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In this paper, an extension of the bilateral Laplace transform on infinite-dimensional Banach spaces is introduced. In order to develop the concept of an analytic bilateral Laplace–Feynman transfor...

This paper is concerned with the existence and asymptotic behavior of traveling wave solutions for a nonlocal dispersal vaccination model with general incidence. We first apply the Schauder's fixed point theorem to prove the existence of traveling wave solutions when the wave speed is greater than a critical speed \begin{document}$ c^* $\end{document} . Then we investigate the boundary asymptotic behaviour of traveling wave solutions at \begin{document}$ +\infty $\end{document} by using an appropriate Lyapunov function. Applying the method of two-sided Laplace transform, we further prove the non-existence of traveling wave solutions when the wave speed is smaller than \begin{document}$ c^* $\end{document} . From our work, one can see that the diffusion rate and nonlocal dispersal distance of the infected individuals can increase the critical speed \begin{document}$ c^* $\end{document} , while vaccination reduces the critical speed \begin{document}$ c^* $\end{document} . In addition, two specific examples are provided to verify the validity of our theoretical results, which cover and improve some known results.

The concept of iterated bilateral Laplace transform (ILT) is used to obtain the frequency response of linear time-varying (LTV) causal circuits and systems. It is shown that the application of this commonly overlooked transform is essential in the analysis of time-varying systems.

This paper is concerned with traveling wave solutions in a discrete diffusion epidemic model with delayed transmission. Employing the way of contradictory discussions and the bilateral Laplace transform, we obtain the nonexistence of nontrivial positive bounded traveling wave solutions. Utilizing the super-/sub-solutions method and the fixed point theory, we derive the existence of nontrivial positive traveling wave solutions with both super-critical and critical speeds. Our results indicate that the critical speed is the minimal speed.

The focus of this paper is on the frequency characterization of autonomous bivariate systems using the defined iterated bilateral Laplace transform (ILT). The application of this commonly overlooked transform is emphasized in the analysis of system the linear time-varying (LTV) systems and circuits.

This work deals with the micromechanical study of periodic thermo-diffusive elastic multi-layered materials, which are of interest for the fabrication of solid oxide fuel cells. The focus is on the dynamic regime that is investigating the dispersive wave propagation within the periodic material. In this framework, a generalization of the Floquet–Bloch theory is adopted, able to determine the complex band structure of such materials. The infinite algebraic linear system, obtained by exploiting both bilateral Laplace transform and Fourier transform, is replaced by its finite counterpart, resulting from a proper truncation at a finite number of considered unknowns and equations. A regularization technique is herein useful to get rid of the Gibbs phenomenon. The solution of the problem is, finally, found in terms of complex angular frequencies, corresponding to a finite sequence of eigenvalue problems for given values of the wave vector. The paper is complemented by numerical examples taking into account thermo-mechanical coupling. The frequency band structure of the periodic thermo-diffusive elastic material is found to be strongly influenced by the interaction between thermal and mechanical phenomena.

In this paper we base our study on the application of the Laplace transform to risk preference theory. With a constant measure of absolute risk aversion (Pratt, 1964; Arrow, 1965), the Certainty Monetary Equivalent (CME) of a risky project previously has been developed into an expression involving the logarithm of the bilateral Laplace transform of the probability density of its stochastic economic outcome. The internal risk aversion (IRA) is the break-even level of the absolute risk aversion, between making the project favourable or unfavourable. Below, we apply this methodology to determining an expression for the incremental risk rate of interest in the case of standard investments with payments having probability distributions. A simple approximate formula is derived explaining how the incremental risk rate of interest depends on the absolute risk aversion and the discount rate, i.e. on measures of risk preference and of time preference.

The present paper discusses the solution of nonlinear homogeneous and nonhomogeneous time-fractional gas dynamic equations arising in shock fronts by a new combination of new integral and projected differential transform method. The new integral projected differential transform method (NIPDTM) makes the calculation very simpler and in this method the nonlinear term can be easily handled by projected differential transform without using Adomian’s polynomial and He’s polynomial, which can be taken as a big advantage of this method. This method is more exertive and easy to handle such fractional differential equation in comparison to other methods. The results obtain from illustrative examples shows the competency and accordance of the proposed method. Keywords: New integral transform method, Projected differential transform method, Time-fractional gas dynamic equation, Caputo fractional derivative, Mittag-Leffler function

This brief presents a novel approach to predict the bounds of the time-domain response of a linear system subject to multiple bounded uncertain input parameters. The method leverages the framework of Taylor models in conjunction with the numerical inversion of Laplace transform (NILT). Different formulations of the NILT are reviewed, and their advantages and limitations are discussed. An implementation relying on an inverse fast Fourier transform turns out to be the most efficient and accurate alternative. The feasibility of the technique is validated based on several diverse application examples, namely a control loop, a lossy transmission-line network, and an active low-pass filter.

We show that the asymptotic 1 / N expansion for the averages of linear statistics of the GUE is convergent when the test function is an entire function of order two and finite type. This allows to fully recover the mean eigenvalue density function for finite N from the coefficients of the expansion thus providing a resummation procedure. As an intermediate result, we compute the bilateral Laplace transform of the GUE reproducing kernel in the half-sum variable, generalizing a formula of Haagerup and Thorbjornsen.

The paper reviews the unilateral and bilateral, one- and two-dimensional Laplace transforms. The unilateral and bilateral Laplace transforms are compared in the one-dimensional case, leading to the formulation of the initial-condition theorem. This problem is solved with all generality in the one- and two-dimensional cases with the bilateral Laplace transform. The case of fractional-order systems is also included. General two-dimensional linear systems are introduced and the corresponding transfer function is defined, This work was partially funded by National Funds through the Foundation for Science and Technology of Portugal, under the projects UIDB/00066/2020.

The Laplace transform is a particularly elegant way to solve linear differential equations with constant coefficients. The Laplace transform describes signals and systems not as functions of time, but as functions of a complex variable s s . When transformed into the Laplace domain, differential equations become polynomials of s s . Solving a differential equation in the time domain becomes a simple polynomial multiplication and division in the Laplace domain. However, the input and output signals are also in the Laplace domain, and any system response must undergo an inverse Laplace transform to become a meaningful time-dependent signal. In this chapter, the Laplace transform is introduced, and the manipulation of signals and systems in the Laplace domain explained. Tools to find time-domain and Laplace-domain correspondences are presented.

We study the integral transform which appeared in a different form in Akhiezer’s textbook “Lectures on Integral Transforms”.

For two real numbers $c>0, \alpha> -1,$ we study some spectral properties of the weighted finite bilateral Laplace transform operator, defined over the space $E=L^2(I,\omega_{\alpha}),$ $I=[-1,1],$ $\omega_{\alpha}(x)=(1-x^2)^{\alpha},$ by ${\displaystyle \mathcal L_c^{\alpha} f(x)= \int_I e^{cxy} f(y) \omega_{\alpha}(y)\, dy}.$ In particular, we use a technique based on the Min-Max theorem to prove that the sequence of the eigenvalues of this operator has a super-exponential decay rate to zero. Moreover, we give a lower bound with a magnitude of order $e^c,$ for the largest eigenvalue of the operator $\mathcal L_c^{\alpha}.$ Also, we give some local estimates and bounds of the eigenfunctions $\varphi_{n,c}^{\alpha}$ of $\mathcal L_c^{\alpha}.$ Moreover, we show that these eigenfunctions are good candidates for the spectral approximation of a function that can be written as a weighted finite Laplace transform of an other $L^2(I,\omega_{\alpha})-$function. Finally, we give some numerical examples that illustrate the different results of this work. In particular, we provide an example that illustrate the Laplace based spectral method, for the inversion of the finite Laplace transform.

The recently developed method by the authors, known as the extended Laplace transform method (ELTM), is used to obtain an explicit semi-analytical, semi-numerical solution for the Thomas–Fermi (TF) equation for neutral atoms. The method makes it possible to calculate the Laplace transform of the intrinsic nonlinearity of the TF equation namely N ( ϕ ) = ϕ 3 , as a novel and unique advantage. We demonstrate that the obtained solution is remarkably accurate and it is superior to the previous solutions in the literature, particularly those by the homotopy analysis method, in terms of the computational efficiency.

For Laplace transform inversion, a method for constructing quadrature rules of the highest degree of accuracy based on an asymptotic distribution of roots of special orthogonal polynomials on the complex plane is proposed.

One of the important problems of stochastic process theory is to define the Laplace transformations for the distribution of this process. With this purpose, we will investigate a semi-Markov random processes with positive tendency and negative jump in this article. The first falling moment into the zero-level of this process is constructed as mathematically and the Laplace transformation of this random variable is obtained.

The fractional Radon transform defined, based on the Fourier slice theorem and the fractional Fourier transform, has many potential applications in optics and the pattern-recognition field. Here we study many properties of the fractional Radon transform using existing theory of the regular Radon transform: the inversion formulas, stability estimates, uniqueness and reconstruction for a local data problem, and a range description. Also, we define the fractional exponential Radon transform and present its inversion.

In [8] , Hu and Qi studied the normalized Laplace tensors and normalized Laplace spectra of k-uniform hypergraphs. They also mentioned the question about whether or not 2 is also an H-eigenvalue of the normalized Laplace tensor of a k-uniform hypergraph, when 2 is an eigenvalue of the normalized Laplace tensor (in this case, k is necessarily even). In this paper, we use an expression for the normalized Laplace tensor in terms of the tensor product, together with the diagonal similarity of tensors, the Perron–Frobenius Theorem for nonnegative tensors and nonnegative weakly irreducible tensors, and the concept and properties of odd-colorable hypergraphs introduced in [13] , to give a complete answer to this question. We show that: (i). When k ≡ 2 ( mod 4 ) , then the answer to this question is affirmative. (ii). When k ≡ 0 ( mod 4 ) , then the answer to this question is negative, and in this case, we give an infinite family of counterexamples. We also study the signless normalized Laplace spectra and the signless normalized Laplace H-spectra of hypergraphs. We give structural characterizations of the hypergraphs having the same normalized Laplace spectrum and signless normalized Laplace spectrum, or having the same normalized Laplace H-spectrum and signless normalized Laplace H-spectrum, or both. Finally, we determine the first two k-uniform supertrees of order n with the largest Laplace spectral radii, and also determine the unique k-uniform hypertree of order n with the smallest Laplace spectral radii, in the case when k is even.

In this paper, we are concerned with existence/non-existence of traveling waves of a diffusive SIR epidemic model with general incidence rate of the form of f ( S ) g ( I ) and infinitely distributed latency but without demography. We show that the existence of traveling waves only depends on the basic reproduction number of the corresponding spatial-homogeneous system of delay differential equations, which is determined by the recovery rate, the local properties of f and g and a minimal wave speed c ∗ that is affected by the distributed delay. The proof of existence of traveling waves is by employing Schauder’s fixed point theorem, and the proof of nonexistence is completed with the aid of the bilateral Laplace transform.

This paper is concerned with the existence, monotonicity, asymptotic behavior and uniqueness of traveling wave solutions for a three-species competitive–cooperative system with nonlocal dispersal and bistable dynamics. By considering a related truncated problem, we first establish the existence and strict monotonicity of traveling waves by means of a limiting argument and a comparative lemma. Then the asymptotic behavior of traveling waves is investigated by using Ikehara’s lemma and bilateral Laplace transform. Finally, we obtain the uniqueness of wave speed and traveling wave by sliding method.

The paper introduces a recursive procedure to invert the multivariate Laplace transform of probability distributions. The procedure involves taking independent samples from the Laplace transform; these samples are then used to update recursively an initial starting distribution. The update is Bayesian driven. The final estimate can be written as a mixture of independent gamma distributions, making it the only methodology which guarantees to numerically recover a probability distribution with positive support. Proof of convergence is given by a fixed point argument. The estimator is fast, accurate and can be run in parallel since the target distribution is evaluated on a grid of points. The method is illustrated on several examples and compared to the bivariate Gaver–Stehfest method.

The problem of one-dimensional non-stationary wave propagation in viscoelastic half space is considered. Relaxation kernel is considered exponential. Initial conditions are set to zero, displacement is determined on the half space boundary. The solution is represented in the form of perturbation and surface Green function resultant. For determination of Green function time Laplace transform is used. Its inverse transform is carried out both analytically expanding Green function in series and numerically. A good coincidence of analytical and numerical Green function calculation results is shown. The final solution is determined analytically. Examples of calculations are represented.

This paper devotes the study of pre-trigonometric and pre-hyperbolic functions. Starting with basic definitions, these functions are further generalized. It is shown that these generalized functions are related with well known Mittag-Leffler function which plays very important role in fractional calculus. We illustrate some properties of these new family of functions. Using Laplace transform, we have shown that they are solutions of differential equations.

This paper presents a new pressure-rate deconvolution algorithm to estimate reservoir properties and predict well performance from measured well responses. The algorithm combines the conveniences of deconvolution in Laplace domain with a new approach to transform sampled data from time domain to Laplace domain, which removes the necessity to extrapolate the data beyond the sampling interval imposed by Laplace transformation. This technique leads to a series form of the deconvolution expression in Laplace domain. Appropriate algorithms are presented to calculate the coefficients of the series to evaluate the results either in time domain or the Laplace space. The time domain computations, which are in the form of a closed solution, do not require evaluation or inversion of the Laplace transforms, can be implemented easily, and returns the equivalent constant-rate production data with minimum user interference. The Laplace domain calculations require the implementation of an inversion algorithm, but provide a wide range of applications in the solution of many pressure- or rate-transient analysis problems. For both cases, the computations are robust and accurate. An optimization algorithm in time domain is introduced in order to alleviate the noise effect on the constant-rate pressure behavior and its logarithmic derivative. The outcome of the process is the data converted to equivalent responses at constant-rate production, which is amenable to analysis by standard PTA/RTA models. Similar to the recent time-domain deconvolution algorithms, which are controlled by a user supplied regularization parameter, the proposed approach also requires minimal user interference.

The main objective of this paper is finding new types of discrete transforms with tuning factor t. This is not only analogy to the continuous Laplace transform but gives discrete Laplace-Fibonacci transform (LFt). This type of Laplace-Fibonacci transform is not available in the continuous case. The LFt generates uncountably many outcomes when the parameter t varies on (0,∞). This possibility is not available in the existing Laplace transform. All the formulae and results derived are verified by MATLAB.

This paper is devoted to designing a practical algorithm to invert the Laplace transform by assuming that the transform possesses the Puiseux expansion at infinity. First, the general asymptotic expansion of the inverse function at zero is derived, which can be used to approximate the inverse function when the variable is small. Second, an inversion algorithm is formulated by splitting the Bromwich integral into two parts. One is the main weakly oscillatory part, which is evaluated by a composite Gauss–Legendre rule and its Kronrod extension, and the other is the remaining strongly oscillatory part, which is integrated analytically using the Puiseux expansion of the transform at infinity. Finally, some typical tests show that the algorithm can be used to invert a wide range of Laplace transforms automatically with high accuracy and the output error estimator matches well with the true error.

We study the action of the Laplace transform $$\mathcal L$$ on rearrangement-invariant function spaces. We focus on the optimality of the range and the domain spaces.

There is considerable interest in the study of fractional order derivative integrator but obtaining analytical impulse and step responses is a difficult problem. Therefore all methods reported on to date use approximations for the fractional derivative/integrator both for analytical based computations and more relevantly in simulation studies. In this paper, an analytical formula is first derived for the inverse Laplace transform of fractional order integrator, 1/sα where α∈R and 0

We propose an adapted Laplace transform method that gives the solution of a linear fractional differential equation with constant coefficients in terms of exponential function. After we mention what the utilized transformation, the CTIT transformation, is based on, we explain how it can reduce the problem from fractional form to ordinary form when it is used with Laplace transformation, via some examples for 0<

This paper mainly focus on the front-like entire solution of a classical nonlocal dispersal equation with ignition nonlinearity. Especially, the dispersal kernel function $J$ may not be symmetric here. The asymmetry of $J$ has a great influence on the profile of the traveling waves and the sign of the wave speeds, which further makes the properties of the entire solution more diverse. We first investigate the asymptotic behavior of the traveling wave solutions since it plays an essential role in obtaining the front-like entire solution. Due to the impact of $f'(0)=0$, we can no longer use the common method which mainly depending on Ikehara theorem and bilateral Laplace transform to study the asymptotic rates of the nondecreasing traveling wave and the nonincreasing one tending to 0, respectively, thus we adopt another method to investigate them. Afterwards, we establish a new entire solution and obtain its qualitative properties by constructing proper supersolution and subsolution and by classifying the sign and size of the wave speeds., Comment: 20 pages

The theory of real quaternion differential equations has recently received more attention, but significant challenges remain the non-commutativity structure. They have numerous applications throughout engineering and physics. In the present investigation, the Laplace transform approach to solve the linear quaternion differential equations is achieved. Specifically, the process of solving a quaternion different equation is transformed to an algebraic quaternion problem. The Laplace transform makes solving linear ODEs and the related initial value problems much easier. It has two major advantages over the methods discussed in literature. The corresponding initial value problems can be solved without first determining a general solution. More importantly, a particularly powerful feature of this method is the use of the Heaviside functions. It is helpful in solving problems, which is represented by complicated quaternion periodic functions. Copyright © 2017 John Wiley & Sons, Ltd.

This study introduces the theory of the Laplace wavelet transform (LWT). The Laplace wavelets are a generalization of the second-order under damped linear time-invariant (SOULTI) wavelets to the complex domain. This generalization produces the mother wavelet function that has been used as the Laplace pseudo wavelet or the Laplace wavelet dictionary. The study shows that the Laplace wavelet can be used to transform signals to the time-scale or time-frequency domain and can be retrieved back. The properties of the new generalization are outlined, and the characteristics of the companion wavelet transform are defined. Moreover, some similarities between the Laplace wavelet transform and the Laplace transform arise, where a relation between the Laplace wavelet transform and the Laplace transform is derived. This relation can be beneficial in evaluating the wavelet transform. The new wavelet transform has phase and magnitude, and can also be evaluated for most elementary signals. The Laplace wavelets inherit many properties from the SOULTI wavelets, and the Laplace wavelet transform inherits many properties from both the SOULTI wavelet transform and the Laplace transform. In addition, the investigation shows that both the LWT and the SOULTI wavelet transform give the particular solutions of specific related differential equations, and the particular solution of these linear time-invariant differential equations can in general be written in terms of a wavelet transform. Finally, the properties of the Laplace wavelet are verified by applications to frequency varying signals and to vibrations of mechanical systems for modes decoupling, and the results are compared with the generalized Morse and Morlet wavelets in addition to the short time Fourier transform’s results.

The main object of this paper is to obtain an integral transform involving Bessel's function into Appell's function, which generalize a well known class of hypergeometric function of some Kampe' de Fe'riet, Srivastava functionF^{(3)}, Appell's function F_{2}, F_{4} and Horn function H_{3}. A number of known and new transformations are also discussed as the special cases of our main result.

This paper deals with the Laplace transform of bivariate Schur-constant models. We show that this transform generates a new family of copulas. Some connections with excursion theory and reflecting Brownian motion are investigated.

In the present paper, a new family of integral transforms depending on two parameters and related to the Dunkl transform is introduced. Well-known transforms, such as the fractional Dunkl transform, Dunkl transform, linear canonical transform, canonical Hankel transform, Fresnel transform, etc., can be seen to be special cases of this general transform. Some useful properties of the considered transform such as Riemann–Lebesgue lemma, reversibility property, additivity property, operational formula, Plancherel formula, Bochner type identity and master formula are derived. The intimate connection that exists between this transformation and the quantum harmonic oscillator is developed.

We study the first hitting time of integral functionals of time-homogeneous diffusions, and characterize their Laplace transforms through a stochastic time change. We obtain explicit expressions of the Laplace transforms for the geometric Brownian motion (GBM) and the mean-reverting GBM process. We also introduce a novel probability identity based on an independent exponential randomization and obtain explicit Laplace transforms of the price of arithmetic Asian options and other derivative prices that non-linearly depend on the integral diffusions. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed method.