Solution In Laplace Domain Research Articles

Application of the inverse Laplace transform techniques to solve the generalized Bagley–Torvik equation including Caputo’s fractional derivative

This article employs the Laplace transform approach to solve the Bagley–Torvik equation including Caputo’s fractional derivative. Laplace transform is a powerful method for enabling solving integer and non-integer order ODEs and PDEs in engineering and science. Using the Laplace transform for solving integer and non-integer order ODEs and PDEs, however, sometimes leads to solutions in the Laplace domain that are difficult to convert back into the time domain using analytical techniques. Therefore, the best option to convert the Laplace domain solution back into real domain is to use numerical inversion techniques. But it is important to remember that the inverse Laplace transform is not a well-posed problem. There is not a single, general approach that can address every challenge in this matter. Numerous algorithms are devised for numerically inverting the Laplace transform. In this paper, we have evaluated three numerical inverse Laplace transform techniques. These methods include the Fourier series method, the Gauss–Hermite quadrature method, and Zakian’s method. Different test problems are used to evaluate these inversion methods. The effectiveness and efficiency of these methods are demonstrated through graphical representations and tabular data. The comparison of the obtained results with the results of other methods available in literature led us to conclude that the proposed methods exhibit rapid convergence with optimal accuracy without any time instability.

Effect of leakage on confined non-darcian aquifer considering storage in semipervious layers

A set of analytical solutions have been presented for non-Darcian flow to a well in a leaky confined aquifer system considering leakage from the storage of semipervious layers/adjacent aquifer lying above and below the main aquifer. Flow in the main aquifer has been assumed to be non-Darcian and horizontal, whereas flow in the semipervious layers is assumed to be Darcian and vertical. The Izbash equation has been employed to describe the non-Darcian flow. Six general cases have been considered with/without wellbore storage and different adjacent aquifer conditions. Analytical solutions under unsteady and steady-state conditions have been obtained in the Laplace domain. Talbot method has been utilized for numerical inversion Laplace domain solution to the actual solution domain. The obtained analytical solutions clearly quantifies the effect of storage.

Temperature field of multi-barrier with gap layer in nuclear waste repository

A three-dimensional four-layer thermal conduction model including an air layer was developed to analyze temperature evolution in nuclear waste repository. Laplace-domain temperature solutions for each layer were obtained using finite Fourier transform and Laplace transform. The Laplace-domain solutions were transformed into time-domain semi-analytical solutions by numerical inversion. The surface temperature of central canister was obtained by superimposing the temperature increments of air and buffer layers on rock wall temperature caused by every canister in one disposal panel. The effects of the geometrical parameters, thermal parameters and nuclear waste parameters on surface temperature of canister were studied based on the obtained semi-analytical solution. The results indicate that the presence of air layer has little effect on temperature distribution in bentonite while its effect on temperature at surface of canister is significant. Thickness of air layer has a greater effect on temperature than that of its thermal conductivity. The burn-up value and cooling time also have significant influence on the temperature. Finally, the influences of repository’s geometric size on temperature in this model were analyzed.

One-dimensional consolidation of unsaturated–saturated soil system considering pervious or impervious drainage condition induced by time-dependent loading

Since the ground with unsaturated soil layer and saturated soil layer is very common in most areas, a more comprehensive analytical solution is essential for the accurate prediction of ground consolidation behavior. In this paper, the one-dimensional consolidation theories for unsaturated and saturated soils are introduced to simulate the consolidation characteristics of the different soil layers, and the hypothetical soil interface of two layers that only allows pore water to flow is supposed in this model. After using the decoupling and Laplace transformation techniques to the coupled governing equations, the general Laplace-domain solutions are proposed under the permeable or impermeable surface for ground consolidation that an unsaturated–saturated ground with finite thickness subject to arbitrary time-dependent loading. The results in the temporal domain for the excess pore pressures and soil settlement are numerically obtained by Crump’s algorithm. The methodology and proposed solution are validated by some published solutions in the literature. Based on the proposed solutions under the three representative boundary conditions, the temporal behavior of ground subjected to ramp loading is explored. The results show that boundary condition would significantly affect the evolutions of excess pore pressures and further change the ground settlement.

Dynamic response of the half-space subjected to a moving point load and thermal stress

Dynamic transient response of the half-space subjected to a moving point load and thermal stress is investigated analytically in this study. By employing the Helmholtz decomposition and introducing a moving coordinate system, the corresponding modified partial differential equations of motion for the transient waves in the half-space are firstly obtained. With one-side and two-side Laplace transformation over the new time and space variables, the second-order partial differential equations of motion in the modified system are then simplified as the ordinary differential equations, whose solutions are thereafter obtained when the boundary condition is satisfied. To get the dynamic response in time domain, the analytical solutions in Laplace domain are inverted using the Cagniard-de Hoop method. Some examples are evaluated and discussed in details for the purpose of examining the effect of the moving load and thermal stress on the transient response of the half-space.

Semi-analytical model for solute transport in a three-dimensional aquifer with dual porosity and a volumetric source term

A new solution for advective–dispersive solute transport is developed for three-dimensional aquifer of finite thickness and width, infinite extent along the direction of flow, dual porosity, equilibrium linear sorption and first-order decay of both the sorbed and dissolved mass with coefficients that may differ in the mobile and immobile domains, and with initial concentration distribution or with source represented by a volumetric mass generation term of arbitrary time-dependence. The analytical Laplace domain solution is numericaly inverted into the time domain. The model is benchmarked against MT3DMS and simpler analytical models. The model is sensitive to transport parameters describing the dual domain, suggesting it may be used for evaluating tracer tests. The model is suitable for checking accuracy of numerical models and identifying ranges of key transport parameters.

Axisymmetric transient response of a cylindrical cavity in an unsaturated poroelastic medium

In this study, the transient response of an infinite unsaturated poroelastic medium with an infinitely long uniform cylindrical cavity subjected to axially symmetric transient radial tractions is investigated. Using the Laplace transform, analytical solutions are obtained for three different types of prescribed transient radial pressures (step, gradually applied step, and triangular pulse loads) acting on the surface of a permeable cavity and that of an impermeable cavity. The time histories of the radial displacement, stresses, pore pressure, and discharge are derived by inverting the Laplace domain solutions using a reliable numerical scheme. The solution can be simplified to one in which the infinite medium is assumed to be a saturated poroelastic medium. The validity of the solution was confirmed by comparison with COMSOL numerical solutions. A detailed parametric study is presented to illustrate the influence of the saturation of the unsaturated poroelastic medium on the response of the radial displacement, hoop stress, and pore fluid pressure at the cavity surface under different boundary conditions.

Numerical study for energy performance optimization of hollow concrete blocks for roofing in a hot climate of Morocco

• A numerical approach is performed for calculating the thermal inertia parameters. • The traditional roofs had the most significant total thermal load over 24-hours. • Integrate passive techniques into hollow blocks enhances their energy performance. • The inside heat transfer coefficient strongly impacts the roof's thermal inertia. • The outside heat transfer coefficient has no effect on the roof's thermal inertia. The roof is the component of the envelope more subjected to climatic excitations, and it contributes to a significant part of the total building's cooling load in hot climates. Therefore, the thermal optimization of hollow concrete blocks for roofing can considerably reduce the enormous energy consumption used for cooling buildings. For this purpose, the thermal inertia of different configurations of roofs built with traditional concrete blocks used in Morocco with three and six cavities was studied numerically. First, a two-dimensional coupled heat transfer by conduction, convection, and radiation was investigated in the steady-state using the Galerkin finite element method to generate an equivalent homogeneous monolayer block with similar thermal behaviour to the original hollow-block roof. After that, the transient heat conduction equation under real thermal solicitations is solved using a numerical inversion of the Laplace domain solution by utilizing the De Hoog algorithm to evaluate the dynamic thermal characteristics of the roofs. The results report that the emissivity and inside combined convection and radiation heat transfer coefficients significantly impact the roof's thermal inertia. In addition, it is shown that the configuration based on using a low emissivity coating and inserting an insulating material between the outer surface of the block and the slab could reduce the total thermal load over 24-hours by about 93.1% compared to the traditional configuration, which is beneficial to minimize the energy consumption used for cooling buildings.

Theoretical investigation on the impact of an oscillating time-dependent pressure gradient on Dean flow in a porous annulus

In this article, a theoretical analysis on flow in a curvilinear horizontal coaxial cylinder with permeable walls has been proposed. Specifically, the transient impact of an oscillating pressure gradient has been taken into account. The non-linear time-dependent partial differential equation accountable for the flow has been transformed using the classical Laplace transform technique. Exact solution of the momentum equation has been obtained in Laplace domain. Due to the intricacy of the Laplace domain solutions, a numerical inversing technique which is established upon the Riemann-sum approximation (RSA) has been utilized to transform the Laplace domain solutions to time domain. Findings reveal that the outcome of suction on the porous walls and boosting the frequency of oscillation renders skin frictions on both walls of the cylinder less effective. The instability of the Dean vortices in the annular gap can be suppressed by amplifying the frequency of oscillating pressure gradient while time is maintained.

Hydrodynamic effect of slip boundaries and exponentially decaying/growing time-dependent pressure gradient on Dean flow

Hydrodynamic behaviour of slip flow and radially applied exponential time-dependent pressure gradient in a curvilinear concentric cylinder is examined. A two-step method of solution has been utilized in resolving the governing momentum equation. Accordingly, the exact solution of the time-dependent partial differential equation is derived in terms of the Laplace parameter. Afterwards, the Laplace domain solution is then inverted to time domain using a numerical-based inverting scheme known as Riemann-sum approximation. The effect of various dimensionless parameters involved in the problem on the Dean velocity, shear stresses and Dean vortices is discussed with the aid of graphs. It is found that maximum Dean velocity is due to an exponentially growing time-dependent pressure gradient and slip wall coefficient. Stability of the Dean vortices is achieved by suppressing time, wall slippage and inducing an exponentially decaying time-dependent pressure gradient.

Semi-analytical solutions for transient flow to a partially penetrated well with variable discharge in a general three-layer aquifer system

A general semi-analytical model for transient flow in a three-layered aquifer system with a partial penetration well having a variable discharge of pumping is developed with the consideration of the interface flow on the adjacent layers. This general three-layer system includes the conventional aquitard-aquifer-aquitard system as a subset and does not require that the permeability contrasts of different layers must be greater than a few orders of magnitude and does not ignore any flow components (either vertical or horizontal) in any layer. The pumping well of infinitesimal radius is screened at any portion of the middle layer. Three widely used types of top and bottom boundary conditions are assigned prescribed head (Case 1), zero flux (Case 2), or prescribed head at top and zero flux at bottom (Case 3). Laplace domain solutions for dimensionless drawdown are obtained using Hankel transformation, and associated time-domain solutions are evaluated numerically. The newly obtained solutions include some available solutions for two- or single-layer aquifer systems as subsets. The drawdowns for individual layers caused by a well with an exponentially decreased discharge are explored as an example of illustration. The results indicate that the pumped layer drawdown close to the partially penetrated well is mainly influenced by the variable pumping rate. The late-time drawdown for all layers is remarkably affected by the chosen types of top and bottom boundary conditions, and the drawdown for Case 3 is greater than that for Case 1 and smaller than that for Case 2. Additionally, the effect of the pumped layer anisotropy on drawdowns in the three-layer system is significant, and the anisotropy of the unpumped layers significantly affects the drawdown in the whole aquifer system without large contrast of hydraulic conductivity between the unpumped layers and the pumped layer. The drawdowns in all three layers are greatly affected by the location and length of well screen, and a larger drawdown can be seen at the position that is closer to the middle point of the screen of the partially penetrating pumping well.

Impact of Thermal Stratification on Unsteady Natural Convection Couette Flow Formation in a Vertical Channel Filled with Anisotropic Porous Material

Recently, heat transfer problems where anisotropic porous medium or stably stratified fluid are taken into account have been separately studied. Developing a mathematical model that combines these physical quantities naturally results to complex coupled differential equations. In this paper, a fully developed time dependent natural convection Couette flow of stably stratified fluid between vertical parallel channels filled with anisotropic porous material is investigated. The governing partial differential equations are transformed into ordinary differential equations using Laplace transform techniques and then decoupled using D’Alembert method. Exact solutions in Laplace domain for the velocity and temperature equations are then obtained. A numerical method: Riemann-sum approximation is then used to invert the expressions for the velocity and temperature profiles, as well as the resulting skin friction, rate of heat transfer and volumetric mass flow rate into their corresponding time domain. The research establishes that both the anisotropic and the stratification parameters aid in regulating the fluid temperature and velocity. The research further reveals that the fluid velocity attains its maximum (or minimum) velocity when θ = 900 (or θ = 00) for k*<1 and when k*>1, the fluid velocity is least (or maximum) when θ = 900 (or θ = 00).

Time-Dependent Wave-Structure Interaction Revisited: Thermo-Piezoelectric Scatterers

In this paper, we are concerned with a time-dependent transmission problem for a thermo-piezoelectric elastic body that is immersed in a compressible fluid. It is shown that the problem can be treated by the boundary-field equation method, provided that an appropriate scaling factor is employed. As usual, based on estimates for solutions in the Laplace-transformed domain, we may obtain properties of corresponding solutions in the time-domain without having to perform the inversion of the Laplace-domain solutions.

Role of exponentially decaying/growing time-dependent pressure gradient on unsteady Dean flow: a Riemann-sum approximation approach

Our study probes the impact of an exponentially decaying/growing time-dependent pressure gradient on unsteady Dean flow in a curved concentric cylinder. A two-step method of solution has been employed in the treatment of the governing momentum equation. Accordingly, the exact solution of the time-dependent partial differential equation is derived in terms of the Laplace variable. The Laplace domain solution is then transformed to the time domain using a numerical inversing scheme known as Riemann-sum approximation. The effect of the various dimensionless parameters involved in the problem on the Dean velocity, skin drags and Dean vortex are illustrated graphically. It was established that maximum Dean velocity is due to an exponentially growing time-dependent pressure gradient. However, the instability of the Dean vortex is rendered less effective by reducing time and applying an exponentially decaying time-dependent pressure gradient.

Analytical solution in Laplace domain of Smoluchowski equation for a flat potential with a rectangular sink

We propose a new method for finding the exact analytical solution in Laplace domain for the problem where a particle is diffusing on a flat potential in the presence of a rectangular sink of arbitrary width and height. In our model, diffusive motion is described by the Smoluchowski equation. Our method with this sink of rectangular shape is very general and can be used to deal with other potentials. We have derived exact analytical expression for rate constants using our model. This is the first model where the exact analytical solution in closed form is found in the case of a sink of arbitrary width. This model is more realistic for understanding reaction–diffusion systems that all other existing models available in literature.

Three-dimensional thermal–hydraulic coupled analysis in the nuclear waste repository

In this paper, coupled governing equations were proposed to simulate three-dimensional heat conduction and moisture transport in the nuclear waste repository. Because there will be a large temperature gradient near inner and outer boundaries of the bentonite buffer layer, and the heat driven moisture transport is introduced in the governing equation of moisture transport. By applying the Fourier and Laplace transforms, the governing equations were converted to the Bessel equations, and the Laplace-domain solutions to temperature and moisture distributions were derived through the inverse Fourier transform. The reliability of the proposed solutions was verified through comparative analysis with the line heat source model. These analytical solutions were applied to obtain the evolutions of temperature field and moisture distribution near the waste canister. Finally, a sensitivity study was performed to analyze the effects of relevant parameters on the heat driven moisture transport.

New Simplified Models of Single‐Well Push‐Pull Tests With Mixing Effect

AbstractWhen using single‐well push‐pull (SWPP) tests to characterize in situ aquifer related to reactive transport, approximate analytical solutions were preferred to determine the aquifer dispersivity and other parameters, due to its computational efficiency. Simplified models were proposed in this study to estimate the retardation factor for a SWPP test with a reactive tracer, where the dispersivity had to be determined firstly using a SWPP test with a conservative tracer. Here, we find that previous models associated with SWPP tests contain an untested assumption when dealing with solute transport in the wellbore, probably causing great errors when interpreting the test data. This assumption states that the wellbore mixing effect is negligible in both injection and extraction phases, where the mixing effect refers to the mixing process between the native wellbore water and solute introduced into the wellbore. In this study, new approximate analytical solutions of the SWPP test are proposed for a fully penetrating well and a point‐source well by specifically considering the wellbore mixing effect. The new models of this study are tested by semianalytical solutions in Laplace domain and finite difference solutions. Results indicate that the new models perform substantially better than previous models of ignoring the wellbore mixing effect, which should not be overlooked when interpreting SWPP test data.

A New Efficient Formula for the Thermal Diffusivity Estimation from the Flash Method Taking into Account Heat Losses in Rear and Front Faces

The flash method is a commonly used technique for estimating the thermal diffusivity of solid materials. In this work, we develop a new efficient formula for the thermal diffusivity calculation to process the flash method data in the case of heat losses at the rear and front faces. First, the transient heat equation is analytically solved using the Green functions under time-dependent convection boundary conditions. The property of the descending part of the analytical solution, containing series which converges very quickly for long times, allows to express the thermal diffusivity as a function of the slope of the natural logarithm of the temperature rise and the Biot numbers related to heat losses at the rear and front faces. These Biot numbers are calculated from the ratio between the integrals of the complete temperature trends of the two faces. For this purpose, we developed a novel way to evaluate these integrals using the analytical solutions in Laplace domain. To verify the accuracy of these formulas, we simulated experimental data adding a Gaussian noise to the theoretical temperature results of the flash method data.

Analysis of gradient elution chromatography using the transport model

A transport model is considered to describe gradient elution in liquid chromatography in packed beds with the linear isotherm dependent on the mobile phase modulator. By applying a coordinate transformation, the model is solved analytically using the Laplace transform approach. The moment generating property of the Laplace domain solution is used to derive analytical expressions for the first three moments of the response to rectangular injections. These moments are instructive for analyzing the retention time, band broadening and asymmetry of elution profiles. Compared to isocratic elution, the derivation of analytical solutions and moments for gradient elution is more complicated, because the retention behavior of the solutes depends on the varying mobile phase modulator. Several case studies are evaluated theoretically. To gain confidence on the derived analytical results, a high-resolution finite volume scheme is also applied to solve the same model equations numerically. The analytical solutions and moments provided are utilized to predict the effects of starting and ending times of gradient, magnitude of modulator concentration variation, gradient slopes, and mass transfer coefficient on retention and peak shape. The analytical moment expressions derived can be used to determine retention and mass transfer parameters from experimental peaks and to predict elution behaviors if these parameters are known.

Influence of Biot’s Compressibility Parameters on Transient Response of Saturated Poroelastic Media

Abstract In many branches of engineering, the effect of the compressibility parameters of solid and fluid are significant in describing the realistic behavior of porous media. The principle goal of the present work is to clarify the influence of the compressibility of each constituent on consolidation and dynamic behavior of porous media. In this research, an analytical solution in Laplace domain is obtained for dynamic 1-D problem, and further, the response in time domain is evaluated using the convolution quadrature method. By considering a wide range of compressibility parameters, the consolidation and wave propagation behavior are discussed in terms of pore pressure and solid displacement. These results offer some guidance in choosing appropriate compressibility parameters in practical problems.