分数阶光滑函数线性和二次插值公式余项估计

本文在局部分数阶导数定义的基础上给出了高阶局部分数阶导数定义,并据此得到了一般形式的分数阶Taylor公式.用该公式给出了分数阶光滑函数线性和二次插值公式余项的表达式,并进一步导出了分段线性插值的收敛阶估计.针对分数阶导数临界阶计算困难的问题,本文利用线性插值余项设计了一种外推算法,能够比较准确地求出函数在某点的局部分数阶导数的临界阶.最后通过编写算法的Mathematica程序,验证了理论分析的正确性,并用实例说明了算法的有效性.     

第二类端点奇异Fredholm积分方程的分数阶退化核方法

本文针对第二类端点奇异Fredholm积分方程构造基于分数阶Taylor展开的退化核方法，设计了两种计算格式，一是在全区间上使用分数阶Taylor展开式近似核函数，二是在包含奇点的小区间上采用分数阶插值，在剩余区间上采用分段二次多项式插值逼近核函数.讨论了两种退化核方法收敛的条件，并给出了混合插值法的收敛阶估计.数值算例表明对于非光滑核函数分数阶退化核方法有着良好的计算效果，且混合二次插值法比全区间上的分数阶退化核方法有着更广泛的适用范围.     

第二类两端奇异Fredholm积分方程的分数阶线性插值方法

考虑第二类两端奇异的Fredholm积分方程,假设核函数在区间的两个端点非光滑,存在分数阶的Taylor展开式.对于这种类型的核函数,在包含端点的小区间上采用分数阶插值,在剩余区间上采用分段线性插值逼近,由此得到一种分数阶线性插值退化核方法.本文讨论该方法收敛的条件,给出收敛阶估计.数值算例表明这种分数阶混合线性插值方法对于两端奇异核函数有着较好的计算效果.     

An Element-Free Galerkin Method for Time-Fractional Diffusion-Wave Equations北大核心CSCD

利用无单元Galerkin法,对Caputo意义下的时间分数阶扩散波方程进行了数值求解和相应误差理论分析。首先用L1逼近公式离散该方程中的时间变量,将时间分数阶扩散波方程转化成与时间无关的整数阶微分方程;然后采用罚函数方法处理Dirichlet边界条件,并利用无单元Galerkin法离散整数阶微分方程;最后推导该方程无单元Galerkin法的误差估计公式。数值算例证明了该方法的精度和效果。     

闭区间[0,1]上的分形插值函数的分数阶积分

介绍了分形插值函数和迭代函数系统以及v阶黎曼-刘准尔分数阶积分的概念及相关定理.利用这些概念及定理讨论了分形插值函数的分数阶积分在[0,1]上连续性及判定[0,1]上的分形插值函数的分数阶积分也是[0,1]上的分形插值函数,并给予了证明.     

一类分形插值函数的分数阶微积分

介绍了分形插值函数和迭代函数系统以及v阶黎曼-刘维尔分数阶积分、微分的概念和相关定理.由于分形插值函数满足应用分数阶微积分处理问题的条件,所以利用这些概念及分步积分的方法讨论了折线段分形插值函数的分数阶积分的连续性,可微性及哪些点是不可微的,进一步说明了该插值函数分数阶微分的连续性并指出其不连续点,用黎曼-刘维尔分数阶微积分与分形插值函数结合起来研究,目的是想设法跟经典微积分一样,能找出函数上在该点的微积分的具体的实际应用意义.这些理论为研究分形插值函数的分数阶微积分的实际应用意义提供了一些理论基础.     

基于分数阶热传导方程激光加热瞬态温度场研究

基于分数阶Taylor(泰勒)级数展开原理,建立单相延迟一阶分数阶近似方程,获得分数阶热传导方程.针对短脉冲激光加热问题建立分数阶热传导方程组,并运用Laplace(拉普拉斯)变换方法进行求解,给出非Gauss(高斯)时间分布的激光内热源温度场解析解.针对具体算例数值研究温度波传播特性.结果表明热传播速度与分数阶阶次有关,分数阶阶次增加,热传播速度减小,温度变化幅度增加.分数阶方程可以用于描述介于扩散方程和热波方程间的热传输过程,且对热传播机制与分数阶热传导方程中分数阶项的关系做了深入剖析.     

基于分数阶Fourier变换的尺度函数的刻画

针对分数阶Fourier变换在信号处理中应用的广泛性,引入了分数阶尺度函数与分数阶小波变换的概念.运用分数阶Fourier变换与时频分析方法研究了分数阶多分辨分析与尺度函数的构造方法,刻画分数阶尺度函数的特征.得到分数阶尺度函数存在的充要条件.     

分布阶扩散-波动方程的有限元解的误差估计

本文考虑了分布阶时间分数阶扩散波动方程,其中时间分数阶导数是在Caputo意义上定义的,其阶次$\alpha,\beta$分别属于（0,1）和（1,2）.文中提出了在计算上行之有效的数值方法来模拟分布阶时间分数阶扩散波动方程.在时间上,通过中点求积公式把分布阶项转换为多项的时间分数阶导数项,并且利用$L1$和$L2$公式来近似Caputo分数阶导数;空间上使用Galerkin有限元方法进行离散.给出了基于$H^1$范数的有限元解的稳定性和误差估计的详细证明,最后的数值算例结果说明了理论分析的正确性以及有效性.     

分数阶尺度函数的分解与重构算法

引入分数阶多分辨分析与分数阶尺度函数的概念.运用时频分析方法与分数阶小波变换,研究了分数阶正交小波的构造方法,得到分数阶正交小波存在的充要条件.给出分数阶尺度函数与小波的分解与重构算法,算法比经典的尺度函数与小波的分解与重构算法更具有一般性.     

Fractional Hermite degenerate kernel method for linear Fredholm integral equations involving endpoint weak singularities

In this article, the Fredholm integral equation of the second kind with endpoint weakly singular kernel is considered and suppose that the kernel possesses fractional Taylor''s expansions about the endpoints of the interval. For this type kernel, the fractional order interpolation is adopted in a small interval involving the singularity and piecewise cubic Hermite interpolation is used in the remaining part of the interval, which leads to a kind of fractional degenerate kernel method. We discuss the condition that the method can converge and give the convergence order. Furthermore, we design an adaptive mesh adjusting algorithm to improve the computational accuracy of the degenerate kernel method. Numerical examples confirm that the fractional order hybrid interpolation method has good computational results for the kernels involving endpoint weak singularities.     

The finite element method for fractional diffusion with spectral fractional Laplacian

This paper focuses on the finite element method for Caputo-type parabolic equation with spectral fractional Laplacian, where the time derivative is in the sense of Caputo with order in (0,1) and the spatial derivative is the spectral fractional Laplacian. The time discretization is based on the Hadamard finite-part integral (or the finite-part integral in the sense of Hadamard), where the piecewise linear interpolation polynomials are used. The spatial fractional Laplacian is lifted to the local spacial derivative by using the Caffarelli–Silvestre extension, where the finite element method is used. Full-discretization scheme is constructed. The convergence and error estimates are obtained. Finally, numerical experiments are presented which support the theoretical results.     

A higher dimensional fractional Borel‐Pompeiu formula and a related hypercomplex fractional operator calculus

In this paper, we develop a fractional integro‐differential operator calculus for Clifford algebra‐valued functions. To do that, we introduce fractional analogues of the Teodorescu and Cauchy‐Bitsadze operators, and we investigate some of their mapping properties. As a main result, we prove a fractional Borel‐Pompeiu formula based on a fractional Stokes formula. This tool in hand allows us to present a Hodge‐type decomposition for the fractional Dirac operator. Our results exhibit an amazing duality relation between left and right operators and between Caputo and Riemann‐Liouville fractional derivatives. We round off this paper by presenting a direct application to the resolution of boundary value problems related to Laplace operators of fractional order.     

Fourier and barycentric formulae for equidistant Hermite trigonometric interpolation

We consider the Hermite trigonometric interpolation problem of order 1 for equidistant nodes, i.e., the problem of finding a trigonometric polynomial t that interpolates the values of a function and of its derivative at equidistant points. We give a formula for the Fourier coefficients of t in terms of those of the two classical trigonometric polynomials interpolating the values and those of the derivative separately. This formula yields the coefficients with a single FFT. It also gives an aliasing formula for the error in the coefficients which, on its turn, yields error bounds and convergence results for differentiable as well as analytic functions. We then consider the Lagrangian formula and eliminate the unstable factor by switching to the barycentric formula. We also give simplified formulae for even and odd functions, as well as consequent formulae for Hermite interpolation between Chebyshev points.     

A gray model for increasing sequences with nonhomogeneous index trends based on fractional‐order accumulation

By comparing the class ratio deviation and restoring error of first‐order accumulation with that of fractional‐order accumulation, a gray model for monotonically increasing sequences can obtain optimal simulation accuracy via selecting a proper cumulative order. In this study, a gray model for increasing sequences with nonhomogeneous index trends based on fractional‐order accumulation is proposed. To reduce the modeling error caused by the background value and to improve the prediction accuracy of the model, an optimized model using the 3/8 Simpson formula is constructed. Finally, the 2 proposed models are used to predict the total energy consumption in China and the monthly sales of new products in an enterprise. Compared with the GM(1,1) model based on fractional‐order accumulation, the proposed model exhibits better simulation and prediction accuracy.     

WEIGHTED KOPPELMAN－LERAY－NORGUET FORMULAS ON A LOCAL q－CONCAVE WEDGE IN A COMPLEX MANIFOLD

A weighted Koppelman-Leray-Norguet formula of (r, s) differential forms on a local q-concave wedge in a complex manifold is obtained. By constructing the new weighted kernels, the authors give a new weighted Koppelman-Leray-Norguet formula with-out boundary integral of (r, s) differential forms, which is different from the classical one. The new weighted formula is especially suitable for the case of the local q-concave wedge with a non-smooth boundary, so one can avoid complex estimates of boundary integrals and the density of integral may be not defined on the boundary but only in the domain. Moreover, the weighted integral formulas have much freedom in applications such as in the intervolation of functions.     

A remainder formula of numerical differentiation for the generalized Lagrange interpolation

Through exploiting the generalized Lagrange interpolation for continuous piecewise smooth functions, a remainder term of numerical differentiation is given. Also, the error estimates for the numerical differentiation formula are presented.