.

Statistical Aspects and Dynamical Entanglement for a Two-Level Atom Moving on Along Cavity Length of x-Direction: Atomic Position Distribution

In this work, the problem of the quantum optical model is considered where an one-mode quantized radiation field interacts with a two-level atom (TLA). Also, the atomic position distribution is taken into account, i.e., the atom passing through the length of the optical cavity. We believe that the atomic position affects the matter-field interaction and can be realized in several multiple experiments, such as ultracold atoms and trapped ions. We take into consideration the atom is moving along the cavity field in the x-direction so that the time-position Schrodinger equation for the atom in the x-direction is obtained. We suppose that the field is initially prepared in a coherent state optical field and the atom is initially prepared in an excited state. Also, the atomic motion along the cavity length vanishes in the cavity wall. By using the Laplace transformation method, an exact analytical solution for the coupled partial differential Schrodinger equation for the wave function is calculated. Some non-classical statistical aspects such as the atomic inversion and the photon distributions are discussed in detail. The collapses-revivals, the Poissonian distributions of the photon by Mandel Q parameter, the degrees of the entanglement, and the fidelity have been investigated. The effect of the atomic position distribution in the x-direction on these phenomena is examined. We observed that the atomic position distribution along the cavity has an effective effect on the quantum statistical properties of these phenomena. Minutely, the influence of the atom location distribution on the amount of quantum entanglement has been obtained.