The joint Laplace transforms for killed diffusion occupation times

The approach of Li and Zhou(2014)is adopted to find the Laplace transform of occupation time over interval(0,a)and joint occupation times over semi-infinite intervals(-∞,a)and(b,∞)for a time-homogeneous diffusion process up to an independent exponential time e_(q)for 0<a<b.The results are expressed in terms of solutions to the differential equations associated with the diffusion generator.Applying these results,we obtain explicit expressions on the Laplace transform of occupation time and joint occupation time for Brownian motion with drift.

Laplace NMR谱图重建——从经典正则化到深度学习

拉普拉斯核磁共振(Laplace NMR)可以提供待测样品的扩散系数或弛豫时间等物理参数信息,是用于研究分子化学结构、动力学和相互作用的强大工具.Laplace NMR的适用性很大程度上取决于拉普拉斯逆变换相关的信号处理算法的性能.在本文中,我们首先讨论了Laplace NMR谱图重建问题的不适定性,接着回顾了经典的基于正则化约束的重建算法,并介绍了目前前沿的深度学习算法在处理Laplace反演问题方面的应用,最后总结了这些算法的优缺点,并对Laplace NMR信号处理方法未来改进方向进行了展望.

On Fuzzy Conformable Double Laplace Transform with Applications to Partial Differential Equations

The Laplace transformation is a very important integral transform,and it is extensively used in solving ordinary differential equations,partial differential equations,and several types of integro-differential equations.Our purpose in this study is to introduce the notion of fuzzy double Laplace transform,fuzzy conformable double Laplace transform(FCDLT).We discuss some basic properties of FCDLT.We obtain the solutions of fuzzy partial differential equations(both one-dimensional and two-dimensional cases)through the double Laplace approach.We demonstrate through numerical examples that our proposed method is very successful and convenient for resolving partial differential equations.

QUATERNIONIC SLICE REGULAR FUNCTIONS AND QUATERNIONIC LAPLACE TRANSFORMS

The functions studied in the paper are the quaternion-valued functions of a quaternionic variable.It is shown that the left slice regular functions and right slice regular functions are related by a particular involution,and that the intrinsic slice regular functions play a central role in the theory of slice regular functions.The relation between left slice regular functions,right slice regular functions and intrinsic slice regular functions is revealed.As an application,the classical Laplace transform is generalized naturally to quaternions in two different ways,which transform a quaternion-valued function of a real variable to a left or right slice regular function.The usual properties of the classical Laplace transforms are generalized to quaternionic Laplace transforms.

On the Approximation of Fractal-Fractional Differential Equations Using Numerical Inverse Laplace Transform Methods

Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects.Using the Laplace transform for solving differential equations,however,sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analyticalmeans.Thus,we need numerical inversionmethods to convert the obtained solution fromLaplace domain to a real domain.In this paper,we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with orderα,β.Our proposed numerical scheme is based on three main steps.First,we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense,and then into Caputo sense.Secondly,we transformthe fractional differential equation in Caputo sense to an equivalent equation in Laplace space.Then the solution of the transformed equation is obtained in Laplace domain.Finally,the solution is converted into the real domain using numerical inversion of Laplace transform.Three inversion methods are evaluated in this paper,and their convergence is also discussed.Three test problems are used to validate the inversion methods.We demonstrate our results with the help of tables and figures.The obtained results show that Euler’s and Talbot’s methods performed better than Stehfest’s method.

APPLICATIONS OF HAMILTONIAN AND LAPLACE TRANSFORM IN AN ECONOMIC MODEL

This paper analyzes the long-run effects and short-run effects of foreign aid on the domestic economy by using the Hamilton system and Laplace transform. It is found that an increase in the foreign aid has no long-run effect on the foreigll borrowing, domestic capital accumulation and the foreign direct investment in the home country, but increases the steady-state consumption level the same amount. However, the short-run analysis presents that increasing foreign aid does not affect the initial consumptioll level and the initial consumption increase rate; but it affects the initial savings positively.

Solving <i>nth</i>-Order Integro-Differential Equations Using the Combined Laplace Transform-Adomian Decomposition Method

In this paper, the Combined Laplace Transform-Adomian Decomposition Method is used to solve nth-order integro-differential equations. The results show that the method is very simple and effective.

ON THE GROWTH OF ZERO ORDER LAPLACE-STIELTJES TRANSFORM CONVERGENT IN THE RIGHT HALF-PLANE

In this article, authors study the growth of Laplace-Stieltjes transform with zero order convergent in the right half-plane, define the exponential order and the exponential low order, and find the relations between them. Some results similar to Dirichlet series are obtained.

Boundary Element Analysis (Laplace Transform Solution) of Groundwater Unsteady Flow to a Multiple Well System in a Confined Aquifer

The calculations of unsteady flow to a multiple well system with the application of boundary elementmethod (BEM) are discussed. The mathematical model of unsteady well flow is a boundary value problem ofparabolic differential equation. It is changed into an elliptic one by Laplace transform to eliminate time varia-ble. The image function of water head H can be solved by BEM. We derived the boundary integral equation ofthe transformed variable H and the discretization form of it, so that there is no need to discretize the bounda-ries of well walls and it becomes easier to solve the groundwater head H by numerical inversion.

On the Approximation of an Analytic Function Represented by Laplace-Stieltjes Transformation

In the present paper,we have considered the approximation of analytic functions represented by Laplace-Stieltjes transformations using sequence of definite integrals. We have characterized their order and type in terms of the rate of decrease of E;（F,β） where E;（F,β） is the error in approximating of the function F（s） by definite integral polynomials in the half plane Res≤β<α.

NUMERICAL INVERSION OF MULTIDIMENSIONAL LAPLACE TRANSFORMS USING MOMENT METHODS

This paper develops a numerical method to invert multi-dimensional Laplace transforms. By a variable transform, Laplace transforms are converted to multi-dimensional Hansdorff moment problems so that the numerical solution can be achieved. Stability estimation is also obtained. Numerical simulations show the efficiency and practicality of the method.

On the Laplace transform of delta function

Delta function is an important function in mathematics and physics. In this paper, the Laplace transforms of δ(t)and δ(t-τ)have been discussed in detail. After the Laplace transform of δ(t)is analyzed, the author has found that three aspects should be taken into account, i.e. τ→0+, τ→0- andτ=0; and it is the same with the Laplace transform of δ(t-τ). Then the results of the Laplace transform of Delta function have been obtained in a rigorous and comprehensive sense.

On <i>q</i>-Analogues of Laplace Type Integral Transforms of <i>q</i>²-Bessel Functions

The present paper deals with the evaluation of the q-Analogues of Laplece transforms of a product of basic analogues of q2-special functions. We apply these transforms to three families of q-Bessel functions. Several special cases have been deducted.

The Numerical Inversion of the Laplace Transform in a Multi-Precision Environment

This paper examines the performance of five algorithms for numerically inverting the Laplace transform, in standard, 16-digit and multi-precision environments. The algorithms are taken from three of the four main classes of numerical methods used to invert the Laplace transform. Because the numerical inversion of the Laplace transform is a perturbed problem, rounding errors which are generated in numerical approximations can adversely affect the accurate reconstruction of the inverse transform. This paper demonstrates that working in a multi-precision environment can substantially reduce these errors and the resulting perturbations exist in transforming the data from the s-space into the time domain and in so doing overcome the main drawback of numerically inverting the Laplace transform. Our main finding is that both the Talbot and the accelerated Gaver functionals perform considerably better in a multi-precision environment increasing the advantages of using Laplace transform methods over time-stepping procedures in solving diffusion and more generally parabolic partial differential equations.

Multidimensional Laplace Transforms over Quaternions, Octonions and Cayley-Dickson Algebras, Their Applications to PDE

Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. Applications to partial differential equations including that of elliptic, parabolic and hyperbolic type are investigated. Moreover, partial differential equations of higher order with real and complex coefficients and with variable coefficients with or without boundary conditions are considered.

水合物热分解Stefan相变模型解的存在性及Laplace变换求解

天然气水合物热分解Stefan相变模型为二阶抛物线型偏微分方程(组)。开展了水合物热分解Stefan相变模型解的存在性、唯一性论证和Laplace求解方法的探索。演绎了简化的过程、细节的推导:应用微分方程的迭代格式,导出了最大、最小值引理和参数取值范围;通过边界上的交替条件映射对xn(t)、Tn(x,t)的极限存在性的证明,证实了微分方程解的存在;采用反证法,证明了Stefan微分方程解的唯一性。运用Laplace变换、分离变量和Laplace逆变换法,求得了该Stefan模型的解析解,超越方程的单调性证明了方程解的唯一性。

Laplace-Stieltjes变换所定义的无限级整函数的增长性

该文系统地研究了在全平面上收敛的无限级Laplace-Stieltjes变换的增长性,得到了两个充要条件,推广了全平面上Dirichlet级数的有关结果.

Laplace-Stieltjes变换所定义的右半平面上的解析函数的增长性与型函数(英文)

本文在Laplace-Stieltjes变换的研究中首次引进了精确级和型函数的概念,讨论了右半平面上有限级L-S变换所定义的解析函数F(s)的增长性,完善并推广了Dirichlet级数的关于精确级和型函数的有关结果.

Laplace-Stieltjes变换所定义的有限级整函数的级与型

引入了Laplace-Stieltjes变换所定义的有限级整函数的级与型的定义,得到了Laplace-Stieltjes变换所表示的整函数的级与型的2个充要条件,推广了Dirichlet级数的相关结果.

右半平面上解析的Laplace-Stieltjes变换对数级

本文研究了零级Laplace-Stieltjes变换的增长性问题.利用对数级和对数下级的定义,获得了这类变换具有对数级的特征,即变换的对数级和对数下级与其系数之间的关系,推广了Dirichlet级数的相关结果.