Laplace Integral Research Articles

Mathematical modeling of groundwater contamination with varying velocity field

Abstract In this study, analytical models for predicting groundwater contamination in isotropic and homogeneous porous formations are derived. The impact of dispersion and diffusion coefficients is included in the solution of the advection-dispersion equation (ADE), subjected to transient (time-dependent) boundary conditions at the origin. A retardation factor and zero-order production terms are included in the ADE. Analytical solutions are obtained using the Laplace Integral Transform Technique (LITT) and the concept of linear isotherm. For illustration, analytical solutions for linearly space- and time-dependent hydrodynamic dispersion coefficients along with molecular diffusion coefficients are presented. Analytical solutions are explored for the Peclet number. Numerical solutions are obtained by explicit finite difference methods and are compared with analytical solutions. Numerical results are analysed for different types of geological porous formations i.e., aquifer and aquitard. The accuracy of results is evaluated by the root mean square error (RMSE).

General Laplace Integral Problems: Accuracy Improvement and Extension to Finite Upper Limits

Many problems in engineering and science involve calculation of difficult Laplace integrals of the form: In previous papers [Hanna and Davis (2011), Davis and Hanna (2013), referred to as Papers I and II, respectively], the authors introduced two new complementary analytical methods for the improved asymptotic (x → ∞) approximation of these integrals when the upper limit A equal to ∞ [the general Watson lemma (WL) problem for any A if x → ∞]. A procedure is developed here that extends application of the previous improvement methods to the difficult problem of Laplace integrals having a finite upper limit (FUL) = A. In addition, problems having growing exponential behavior, certain infinite Fourier integrals and problems having large (tα) factors, are also considered. The main result is that, with some modifications, the exponential, expansion-point, and combination procedures developed for infinite integrals (Papers I and II) can be easily applied directly to FUL Laplace integrals. This is accomplished with the aid of a simple new “generalized incomplete gamma function (IGF)” algorithm which itself utilizes an improvement procedure. The new FUL procedure requires only F(t), F′(t), and a few terms of the Taylor expansion. A simple EXCEL program which implements the new procedure is discussed in detail in Appendix A and is freely available to users at http://www.d.umn.edu/~rdavis/CEC/. Many numerical comparisons presented here indicate that good engineering accuracy is achieved for these improved approximations at virtually all positive A values, over a very wide range in x, for various α, β, and F(t) functions. Where comparisons are possible (A = ∞), the new results are far superior to those of the best Watson’s lemma results.

General fractional calculus operators containing the generalized Mittag-Leffler functions applied to anomalous relaxation

In this paper, we address a family of the general fractional calculus operators of Wiman and Prabhakar types for the first time. The general Mittag-Leffler function to structure the kernel functions of the fractional order derivative operators and their Laplace integral transforms are considered in detail. The formulations are as the mathematical tools proposed to investigate the anomalous relaxation.

Coding in the Finite-Blocklength Regime: Bounds Based on Laplace Integrals and Their Asymptotic Approximations

In this paper we provide new compact integral expressions and associated simple asymptotic approximations for converse and achievability bounds in the finite blocklength regime. The chosen converse and random coding union bounds were taken from the recent work of Polyanskyi-Poor-Verdu, and are investigated under parallel AWGN channels, the AWGN channels, the BI-AWGN channel, and the BSC. The technique we use, which is a generalization of some recent results available from the literature, is to map the probabilities of interest into a Laplace integral, and then solve (or approximate) the integral by use of a steepest descent technique. The proposed results are particularly useful for short packet lengths, where the normal approximation may provide unreliable results.

Nonstationary Deformation of an Elastic Layer with Mixed Boundary Conditions

The analytic solution to the plane problem for an elastic layer under a nonstationary surface load is found for mixed boundary conditions: normal stress and tangential displacement are specified on one side of the layer (fourth boundary-value problem of elasticity) and tangential stress and normal displacement are specified on the other side of the layer (second boundary-value problem of elasticity). The Laplace and Fourier integral transforms are applied. The inverse Laplace and Fourier transforms are found exactly using tabulated formulas and convolution theorems for various nonstationary loads. Explicit analytical expressions for stresses and displacements are derived. Loads applied to a constant surface area and to a surface area varying in a prescribed manner are considered. Computations demonstrate the dependence of the normal stress on time and spatial coordinates. Features of wave processes are analyzed

Nonstationary heat transfer in anisotropic half-space under the conditions of heat exchange with the environment having a specified temperature

An analytical solution to the problem of heating anisotropic half-space by the environment with a spatially and temporally variable temperature (boundary conditions of the third kind at an anisotropic body) has been obtained for the first time based on the construction of the boundary influence function (the Green’s function), which is determined using the Fourier and Laplace integral transforms. Nonstationary temperature fields in anisotropic blunt bodies have been found under the conditions of aero-gas-dynamic heating of hypersonic aircrafts with different heat-transfer coefficients and incoming-flow temperatures. The solution obtained is recommended for determining the state of thermal protection fabricated from composites which are generally anisotropic.

The study of thermal field of an object represented in the basis of green functions

Existing means of thermal field modeling for the construction of the real time spatial distribution of the field were analyzed. The expediency of using the method of integral transformations for solving heat conduction problems was shown. Based on the comparison of methods of thermal fields construction method of integral transformations in the basis of the Green's functions was chosen. Construction of Green's functions for the use in the experimental facility is performed by the method of Laplace integral transforms. Software, generated based on the use of Matlab environment, allows to adjust the visualized field pattern by actual measured temperature values. Temperature point values of the experimental facility field were measured by NAT thermocouples, that are connected to a PC through OWEN 150UM controller interface, which makes it possible not only to form an array of measured values of the temperature, but also to adjust the time-varying boundary conditions of the calculated heat equation. The main purpose of the research is to assess the performance of the developed algorithms to reduce the field frame modeling time, which would allow to use the resulting pattern of the thermal field distribution in the industrial thermal facilities control process. The advantage of the thermal field constructing algorithm in the basis of the Green's function is the ability of the independent calculation of all points of the investigated area, allowing in the future to speed up the computation by applying high-performance computing.

Non-stationary stress state of an elastic layer at the mixed boundary conditions

An exact analytical solution has been constructed for the plane problem on action of a non-stationary load on the surface of an elastic layer for conditions of a ‘mixed’ boundary problem when normal stress and tangent displacement (the fourth boundary problem) are specified on one boundary and normal displacement and shear stress (the second boundary problem) for another boundary. Laplace and Fourier integral transforms are used. Their inversions were obtained with the help of tabular relationships and the convolution theorem for a wide range of acting non-stationary loads. Expressions for stresses (displacements) were obtained in explicit form. The obtained expressions allow determining the wave process characteristics in any point of the layer at an arbitrary moment of time. Some variants of non-stationary loads acting on an area with fixed boundaries or an area with boundaries changing by a known function are considered. Distinctive features of wave processes are analyzed.

Reflection–transmission matrix model for the axisymmetric deformation of a layered transversely isotropic saturated soil

The reflection–transmission matrix (RTM) method for the layered transversely isotropic saturated soil (TISS) undergoing axisymmetric consolidation is developed in this study. To this aim, a system of partial differential equations is established based on the governing equations for the TISS first. By means of the Hankel–Laplace integral transform method, the system of partial differential equations is reduced to a system of ordinary differential equations. By using the general solution to the system of ordinary differential equations, the static wave vector corresponding to the state vector of the TISS is introduced, and the transfer matrices for the state and wave vectors are defined. The RTMs for the aforementioned static wave vector of the layered TISS are then established. By means of the RTMs for the layered TISS, solutions for the layered TISS subjected to external sources are derived. Numerical results due to the proposed RTM method are compared with some existing results, validating the proposed RTM method. Parametric analyses are conducted to illustrate the influence of some parameters on the consolidation of the layered TISS.

On the Laplace integral representation of multivariate Mittag‐Leffler functions in anomalous relaxation

In the given paper, a special method of representation of the Mittag‐Leffler functions and their multivariate generalizations in the form of the Laplace integrals is suggested. The method is based on the usage of the generalized multiplication Efros theorem. The possibilities of a new method are demonstrated on derivation of the integral representations for relaxation functions used in the anomalous dielectric relaxation in time domain. Copyright © 2016 John Wiley & Sons, Ltd.

Generalized Binomial Transform Applied to the Divergent Series

The divergent series for a function defined through Lapalce integral and the ground state energy of the quartic anharmonic oscillator to large orders are studied to test the generalized binomial transform which is the renamed version of $\delta$-expansion proposed recently. We show that, by the use of the generalized binomial transform, the values of functions in the limit of zero of an argument is approximately computable from the series expansion around the infinity of the same argument. In the Laplace integral, we investigate the subject in detail with the aid of Mellin transform. In the anharmonic oscillator, we compute the strong coupling limit of the ground state energy and also the expansion coefficients at strong coupling from the weak coupling perturbation series. The obtained result is compared with that of the linear delta expansion.

Analytical solution of the asymmetric transient wave in a transversely isotropic half-space due to both buried and surface impulses

With the aid of a complete set of two scalar potential functions, the problem of transient wave propagation in transversely isotropic half-space, subjected to time dependent tractions applied on a finite patch at an arbitrary depth below the free surface of the half-space is investigated. With the use of the displacement–potential function relationships in a cylindrical coordinate system, the coupled equations of motion are uncoupled; resulting in two separate partial differential equations one of which is second order and the other is fourth order. These two partial differential equations are solved with the aid of both Fourier series expansion and joint Hankel–Laplace integral transforms. The solutions are also investigated in details for tractions varying with time as Heaviside step function, which may be used as a kernel in any integral based method for more complicated elastodynamic initial-boundary value problems. Moreover, some displacement Green׳s functions are numerically evaluated for a synthetic transversely isotropic material to graphically demonstrate the transient motion of the free surface of the half-space.

Stresses induced in saturated asphalt pavements by moving wheel pressure

The purpose of this paper is to investigate the stresses and excess pore fluid pressure induced by the moving wheel pressure on saturated asphalt pavements, which is one of the reasons for a damage phenomenon in flexible pavements denoted by stripping. The saturated asphalt pavement is modelled as a multi-layered poroelastic half space exerted by a wheel pressure, which is moving at a constant velocity along the surface of the pavement. The governing equations for the proposed analysis are based on Biot's theory of dynamics in saturated poroelastic media. The governing partial differential equations are solved by using Laplace and Hankel integral transforms. The solutions for the stresses and excess pore pressure are expressed in the form of numerical inversion Laplace and Hankel integral transforms. The numerical simulation results clearly demonstrate the induced deformation and water flow in the asphalt pavement.

On the asymptotic Laplace method and its application to random chaos

The asymptotics of the multidimensional Laplace integral for the case in which the phase attains its minimum on an arbitrary smooth manifold is studied. Applications to the study of the asymptotics of the distribution of Gaussian and Weibullian random chaoses are considered.

Nonstationary distributed axisymmetric load on an elastic half-space

This paper presents analytical and numerical-analytic approaches to solving the problem of the action of an arbitrarily distributed axisymmetric load applied instantly to the surface of an isotropic elastic half-space. The first approach is built around the Laplace and Hankel integral transforms whose inversion is performed jointly with Cagniard’s technique, and as a result, exact analytical expressions are obtained for computing stresses along an axis of symmetry. The second approach uses the Laplace integral transform and the expansion of sought for values into the Fourier–Bessel series to reduce the problem to a numerical solution of a series of Volterra integral equations. Concrete numerical analysis was performed for cases where the domain of application of a distributed load is fixed or expands in time with both constant and variable velocity.

On a non‐stationary load on the surface of a semiplane with mixed boundary conditions

An exact analytical solution has been constructed for the plane problem on action of a non‐stationary load on the surface of an elastic semiplane for conditions of a 'mixed' boundary problem when normal stress and tangent displacement (the fourth boundary problem) are specified on the boundary. Laplace and Fourier integral transforms are used. Their inversions were obtained with the help of tabular relationships and the convolution theorem for a wide range of acting non‐stationary loads. Expressions for stresses (displacements) were obtained in explicit form. The obtained expressions allow determining the wave process characteristics in any point of the object at an arbitrary point of time. Some variants of non‐stationary loads acting on an area with fixed boundaries or an area with boundaries changing by a known function are considered. For a particular case, computed numerical results are compared with the solution of the first boundary problem. Constructing exact analytical solutions, even if infrequently used in practice, besides being significant on their own, can also help refine various numerical and approximate approaches, for which the types of boundary conditions are not critical.

Comparative Analysis of 3D Viscoelastic and Poroelastic Dynamic Boundary-Element Modelling

3D dynamic boundary-value problems of linear viscoelasticity and poroelasticity are considered. Laplace integral transform and its numerical inversion are used. Classical viscoelastic models, such as Maxwell model, KelvinVoigt model, standard linear solid model, and model with weakly singular kernel (Abel type) are considered. Boundary integral equations (BIE) method is developed to solve three-dimensional boundary-value problems. A numerical modelling of wave propagation is done by means of boundary element approach.

Nucleation and crystal growth kinetics during solidification: The role of crystallite withdrawal rate and external heat and mass sources

A complete analytical solution of the integro-differential model describing the transient nucleation and growth of the crystals at the intermediate stage of phase transitions is constructed. The roles of external heat/mass sources appearing in the balance equations and the crystallite withdrawal rate entering in the Fokker–Planck equation are detailed. An exact analytical solution of the Fokker–Planck equation is found for arbitrary nucleation mechanisms and growth kinetics. Two important cases of the Weber–Volmer–Frenkel–Zel׳dovich and Meirs kinetics are considered in some detail. A non-linear time-dependent integral equation with memory kernel for the metastability level is analytically solved on the basis of the saddle-point method for the Laplace integral in the case of mixed kinetic-diffusion regime of crystal growth, which is of frequent occurrence. It is shown that the desupercooling/desupersaturation rate decreases with increasing the crystal withdrawal rate and intensities of external sources. The density distribution function becomes more and more broad with time. In addition, this function increases with decreasing the crystallite withdrawal rate and with increasing intensities of external sources.

A fractional differential constitutive model for dynamic stress intensity factors of an anti-plane crack in viscoelastic materials

Fractional differential constitutive relationships are introduced to depict the history of dynamic stress intensity factors (DSIFs) for a semi-infinite crack in infinite viscoelastic material subjected to anti-plane shear impact load. The basic equations which govern the anti-plane deformation behavior are converted to a fractionalwave-like equation. By utilizing Laplace and Fourier integral transforms, the fractional wave-like equation is cast into an ordinary differential equation (ODE). The unknown function in the solution of ODE is obtained by applying Fourier transform directly to the boundary conditions of fractional wave-like equation in Laplace domain instead of solving dual integral equations. Analytical solutions of DSIFs in Laplace domain are derived by Wiener-Hopf technique and the numerical solutions of DSIFs in time domain are obtained by Talbot algorithm. The effects of four parameters α, β, b1, b2 of the fractional differential constitutive model on DSIFs are discussed. The numerical results show that the present fractional differential constitutive model can well describe the behavior of DSIFs of anti-plane fracture in viscoelastic materials, and the model is also compatible with solutions of DSIFs of anti-plane fracture in elastic materials.

Elastodynamic Interaction of Two Offset Interfacial Cracks in Bonded Dissimilar Media With a Functionally Graded Interlayer Under Antiplane Shear Impact

The impact response of bonded media with a functionally graded interlayer weakened by a pair of two offset interfacial cracks is investigated under the condition of antiplane deformation. The material nonhomogeneity in the graded interlayer is represented in terms of power-law variations of shear modulus and mass density between the dissimilar, homogeneous half-planes. Laplace and Fourier integral transforms are employed to reduce the crack problem to solving a system of Cauchy-type singular integral equations in the Laplace domain. The crack-tip behavior in the physical domain is recovered through the inverse Laplace transform to evaluate the dynamic mode III stress intensity factors as a function of time. As a result, the transient interaction of the offset interfacial cracks spaced apart by the graded interlayer is illustrated. The peak values of the dynamic stress intensity factors are also presented versus offset crack distance, elaborating the effects of various material and geometric parameters of the bonded system on the overshoot characteristics of the transient behavior in the near-tip regions, owing to the impact-induced interaction of singular stress fields between the two cracks.