A Function Related to a Lagrange-Bürmann Series

An infinite series which arises in certain applications of the Lagrange-Bürmann formula to exponential functions is investigated. Several very exact estimates for the Laplace transform and higher moments of this function are developed.     

Discussion on the Leibniz rule and Laplace transform of fractional derivatives using series representation

ABSTRACT

Taylor series is a useful mathematical tool when describing and constructing a function. With the series representation, some properties of fractional calculus can be revealed clearly. On this basis, the Lebiniz rule and Laplace transform of fractional calculus is investigated. It is analytically shown that the commonly used Leibniz rule cannot be applied for Caputo derivative. Similarly, the well-known Laplace transform of Riemann–Liouville derivative is doubtful for n-th continuously differentiable function. After pointing out such problems, the exact formula of Caputo Leibniz rule and the explanation of Riemann–Liouville Laplace transform are presented. Finally, three illustrative examples are revisited to confirm the obtained results.     

Real inversion formulas with rates of convergence

Some classical real inversion formulas, such as those concerning Fourier, Laplace and Stieltjes transforms, are unified in a way which allows us to give rates of convergence. As illustration, the case of the Fourier transform is considered. This revised version was published online in June 2006 with corrections to the Cover Date.     

Solving nonlinear ordinary differential equations using the NDM

In this research paper, we examine a novel method called the Natural Decomposition Method (NDM). We use the NDM to obtain exact solutions for three different types of nonlinear ordinary differential equations (NLODEs). The NDM is based on the Natural transform method (NTM) and the Adomian decomposition method (ADM). By using the new method, we successfully handle some class of nonlinear ordinary differential equations in a simple and elegant way. The proposed method gives exact solutions in the form of a rapid convergence series. Hence, the Natural Decomposition Method (NDM) is an excellent mathematical tool for solving linear and nonlinear differential equation. One can conclude that the NDM is efficient and easy to use.     

On representations,inversions, and approximations of laplace transforms in banach spaces

The paper extend's some of the basic results of classical Laplace transform theory to the Laplace transform of bounded Bochner integrable functions and the Laplace-Stieltjes transform of Lipschitz continuous functions with values in a Banach space.     

一种基于小波尺度函数变换的Laplace反演方法

该文基于Ｄａｕｂｅｃｈｉｅｓ小波尺度函数变换建立了关于Ｌａｐｌａｃｅ变换的一种反演数值方法．通过对小波尺度函数的低带通谱特性的定性与定量讨论,给出了这一反演方法所得原像函数的适用域．结果发现：其区域大小随着小波尺度函数的分辨指标（ｒｅｓｏｌｕｔｉｏｎｌｅｖｅｌ）选取的升高而增大．最后,以颤振曲线、具有指数增长的复函数、和一维振动弦的初边值问题等为例,定量给出了其反演方法的数值结果．通过与相应的原像精确结果对比发现：在反演的有效区域内,其数值反演的原像几乎与精确的原像图象重合．这表明这一Ｌａｐｌａｃｅ反演数值方法是有效和可靠的．     

Numerical computation of nonlinear fractional Zakharov–Kuznetsov equation arising in ion-acoustic waves

The main aim of the present work is to propose a new and simple algorithm for fractional Zakharov–Kuznetsov equations by using homotopy perturbation transform method (HPTM). The Zakharov–Kuznetsov equation was first derived for describing weakly nonlinear ion-acoustic waves in strongly magnetized lossless plasma in two dimensions. The homotopy perturbation transform method is an innovative adjustment in Laplace transform algorithm (LTA) and makes the calculation much simpler. HPTM is not limited to the small parameter, such as in the classical perturbation method. The method gives an analytical solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. The numerical solutions obtained by the proposed method indicate that the approach is easy to implement and computationally very attractive.     

Laguerre polynomials and the inverse Laplace transform using discrete data

We consider the problem of finding a function defined on (0,∞) from a countable set of values of its Laplace transform. The problem is severely ill-posed. We shall use the expansion of the function in a series of Laguerre polynomials to convert the problem in an analytic interpolation problem. Then, using the coefficients of Lagrange polynomials we shall construct a stable approximation solution. Error estimate is given. Numerical results are produced.     

第二拉普拉斯变换及其应用

本文提出了一种新的积分变换——第二拉普拉斯变换,大家所熟悉的单边拉普拉斯变换为本变换的一种特殊情况．关于第二拉普拉斯变换的应用,本文举例说明     

Beyond sumudu transform and natural transform: J-transform properties and applications

In this paper, we introduce an efficient integral transform called the ${\mathbb J}$-transform which is a modification of the well-known Sumudu transform and the Natural transform for solving differential equations with real applications in applied physical sciences and engineering. The ${\mathbb J}$-transform is more advantageous than both the Sumudu transform and the Natural transform. Interestingly, our proposed ${\mathbb J}$-transform can be applied successfully to solve complex problems that are ordinarily beyond the scope of either Sumudu transform or Natural transform. As a proof of concept, we consider some classic examples and highlight the limitations of the previously reported integral transforms and lastly demonstrate the superiority of the proposed ${\mathbb J}$-transform in addressing those limitations.     

Shehu Transform: Extension to Distributions and Measures

This paper improves the computational aspect of the Shehu transform. An inversion formula is given. Finally the Shehu transform is extended to distributions and measures.     

The use of partition polynomial series in Laplace inversion of composite functions with applications in fractional calculus

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.     

应用拉普拉斯变换解决高等数学中的一些问题

拉普拉斯变换是积分变换的重要内容,不仅是学习后续专业课的重要数学工具,同时在数学的其他分支中也有重要的应用.本文利用拉普拉斯变换解决高等数学中的两类问题,并给出例题.     

一类索赔相关且带干扰的风险模型

该文讨论了索赔时间间隔与索赔量相关且带干扰的风险模型. 借助拉普拉斯变换研究了此模型的破产时刻、破产前瞬间盈余及破产时赤字三者的联合分布,得到了此联合分布拉普拉斯变换的一个分析表达式并讨论了当初始盈余值趋于无穷大时,此联合分布的渐近表达式.     

A Laplace transform technique for the analytical solution of a diffusion- convection equation over a finite domain

A Laplace transform technique has been utilized to obtain two different analytic solutions to a single diffusion-convection equation over a finite domain. One analytic solution is continuous at both ends of the domain of interest, while the other solution is discontinuous at the origin. This difference in the two solutions is explained. An application of the Laplace transform technique to a more complex system of equations, on a finite domain, is noted and an error apparent in a previous paper is corrected.     

Henstock-Kurzweil 可积函数的Laplace 变换的反演定理

该文建立了Henstock-Kurzweil 可积函数的 Laplace变换, 讨论了其基本性质及解析性质, 得到Henstock-Kurzweil可积意义下的反演公式, 并给出反例说明这一结果不能改进     

A Method of Solution for the One-Dimensional Heat Equation Subject to Nonlocal Conditions

The problem of solving the one-dimensional heat equation /t - ²/x² = f(x, t) subject to given initial and nonlocal conditions is considered. It is solved in the Laplace transform domain by taking the Laplace transform of the unknown function with respect to time t. The physical solution is recovered with the help of a numerical technique for inverting the Laplace transform.AMS Subject Classification (1991): 35K20.     

Application of differential transform method to non-linear oscillatory systems

In this paper, the differential transform method is proposed for solving non-linear oscillatory systems. These solutions do not exhibit periodicity, which is the characteristic of oscillatory systems. A modification of the differential transform method, based on the use of Padé approximants, is proposed. We use alternative technique by which the solution obtained by the differential transform method is made periodic. The method is described and illustrated with examples. The results reveal that the method is very effective and convenient.     

Laplace Transform of Temperate Holomorphic Functions

Let V be n-dimensional complex vector space. The aim of this paper is to give an elementary proof of the isomorphism O ^t _V ^{^}[n] O ^t _V*, which quantizes the Fourier–Sato transform of the conic sheaf O ^t _V of temperate holomorphic functions     

An extension of the Euler Laplace transform inversion algorithm with applications in option pricing

We show that the Euler algorithm for Laplace transform inversion can be extended to functions defined on the entire real line, if they have specific decay features. Our objective is to apply the method to option pricing problems, specifically when inverting Laplace transforms of the option price in the logarithm of the strike.