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The paper deals with the stochastic analysis of a two identical unit cold standby system model. As soon as a unit fails it is attended first by assistant repairman, who completes its repair to make it as good as new with fixed known probability p. Otherwise, with probability q( =1-p), the unit repaired by assistant repairman goes to expert repairman so as to make it as good as new. Both the repairmen are always available with the system and they work their jobs simultaneously. The failure time and repair time by assistant repairman are correlated random variables having their joint distribution as Bivariate exponential. Various measures of system effectiveness useful to system designers and operation managers are obtained by using regenerative point technique. The MTSF and steady state profit have also been studied through graphs in respect of parameter alpha for three different values of repair rate by expert repairman and two different values of correlation coefficient between failure and repair times and important conclusions are drawn. [ABSTRACT FROM AUTHOR]

In this research, we proposed coupling the Laplace transform method and the homotopy Perturbation Method (LHPM). We employed the fusion of the Laplace method to make up for the shortcomings of other semi-analytical approaches like the homotopy perturbation method, variation iteration method, and the Adomian decomposition method. We aim to obtain an approximate and semi-analytic solution of the n-dimensional system of nonlinear partial differential equations. N-dimensional partial differential equations with nonlinear terms are known as nonlinear partial differential equations. They have been used to solve mathematical problems like the Poincare conjecture and the Calabi-Yau conjecture and describe physical systems, from gravity to fluid dynamics. Therefore, we proffer a semi-analytic solution in the form of a Taylor multivariate series of displacements x, y, and time t using the proposed method. A sideby-side comparison was carried out to compare the exact solution with the new solution using 3-dimensional graphs, and thus the graph analysis followed. Results show excellent agreement, and the emergence of this method as a viable alternative demonstrates its viability by requiring fewer computations and being much easier and more convenient than others, making it suitable for widespread use in engineering as well. [ABSTRACT FROM AUTHOR]

In this study, the suggested residual power series transform method is used to compute the numerical solution of the fractional-order nonlinear Gardner and Cahn-Hilliard equations and the result is discovered in a fast convergent series. The leverage and effcacy of the suggested technique are demonstrated by the test examples provided. The achieved results are proved graphically. The current method handles the series solution in a sizable admissible domain in a powerful way. It provides a simple means of modifying the solution's convergence zone. Results with graphs expressly demonstrate the effectiveness and abilities of the suggested method. [ABSTRACT FROM AUTHOR]

This paper deals with the maximum likelihood estimator for the parameter of first-order autoregressive models driven by the stationary Gaussian noises (Colored noise) together with an input. First, we will find the optimal input that maximizes the Fisher information, and then, with the method of the Laplace transform, both the asymptotic properties and the asymptotic design problem of the maximum likelihood estimator will be investigated. The results of the numerical simulation confirm the theoretical analysis and show that the proposed maximum likelihood estimator performs well in finite samples. [ABSTRACT FROM AUTHOR]

In this paper, the lognormal distribution is studied, and a new series representation is proposed. This series uses the powers of the bilinear function. From it, a simplified form is obtained and used to compute the Laplace transform of the distribution. [ABSTRACT FROM AUTHOR]

The current manuscript examines the effect of the fractional temporal variation on the vibration of waves on non-homogeneous elastic substrates by applying the Laplace integral transform and the asymptotic approach. Four different non-homogeneities, including linear and exponential forms, are considered and scrutinized. In the end, it is reported that the fractional temporal variation significantly affects the respective vibrational fields greatly as the vibrations increase with a decrease in the fractional-order µ. Besides, the two approaches employed for the cylindrical substrates are also shown to be in good agreement for very small non-homogeneity parameter α. More so, the present study is set to play a vital role in the fields of material science, and non-homogenization processes to state a few. [ABSTRACT FROM AUTHOR]

It is well known that the Laplace–Stieltjes transform of a nonnegative random variable (or random vector) uniquely determines its distribution function. We extend this uniqueness theorem by using the Müntz–Szász Theorem and the identity for the Laplace–Stieltjes and Laplace–Carson transforms of a distribution function. The latter appears for the first time to the best of our knowledge. In particular, if X and Y are two nonnegative random variables with joint distribution H, then H can be characterized by a suitable set of countably many values of its bivariate Laplace–Stieltjes transform. The general high-dimensional case is also investigated. Besides, Lerch's uniqueness theorem for conventional Laplace transforms is extended as well. The identity can be used to simplify the calculation of Laplace–Stieltjes transforms when the underlying distributions have singular parts. Finally, some examples are given to illustrate the characterization results via the uniqueness theorem. [ABSTRACT FROM AUTHOR]

This paper proposes a method for solving the stress field of a deep circular tunnel affected by an unsteady temperature field. The tunnel is idealized as an infinite domain problem, and the solution of the unsteady temperature field can be regarded as solving the temperature field in the infinite domain under the first type of boundary conditions. The temperature field distribution inside the tunnel surrounding rock is obtained by the Laplace transform method. On this basis, through the stress condition at the boundary of the tunnel and the zero displacement boundary condition at infinity, the solution equations expressed by the analytic functions are listed, and then the power series method of the complex variable function method is used to obtain the desired analytic functions. From this, the temperature stress field caused by the temperature change can be calculated (the displacement field caused by the temperature change is also derived). By superimposing this stress field upon the surrounding rock stress caused by in situ stress, the total stress field inside the surrounding rock can be obtained. The example shows that when the time t is small and the range of the numerical model is suitably large, the results of the analytical method and the numerical method agree well. This paper also discusses two cases in which the tunnel boundary temperature is greater than and less than the initial temperature of the surrounding rock. [ABSTRACT FROM AUTHOR]

The Laplace transforms of the transition probability density and distribution functions of the Feller process contain products of a Kummer and a Tricomi confluent hypergeometric function. The intricacies caused by the singularity at 0 of the Feller process imply that ultimately seven new inverse Laplace transforms can be derived of which four contain the Marcum Q function. The results of this paper together with a scarcely used link between the Marcum and Nuttall Q functions also provide two alternative proofs for an existing identity involving two Marcum Q functions with reversed arguments. The paper also expands the existing expression for the Marcum Q function with identical arguments and order 1. In particular, the new formula applies to all integer and fractional values of the order and is expressed in terms of the generalized hypergeometric function. [ABSTRACT FROM AUTHOR]

• We derive Benford's law with a Benford term together with an additional err term from the Laplace transform. • The Benford term originates from the structure of the number system, as a result of the way that we write numbers. • The err term leads to deviation from Benford's law with the inverse Laplace transform of probability density functions. • The whole family of completely monotonic distributions can all satisfy Benford's law within a small bound. • Distributions with violently oscillating inverse Laplace spectrum generally break Benford's law. The occurrence of digits 1 through 9 as the leftmost nonzero digit of numbers from real-world sources is distributed unevenly according to an empirical law, known as Benford's law or the first digit law. It remains obscure why a variety of data sets generated from quite different dynamics obey this particular law. We perform a study of Benford's law from the application of the Laplace transform, and find that the logarithmic Laplace spectrum of the digital indicator function can be approximately taken as a constant. This particular constant, being exactly the Benford term, explains the prevalence of Benford's law. The slight variation from the Benford term leads to deviations from Benford's law for distributions which oscillate violently in the inverse Laplace space. We prove that the whole family of completely monotonic distributions can satisfy Benford's law within a small bound. Our study suggests that the origin of Benford's law is from the way that we write numbers, thus should be taken as a basic mathematical knowledge. [ABSTRACT FROM AUTHOR]

This paper presents an analytical method towards Laplace transform inversion of composite functions with the aid of Bell polynomial series. The presented results are used to derive the exact solution of fractional distributed order relaxation processes as well as time‐domain impulse response of fractional distributed order operators in new series forms. Evaluation of the obtained series expansions through computer simulations is also given. The results are then used to present novel series expansions for some special functions, including the one‐parameter Mittag‐Leffler function. It is shown that truncating these series expansions when combined with using potential partition polynomials provides efficient approximations for these functions. At the end, the results are shown to be also useful in studying asymptotical behavior of partial Bell polynomials. Numerical simulations as well as analytical examples are provided to verify the results of this paper. [ABSTRACT FROM AUTHOR]

Abstract An approach to use the method of Laplace transform for the perturbative solution of the Schrödinger equation at any order of the perturbation for a system of N qubits coupled to a cavity with n photons is suggested. We investigate the dynamics of a system of N superconducting qubits coupled to a common resonator with time-dependent coupling. To account for the contribution of the dynamical Lamb effect to the probability of excitation of the qubit, we consider counter-rotating terms in the qubit-photon interaction Hamiltonian. As an example, we illustrate the method for the case of two qubits coupled to a common cavity. The perturbative solutions for the probability of excitation of the qubit show excellent agreement with the numerical calculations. Highlights • The Laplace transform is used to solve Schrödinger equation of N qubits and n photons. • The Laplace transform allows to treat time-dependent qubit-photon coupling. • The results for the case of two qubits coupled to a resonator are given. [ABSTRACT FROM AUTHOR]

Highlights • Eighty-five batch flotation tests were conducted to obtain the flotation kinetics. • Six one-parameter models were used to estimate the flotation rate distribution. • For each test, the best of these 6 models was compared with the Gamma distribution. • A regularized Inverse Laplace Transform was also used to consolidate the results. • All estimates were in good agreement, showing the flexibility of the Gamma model. Abstract In this paper, different methodologies to characterize the flotation rate distribution at laboratory batch scale are presented. Conventional model fitting is compared with a selectively chosen set of one-parameter models to describe the flotation rate distribution F (k). In addition, a numerical inversion of the Laplace Transform, which is applicable to first-order systems, is shown. A wide range of time-recovery curves and probability density functions (PDF) for F (k) are presented based on 85 flotation tests. The Single Rate Constant, Rectangular, Exponential, Rayleigh, Unilateral Raised Cosine and χ 2 models are applied to fit the time-recovery curves. The best model per test is chosen and compared to the model fitting obtained from the Gamma distribution, showing a good agreement between these two approaches. The one-parameter PDFs are compared with F (k) estimates obtained from a regularized Inverse Laplace Transform. In this approach, no assumptions on the F (k) shape are made. The flotation rate distributions estimated from the Laplace Inversion show similar shapes to those obtained from the Gamma and the optimal one-parameter approach. In addition, the F (k) estimates present positive skews in all the evaluated tests. The use of the Laplace Transform Inversion along with conventional model fitting is a powerful tool to validate the F (k) estimates at laboratory scale. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we considered nonlinear systems of fractional order differential equations. They have been solved by a computational methods which are so-called Laplace Adomian decomposition method (LADM) and modified Laplace decomposition method (MLDM). The fractional derivatives are described in the Caputo sense. The (LADM) and the (MLDM) are a combination of the Laplace transform and the Adomian decomposition method and iterative method respectively. These techniques were applied for some illustrative examples in order to solve nonlinear systems of fractional order differential equations. From the results of the illustrative examples we conclude that these methods are computationally efficient. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analysis of the time‐domain elastic wave scattering problem in an unbounded structure. The transparent boundary condition is developed to reformulate the scattering problem into an initial‐boundary value problem in an infinite slab. The well posedness and stability are established for the reduced problem in both the frequency and time domains. Our proofs are based on the energy method, the Lax‐Milgram theorem, and the inversion theorem of the Laplace transform. Moreover, a priori estimates with explicit dependence on the time are achieved for the elastic displacement by taking special test functions for the time‐domain variational problems of the Navier equation. [ABSTRACT FROM AUTHOR]

The paper presents a computationally efficient method for modeling and simulating distributed systems with lossy transmission line (TL) including multiconductor ones, by a less conventional method. The method is devised based on 1D and 2D Laplace transforms, which facilitates the possibility of incorporating fractional-order elements and frequency-dependent parameters. This process is made possible due to the development of effective numerical inverse Laplace transforms (NILTs) of one and two variables, 1D NILT and 2D NILT. In the paper, it is shown that in high frequency operating systems, the frequency dependencies of the system ought to be included in the model. Additionally, it is shown that incorporating fractional-order elements in the modeling of the distributed parameter systems compensates for losses along the wires, provides higher degrees of flexibility for optimization and produces more accurate and authentic modelling of such systems. The simulations are performed in the Matlab environment and are effectively algorithmized. [ABSTRACT FROM AUTHOR]

In the paper, we investigate a single-server queueing system with unlimited memory space and non-homogeneous customers (calls) of the two following types: 1) external customers that are served by the system under consideration, 2) internal customers that arrive and interrupt the service process only when an external customer is being served. The external customers appear according to a stationary Poisson process. Customers of each of the above-mentioned types are characterized by some random volume. The customer service time depends arbitrarily on its volume. Two schemes of customer service organization are analyzed. The non-stationary and stationary distributions of the total volume of customers present in the system are determined in terms of Laplace and Laplace-Stieltjes transforms. The stationary first and second moments of total customers volume are also calculated. The obtained results are used to approximate loss characteristics in analogous systems with limited buffer space. Numerical examples illustrating theoretical results are attached as well. [ABSTRACT FROM AUTHOR]

In this paper, we analyze an alcoholism model which involves the impact of Twitter via Liouville–Caputo and Atangana–Baleanu–Caputo fractional derivatives with constant- and variable-order. Two fractional mathematical models are considered, with and without delay. Special solutions using an iterative scheme via Laplace and Sumudu transform were obtained. We studied the uniqueness and existence of the solutions employing the fixed point postulate. The generalized model with variable-order was solved numerically via the Adams method and the Adams–Bashforth–Moulton scheme. Stability and convergence of the numerical solutions were presented in details. Numerical examples of the approximate solutions are provided to show that the numerical methods are computationally efficient. Therefore, by including both the fractional derivatives and finite time delays in the alcoholism model studied, we believe that we have established a more complete and more realistic indicator of alcoholism model and affect the spread of the drinking. [ABSTRACT FROM AUTHOR]

The Greek parameters in option pricing are derivatives used in hedging against option risks. In this paper, the Greeks of the continuous arithmetic Asian option pricing model are derived. The derivation is based on the analytical solution of the continuous arithmetic Asian option model obtained via a proposed semi-analytical method referred to as Laplace-Adomian decomposition method (LADM). The LADM gives the solution in explicit form with few iterations. The computational work involved is less. Nonetheless, high level of accuracy is not neglected. The obtained analytical solutions are in good agreement with those of Rogers & Shi (J. of Applied Probability 32: 1995, 1077-1088), and Elshegmani & Ahmad (ScienceAsia, 39S: 2013, 67-69). The proposed method is highly recommended for analytical solution of other forms of Asian option pricing models such as the geometric put and call options, even in their time-fractional forms. The basic Greeks obtained are the Theta, Delta, Speed, and Gamma which will be of great help to financial practitioners and traders in terms of hedging and strategy. [ABSTRACT FROM AUTHOR]

Purpose This paper aims to use the Laplace Adomian decomposition method (LADM) to investigate the effects of thermal convection, thermal conduction, surface emissivity and thermal radiation on the heat dissipated by a continuously moving plate undergoing thermal processing.Design/methodology/approach In performing the analysis, it is assumed that the thermal conductivity and surface emissivity of the plate are both temperature-dependent. The accuracy of the LADM solutions is confirmed by comparing the results obtained for the temperature distribution within the plate with those reported in the literature based on the differential transformation method.Findings It is shown that the heat dissipated from the plate reduces as the Peclet number increases. By contrast, the dissipated heat increases as any one of the non-dimensionalized parameters of the system, i.e. Nc, Nr and B, increases. In addition, the temperature drop along the length of the plate reduces as parameter A increases owing to a more rapid heat transfer.Originality/value The results provide a useful source of reference for the choice of suitable materials and cooling fluids in a variety of practical applications. [ABSTRACT FROM AUTHOR]

In this work, we study the thermoelastic properties of an isotropic and homogeneous one-dimensional semi-infinite elastic medium subjected to a laser short-pulse heating with time exponentially decaying pulse type in light of the new theory of fractional order strain thermoelasticity. The solution for temperature, stress, and strain distribution functions has been obtained in the Laplace domain. To obtain the different inverse field functions numerically we used a complex inversion formula of Laplace transform based on a Fourier expansion. The effects of different parameters, namely, the pulse intensity, time, fractional order, and relaxation time on the thermodynamical temperature, stress, and on the strain distribution, are presented graphically. [ABSTRACT FROM AUTHOR]

In order to study the distribution characteristics and related parameters of the temperature field in underground coal gasification stope, Laplace transform and inverse Laplace transform were used to determine the analytical solution to temperature propagation in the stope under type one boundary conditions. Additionally, Mathematic software was used to conduct numerical simulation. Results indicate that in an unsteady stage, the temperature field gradually propagated in the coal seam, and that the propagation rate became steady after a gradual increase. The time required for the temperature field in the coal seam undergoing gasification to transform from the unsteady to the steady state was inversely proportional to the advance rate of the flame working face. During the stages of temperature increase and decrease, the temperature field in the roof, floor, and surrounding coal seam gradually moved towards the interior. During the temperature increase stage, the peak of the temperature field occurred on the surface of the surrounding rock and gradually increased; it also moved inward from the surface and gradually decreased during the temperature decrease stage. The temperature decrease stage had greater influence on the temperature field in the roof, floor, and surrounding seam than the temperature increase stage. The influence of the temperature increase stage was related to the advance rate of the flame working face; this influence is negligible when the advance rate of the flame working face exceeded a threshold value. Finally, based on the features of the envelope curves of temperature curve families, the ranges of the surrounding coal seam with no load-bearing capacity and coking cycle were obtained, as were after determination of their respective criteria and temperature threshold values. [ABSTRACT FROM AUTHOR]

Reaction–diffusion equations describe a number of physical, chemical, and biological phenomena, many of which occur in composite environments with piece-wise constant diffusion coefficients. We develop semi-analytical solutions of axisymmetric reaction–diffusion equations with first-order reaction kinetics and continuous transient boundary conditions. These solutions are directly applicable to heat conduction in composite media with transient boundary conditions and heat generation. The solutions lose their robustness in the long time regime, when the Laplace variable tends to zero. This limitation is overcome by the use of corresponding steady-state solutions. [ABSTRACT FROM AUTHOR]

Letbe the survival function of a non negative random variable satisfying thatis of second-order regular variation with the regular variation index − α < 0 and the second-order parameter ρ < 0. Let ϕ(s) denote the Laplace–Stieltjes transform ofF. For α ∈ (m,m+ 1) and α − ρ ∈ (k,k+ 1) with non negative integersm, k, ifm=k, it is proved that the second-order regular variation ofimplies that of ( − 1)k(φ(k)(0) − φ(k)(1/ · )), while form

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. [ABSTRACT FROM AUTHOR]

This paper is about methods for expressing infinite series in closed form by using Laplace transforms and their inverses where resulting integrals are to be evaluated instead. If needed, parametrizing the series and applying an additional Laplace transform offer more options for the summation of infinite series. [ABSTRACT FROM AUTHOR]

Turbo warrant is a special type of barrier options in which the rebate is calculated as another exotic option. In this paper, using Laplace transforms we obtain the valuation of turbo warrant under the mixed-exponential jump diffusion model, which is able to approximate any jump size distribution. The numerical Laplace inversion examples verify that the analytical solutions are accurate. The results of simulation confirm the argument that jump risk should not be ignored in the valuation of turbo warrants. [ABSTRACT FROM AUTHOR]

We compute the exponential decay of the probability that a given multi-dimensional random walk stays in a convex cone up to time n, as n goes to infinity. We show that the latter equals the minimum, on the dual cone, of the Laplace transform of the random walk increments. As an example, our results find applications in the counting of walks in orthants, a classical domain in enumerative combinatorics. [ABSTRACT FROM AUTHOR]

Given a (not necessarily regular) holonomic $$\fancyscript{D}$$ -module $$\fancyscript{L}$$ defined on the product of two complex manifolds, we prove that the correspondence associated with $$\fancyscript{L}$$ commutes (in some sense) with the De Rham functor. We apply this result to the study of the classical Laplace transform. The main tools used here are the theory of ind-sheaves and its enhanced version. [ABSTRACT FROM AUTHOR]

This study is to investigate the effect of anisotropy on the forced vibration behavior of hollow cylinders under dynamic internal pressure. The problems are solved analytically in the Laplace domain, and the results obtained are transformed to the real-time space using the modified Durbin's numerical inversion method. Durbin's numerical inverse method into the analysis of transient thermal stresses in annular fins is a novel approach. Durbin's numerical inverse method successfully implements the boundary value problem which can be solved in Laplace space. Various material models from the literature are used and corresponding radial displacement distributions and stress distributions are computed. Verification of the proposed method is done using benchmark solutions available in the literature for some special cases and virtually exact results are obtained. The anisotropy constant is a useful parameter from a design point of view in that it can be tailored for specific applications to control the stress distribution. [ABSTRACT FROM AUTHOR]

We have obtained solutions of integral equations such as Volterra convolution type of first and second type and Abel integral equation using Laplace - Stieltjes transform and applications are mentioned through examples. The convolution property for Laplace - Stieltjes transform is obtained. The solution obtained is considered in distributional sense. [ABSTRACT FROM AUTHOR]

This paper presents a high-order homogenization model for wave propagation in viscoelastic composite structures. Asymptotic expansions with multiple spatial scales are employed to formulate the homogenization model. The proposed multiscale model operates in the Laplace domain allowing the representation of linear viscoelastic constitutive relationship using a proportionality law. The high-order terms in the asymptotic expansion of response fields are included to reproduce micro-heterogeneity-induced wave dispersion and formation of bandgaps. The first and second-order influence functions and the macroscopic deformation are evaluated using the finite element method with complex coefficients in the Laplace domain. The performance of the proposed model is assessed by investigating wave propagation characteristics in layered and particulate composites and verified against direct numerical simulations and analytical solutions. The analysis of dissipated energy revealed that material dispersion may contribute significantly to wave attenuation in dissipative composite materials. The wave dispersion characteristics are shown to be sensitive to microstructure morphology. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]