使用解析方法获得FitzHugh-Nagumo方程新的peakon解

依据对FitzHugh-Nagumo方程的研究,通过微分变化法近似分析出FitzHugh-Nagumo方程,获得了这个方程的尖峰孤立波（peakon soliton）的解,从而获得了更多形式的peakon解,同时也分析了微分变换法（differential transform method, DTM）收敛区域和收敛速度．构建的微分变换法,结合帕德(Padé)逼近,构建一个明确的,完全解析,对FitzHugh-Nagumo方程全部有意义的尖波解．其主要思想是限制边界条件而令导数在孤立波不存在峰值,但导数的孤立波在两侧存在．结果表明,微分变换法在参数很小的情况下可以避免摄动的限制．表明这种方法提供了一种强大而有效地获得FitzHugh-Nagumo方程新的peakon解的数学方法．     

从弦振动柯西问题的解法谈学生创新思维的培养

针对学生认识的误区,对于无界弦振动的柯西问题,除了达朗贝尔解法外,给出了傅立叶变换法、拉普拉斯变换法、格林函数法和微分算子法四种解法,并倡议教师在教学过程中要充分利用学生的"好奇"、"好想"、"好动"的心理,采用提问式教学,要求学生一题多解,培养其发散性思维和创造性思维.     

Ungar微分变换在弹性动力学中的一个应用

近些年来,在求解弹性波方面的一些问题时,许多著者都应用了Cagniard—de Hoop方法[1][2].但是,在使用该法时,定要进行一些比较复杂的改变积分路径的工作.A.Ungar所提出的一种微分变换[3~6]可以避免这种困难.本文应用Ungar微分变换来求解Lamb问题[1][2]的一情形.     

Laplace积分的几种计算方法

主要给出一类Laplace积分的五种求解方法,这五种方法是:积分号下积分法、积分号下微分法、解微分方程法、Laplace变换法以及Fourier变换法,并在最后给出与Laplace积分相关的含参量广义积分的结果.     

Riemann—Liouville型分数阶微分方程的微分变换方法

本文在Riemann-Liouville分数阶导数的广义Taylor公式的基础上,建立了求解Riemann-Liouville型分数阶微分方程的微分变换方法.本文所建立的基于Riemann-Liouville分数阶导数微分变换方法给求解Riemann-Liouville分数阶导数的微分方程提供了一种新工具。     

关于微分熵的一类强偏差定理

利用关于乘积分布密度的相对熵和相对熵率的概念,建立了相依连续型随机变量序列关于参考微分熵的一类强偏差定理,证明中给出了将Laplace变换应用于微分熵强偏差定理的研究的一种途径．     

广义的二维微分变换法在分数阶偏微分方程中的应用

分数阶偏微分方程的解析近似解是近年来国内外重要的研究工作之一.借助于符号计算软件Maple,应用广义的二维微分变换法求解Caputo型分数阶导数定义下的时间分数阶偏微分方程、空间分数阶偏微分方程和时空分数阶偏微分方程.在获得三种分数阶偏微分方程解析近似解的同时,验证广义的二维微分变换法的可行性和有效性,说明此解析技术可以用于求解复杂的分数阶偏微分方程系统.     

学习理论中核函数逼近的Jackson型不等式（英文）

从微分算子角度理解核函数空间,借助经典Fourier变换研究核函数逼近问题.应用Fourier乘子算子和算子半群定义了一种光滑模,证明其与一种基于微分算子的K-泛函的等价性,由此给出了刻画核函数逼近收敛性的Jackson不等式.进一步证明,如果微分算子为Riesz势算子或Bessel势算子,逼近的收敛性可以转化为卷积算子逼近.特别地,给出了再生核Hilbert空间逼近的一种上界估计.     

论算符微积的基础

介绍算符微积——亦称运算微积——的方法通常有下列几种 1°代数法, 2°实验法, 3°积分变换法, 4°抽象空间法. Lagrange首先应用了代数法.用抽象代数的语言可以很容易地标明此法的特徵,即对复数域添入一微分算符s後形成一个算符域,然後以此作用於一个函数环C     

高层大气结构与动力学若干问题的解析理论(Ⅰ)

本文首次推导并讨论了热层大气影响函数,潮汐微分算子及其本征函数与扩散波等新函数和概念。 对于热层大气浓度和温度变化,找出了一种新的解析解法,把周期变化方程变换成二阶自共轭非齐次微分方程,并由此推导出热层大气影响函数这一类新函数。对于三维情形,用分层模式法和积分变换法得到了分层影响函数与热层大气潮汐微分算子及其本征函数。这一新算子是广泛应用的球谐波算子和Laplacc潮汐微分算子的推广;新本征函数是包括球函数和Hough函数在内的一类更加广泛的函数。 文中还讨论了这些新函数与概念在热层大气结构和动力学研究中的意义;并求解了中、小尺度热层大气波动的色散方程,从理论上推出了一类新波动——扩散波。     

The modified algorithm for the differential transform method to solution of Genesio systems

In this article, approximate analytical solution of chaotic Genesio system is acquired by the modified differential transform method (MDTM). The differential transform method (DTM) is mentioned in summary. MDTM can be obtained from DTM applied to Laplace, inverse Laplace transform and Padé approximant. The MDTM is used to increase the accuracy and accelerate the convergence rate of truncated series solution getting by the DTM. Results are given with tables and figures.     

Differential transform method for solving Volterra integral equation with separable kernels

In this paper, Volterra integral equations with separable kerenels are solved using the differential transform method. The approximate solution of this equation is calculated in the form of a series with easily computable terms. Exact solutions of linear and nonlinear integral equations have been investigated and the results illustrate the reliability and the performance of the differential transform method.     

Laplace transform for the solution of higher order deformation equations arising in the homotopy analysis method

A new analytic approach for solving nonlinear ordinary differential equations with initial conditions is proposed. First, the homotopy analysis method is used to transform a nonlinear differential equation into a system of linear differential equations; then, the Laplace transform method is applied to solve the resulting linear initial value problems; finally, the solutions to the linear initial value problems are employed to form a convergent series solution to the given problem. The main advantage of the new approach is that it provides an effective way to solve the higher order deformation equations arising in the homotopy analysis method.     

Analytical and approximate solutions of fractional‐order susceptible‐infected‐recovered epidemic model of childhood disease

In this work, we make use of the conformable fractional differential transform method (CFDTM) in order to compute an approximate solution of the fractional‐order susceptible‐infected‐recovered (SIR) epidemic model of childhood disease. The method provides the solution in the form of a rapidly convergent series. Two explanatory and illustrative examples are given to represent the efficacy of the obtained results.     

The Laplace transform and polynomial Trefftz method for a class of time dependent PDEs

In this paper, we present a new method for solving 1D time dependent partial differential equations based on the Laplace transform (LT). As a result, the problem is converted into a stationary boundary value problem (BVP) which depends on the parameter of LT. The resulting BVP is solved by the polynomial Trefftz method (PTM), which can be regarded as a meshless method. In PTM, the source term is approximated by a truncated series of Chebyshev polynomials and the particular solution is obtained from a recursive procedure. Talbot’s method is employed for the numerical inversion of LT. The method is tested with the help of some numerical examples.     

Exact solutions for non-linear Schrödinger equations by differential transformation method

In this paper, we implemented relatively new, exact series method of solution known as the differential transform method for solving linear and non-linear Schrödinger equations with initial profile. The method can easily be applied to many linear and non-linear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. The results obtained are in good agreement with the exact solution. These results show that the technique introduced here is accurate and easy to apply.     

Solution of free vibration equations of beam on elastic soil by using differential transform method

Free vibration differential equations of motion of one end fixed, the other simply supported and axial loaded beams on elastic soil is solved using differential transform method (DTM), analytical solution and frequency factors are obtained.     

A study on the convergence conditions of generalized differential transform method

This paper deals with constructing generalized ‘fractional’ power series representation for solutions of fractional order differential equations. We present a brief review of generalized Taylor's series and generalized differential transform methods. Then, we study the convergence of fractional power series. Our emphasis is to address the sufficient condition for convergence and to estimate the truncated error. Numerical simulations are performed to estimate maximum absolute truncated error when the generalized differential transform method is used to solve non‐linear differential equations of fractional order. The study highlights the power of the generalized differential transform method as a tool in obtaining fractional power series solutions for differential equations of fractional order. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical solutions of the space- and time-fractional coupled Burgers equations by generalized differential transform method

In this paper, by introducing the fractional derivative in the sense of Caputo, the generalized two-dimensional differential transform method (DTM) is directly applied to solve the coupled Burgers equations with space- and time-fractional derivatives. The presented method is a numerical method based on the generalized Taylor series formula which constructs an analytical solution in the form of a polynomial. Several illustrative examples are given to demonstrate the effectiveness of the generalized two-dimensional DTM for the equations.     

Equations of anisotropic elastodynamics as a symmetric hyperbolic system: Deriving the time-dependent fundamental solution

The dynamic system of anisotropic elasticity from three second order partial differential equations is written in the form of the time-dependent first order symmetric hyperbolic system with respect to displacement velocity and stress components. A new method of deriving the time-dependent fundamental solution of the obtained system is suggested in this paper. This method consists of the following. The Fourier transform image of the fundamental solution with respect to a space variable is presented as a power series expansion relative to the Fourier parameters. Then explicit formulae for the coefficients of these power series are derived successively. Using these formulae the computer calculation of fundamental solution components (displacement velocity and stress components arising from pulse point forces) has been made for general anisotropic media (orthorhombic and monoclinic) and the simulation of elastic waves has been obtained. These computational examples confirm the robustness of the suggested method.