复合型分数阶微分方程初值问题的解

首先给出分数阶积分和微分的定义和性质;然后利用Laplace变换、Laplace逆变换公式和T函数汉克尔积分表达式改正了文献[1]和[9]中错误,给出了解的存在性和非存在性定理;最后用Laplace变换和逐次逼近法给出复合型分数阶变系数微分方程的初值问题的显式解.     

Ostrowski不等式的改进及其应用

通过对Ostrowski不等式的改进,扩大了Ostrowski积分公式的适用范围,将该积分公式应用于数值积分推广了经典的中点积分公式、梯形积分公式和Simpson积分公式,同时得到相应的最佳误差限,并给出了具体的数值应用.     

紧支撑样条小波插值及其应用

基于紧支撑样条小波函数插值与定积分的思想,给出了由紧支撑样条小波插值函数构造数值积分公式的方法．并将该方法应用于二次、三次、四次和五次紧支撑样条小波函数,得到了相应的数值积分公式．最后,通过数值例子验证,发现该方法得到的数值积分公式是准确的,且具有较高精度．     

Simpson不等式的改进及其应用

通过一阶导数对Simpson不等式进行了改进,并推导出了相应的数值积分公式和最佳误差限,扩大了Simpson积分公式的适用范围,最后给出了具体的数值应用.     

数值积分的双侧逼近及应用

通过分析基本数值求积公式的双侧逼近现象,利用加权平均的方法构造出了比原来求积公式至少高二次代数精度新的混合型求积公式,使得积分近似值精度得到大幅度提高,并给出应用它们求数值积分的具体实例.     

利用Laplace变换研究基于一个双跳模型的脆弱期权定价问题

本文针对欧式脆弱期权首先给出一个定价模型.在该模型中,期权对手方的企业资产价值服从双指数跳跃-扩散过程并且与期权标的资产的价格相关.双跳过程能够刻画对手方资产价值的突然提高或下降,从而对脆弱期权的定价提供更深层次的经济学解释.基于我们推导出的关于双跳过程的首次到达时间与相关Brownian运动的联合Laplace变换的显性表达式,并结合提前违约条件,本文通过二维Laplace变换给出关于欧式脆弱期权价格的的一个简单公式.采用数值Laplace逆变换方法,可实现利用该公式对欧式脆弱期权的定价.数值计算的结果表明,我们得到的定价公式是正确和有效的.     

一类新的高精度数值积分公式的构造

利用积分区间上的四个节点处的函数值及其一阶导数值作加权平均,构造高精度的数值积分公式,并对公式进行复化和加速.然后,推广到二重积分情形.最后,给出了几个数值算例,验证公式的有效性.     

有限土层轴对称Biot固结的一个新的解析解

提出一个新的解析方法来研究有限土层的轴对称Biot固结.从轴对称Biot固结的控制方程出发,结合Laplace变换的微分性质,建立了Laplace和Hankel变换域内有限土层地基表面(z=0)和任意深度z处基本变量之间的关系.然后结合有限土层的边界条件,推导出Laplace和Hankel变换域内任意一点的解析解.通过进行Laplace逆变换和Hankel逆变换得到了物理域内的解.编制了计算程序,并对有限土层轴对称固结进行了数值分析.     

一组一点外推的数值积分公式的提出与验证

通过利用拉格朗日多项式对等间距的离散采样数据进行函数逼近,就相等间距外推一点进行积分,得到了具有一定未来预测能力的一组一点外推的数值积分公式.计算机数值实验的结果表明了公式的有效性,即应用公式预测的积分结果可以达到一个较高的精度.     

积分第一中值定理中间点的一般渐近性质与求积公式

证明关于积分第一中值定理的中间点ξ的渐近性质的一般结果.而且,由此自然地推导出单节点数值积分公式.此求积公式具有高精度,还适于瑕积分的数值计算.     

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 solution of the impact problem for a rigid mass and a viscoelastic rod of finite length

The problem of the propagation and interaction of longitudinal waves in a rod struck by a rigid mass is considered. A numerical computer solution is obtained for the case when the mechanical behavior of the rod material is expressed by a rectangular relaxation spectrum. A numerical method of integral transform inversion is developed and applied to problems of polymer dynamics.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 450–456, May–June, 1971.     

The backus-gilbert method revisited: background,implementation and examples

The paper deals with the Backus-Gilbert or averaging kernel inversion of linear integral equations. The theoretical background of the method is developed: it is shown that the method leads to a sequence of linear pointwise estimates, which are asymptotically unbiased when no error is present. Anumerical implementation is given. Finally, the algorithm is applied to numerical differentiation, Laplace transform inversion and to a geophysical inverseproblem arising in electromagnetic sounding.     

Numerical inversion of Mellin transforms

A method is presented for the numerical inversion of Mellin transforms in which the inverse is obtained as an expansion in terms of Laguerre polynomials. The coefficients of this expansion are obtained as linear combinations of values of the transformed function or, equivalently, in terms of forward differences of this function. Thus, the Mellin transform of the series can be written as a forward interpolation series. Consequently the error of the numerical inversion procedure can be estimated. The practical advantage of the method is that values are needed for real arguments only.     

Numerical solution of fourth-order time-fractional partial differential equations with variable coefficients

In this paper, a numerical method for fourth-order time-fractional partial differential equations with variable coefficients is proposed. Our method consists of Laplace transform, the homotopy perturbation method and Stehfest's numerical inversion algorithm. We show the validity and efficiency of the proposed method (so called LHPM) by applying it to some examples and comparing the results obtained by this method with the ones found by Adomian decomposition method (ADM) and He's variational iteration method (HVIM).     

Numerical approximation of weakly singular integrals on the half line

This paper deals with the numerical approximation of a weakly singular integral transform by means of Laguerre nodes. Error estimates in a weighted uniform norm and some numerical tests are given.     

Numerical solution of Volterra integral and integro-differential equations of convolution type by using operational matrices of piecewise constant orthogonal functions

In this paper, we use operational matrices of piecewise constant orthogonal functions on the interval [0,1)$[0,1)$ to solve Volterra integral and integro-differential equations of convolution type without solving any system. We first obtain Laplace transform of the problem and then we find numerical inversion of Laplace transform by operational matrices. Numerical examples show that the approximate solutions have a good degree of accuracy.     

Numerical inversion of the Mellin transform on the real line for heavy-tailed probability density functions

A method for numerical inversion on the real line of the Mellin transform, without reduction of the problem to the inversion of Laplace transform is described. Maximum entropy technique is invoked in choosing the analytical form of the approximant function. Entropy-convergence and then L₁-norm convergence is proved. A stability analysis in evaluating entropy and expected values is illustrated. An upper bound of the error in the expected values computation is provided in terms of entropy.     

Numerical Inversion of the Mellin Transform

A method is described for the numerical inversion of the Mellintransform, without reduction of the problem to the inversionof the Laplace transform. The expansion of the original function(t) in a series of orthogonal Laguerre functions is used andthe coefficients of this expansion are obtained by means ofa collocation on the real axis of the transformed plane. Theperformance of the method is illustrated by the inversion offive test functions.     

Numerical inversion of the Laplace transform by accelerating the convergence of Bromwick's integral

A direct generalization of a convergence acceleration algorithm due to the author to infinite integrals is presented, and is used to obtain an efficient method for numerical inversion of the Laplace transform. Some numerical examples are given and compared with results of methods due to Salzer and Longman.