查询词典 fractional derivative

By discussing the position hypothesis of fractional-dimension derivative about general function and the formula form the hypothesis of fractional-dimension derivative about power function, the concrete equation formulas of fractional-dimension derivative, differential and integral are described distinctly further, and the difference between the fractional-dimension derivative and the fractional-order derivative are given too. Subsequently, the concrete forms of measure calculation equations of self-similar fractal obtaining by based on the definition of form in fractional-dimension calculus about general fractal measure are discussed again, and the differences with Hausdorff measure method or the covering method at present are given. By applying the measure calculation equations, the measure of self-similar fractals which include middle-third Cantor set, Koch curve, Sierpinski gasket and orthogonal cross star are calculated and analyzed.

通过讨论一般函数的分维导数的位置假设及幂函数的分维导数的形式假设，进一步明晰了幂函数的分维导数、分维微分及分维积分的具体方程形式，给出分维导数与分数阶导数的区别，随后讨论了基于一般分形测度的分维微积分形式定义导出的自相似分形的测度计算方程具体形式，给出了其与目前 Hausdorff 测度方法的区别，并对包括三分 Cantor 集合、 Koch 曲线、 Sierpinski 垫片及正交十字星形等自相似分形在内的测度进行了计算分析。

In recent years, systems which based on fractional differential and integral fractional, have been researched extensively, involving fractional circuit, fractional digital signal processing, fractional dynamics control systems and fractional-Chaos and super-chaos, Chaos Control and fractional chaos synchronization, and security communications. And a lot of theoretical and practical results are obtained.

故而近年来基于分数阶微分和积分的分数阶系统已在动力学系统中得以较为广泛的研究，其中涉及分数阶电路、分数阶数字信号处理、分数阶动力学控制系统以及分数阶混沌和超混沌、分数阶混沌控制与混沌同步、保密通信等多个领域，取得了不少的理论和实际结果。

The main results are as follows: the relations between local fractional integrated semigroups and the corresponding Cauchy problem, global fractional integrated semigroups and regularized semigroups are given; introduction of the notion of regularized resolvent families, and the generation theorem and analyticity criterions for regularized resolvent families are obtained; the spectral inclusions between fractional resolvent family and its generator, and the approximation for fractional resolvent families in the cases of generators approximation and fractional orders approximation; elliptic operators with variable coefficients generating fractional resolvent family on L^2 by using numerical range techniques; and the L^p theory for elliptic operators with real coefficients highest order are obtained by Sobolev''s inequalities and the a priori estimates for elliptic operators; and a kind of coercive differential operators generates fractional regularized resolvent family by applying the Fourier multiplier method, functional calculus and some basic properties of Mittag-Leffler functions.

主要结论是：给出了局部分数次积分半群和相应的Cauchy问题的关系以及分数次积分半群和正则半群的关系；引入了正则预解族的概念，并给出了其生成定理和解析生成法则；给出了分数次预解族与其生成元的谱包含关系，并研究了在生成元逼近和分数阶逼近两种情况下相应的预解族的逼近问题；利用数值域方法证明了具变系数的椭圆算子在L^2上生成分数次预解族；利用Sobolev不等式和椭圆算子的先验估计证明了具变系数的椭圆算子在其最高项系数为实数时在L^p上生成分数次预解族；运用Fourier乘子理论、泛函演算和Mittag-Leffler函数证明了一类强制微分算子可以生成分数次正则预解族，并给出了该预解族的范数估计。

Combining the definition of CWT and the derivative property of convolution, we constructed a general method to calculate the approximate derivative of signal through CWT by using the first and second derivative of Gaussian function, Haar, and the first derivative of three-order-Spline function as wavelets. As compared with the other approaches of calculating derivative, which include the numerical differentiation, polynomial filters, Fourier transform, and the recently proposed DWT method, fast calculation and simple mathematical operation were remarkable advantages of CWT method. For the signal corrupted by severe noise (Signal-toNoise Ratio=5), the satisfactory results could also obtained via CWT method through appropriately adiusting the dilations.

在此基础上，(1)结合连续小波变换的特点和卷积的微分性质，提出了使用Gaussian函数的一阶和二阶导数，Haar和三次样条函数的一阶导数作为小波函数的连续小波变换计算信号近似导数的一般性方法，与其他导数计算方法(包括数字微分法，多项式滤波法，Fourier变换法和离散小波变换法)相比，本法简单便捷，计算速度快，对于噪声含量较高的信号(S/N为5)，只要适当调节尺度即可获得比较满意的结果。

Finally, in the third section, by constructing some functional which similar to the conservation law of evolution equation and the technical estimates, we prove that in the inviscid limit the solution of generalized derivative Ginzburg—Landau equation converges to the solution of derivative nonlinear Schrodinger equation correspondently in one-dimension; The existence of global smooth solution for a class of generalized derivative Ginzburg—Landau equation are proved in two-dimension, in some special case, we prove that the solution of GGL equation converges to the weak solution of derivative nonlinear Schr〓dinger equation; In general case, by using some integral identities of solution for generalized Ginzburg—Landau equations with inhomogeneous boundary condition and the estimates for the L〓 norm on boundary of normal derivative and H〓 norm of solution, we prove the existence of global weak solution of the inhomogeneous boundary value problem for generalized Ginzburg—Landau equations.

第三部分：在一维情形，我们考虑了一类带导数项的Ginzburg—Landau方程，通过构造一些类似于发展方程守恒律的泛函及巧妙的积分估计，证明了当粘性系数趋于零时，Ginzburg—Landau方程的解逼近相应的带导数项的Schr〓dinger方程的解，并给出了最优收敛速度估计；在二维情形，我们证明了一类带导数项的广义Ginzburg—Landau方程整体光滑解的存在性，以及在某种特殊情形下，GL方程的解趋近于相应的带导数项的Schr〓dinger方程的弱解；在一般情形下，我们讨论了一类Ginzburg—Landau方程的非齐次边值问题，通过几个积分恒等式，同时估计解的H〓模及法向导数在边界上的模，证明了整体弱解的存在性。

this paper proposes a design method which can improve the performance of fractional order differential filter obviously under the premise of not increasing the structure complexity of the filter.this scheme which bases on mutually compensatory characters of typical differentiator and uses interpolated method to improve performance of iir digital fractional differential filter.the improved frequency response of fractional differential filter is more close to ideal fractional differential filter,it indicates the validity of proposed method.

摘 要：提出一种在不增加分数阶微分滤波器复杂度的前提下，能有效提高分数阶微分滤波器性能的方法。该方法利用几种基于典型微分算子的分数阶微分滤波器之间的互补性，通过相互内插结合的方式，用于提高iir分数阶数字滤波器的性能。改进后的分数阶微分滤波器频率响应更接近理想分数阶微分滤波器，表明所提方法的有效性。

In Chapter 3, considering fractional relaxation equation, we make use of directlythe Grunwald-Letnikov definition to discrete fractional derivative, obtain a numericalmethod of fractional relaxation equation,and give the proof of consistency ,conver-gency and stability.

第三章对于一个推广到分数阶的松驰方程，直接利用Gru¨nwald-Letnikov分数阶导数定义进行离散，得到分数阶松驰方程一个数值方法，并给出了相容性，收敛性和稳定性的证明。

We construct an explicit finite difference approximation for the equation by using the equivalence relation between Riemann-Liouville fractional derivative and Grünwald-Letnikovmake fractional derivative.

再利用Riemann-Liouville分数阶导数与Grtinwald-Letnikov分数阶导数之间的等价关系，构造一显式有限差分近似，这一离散格式可以解释为一个随机游走模型，并且收敛于稳定的概率分布。

In this paper,a space fractional differential equation is considered.The equation is obtained from the advection-diffusion equation by replacing the second order derivative in space by a fractional derivative in space of order.

考虑空间分数阶微分方程(即在一个标准的扩散-对流方程中，用分数阶导数代替空间二阶导数)，给出了该分数阶微分方程的显式和隐式有限差分格式。

The main research contents and results are described as follows:The constitutive equation of the elastomer modeled by the fractional derivative Maxwell model has been investigated. Based on the thermodynamics constraints analysis of the physical parameters of the fractional derivative Maxwell model, the relaxation modulus and the creeping modulus of the model are analyzed.

本论文针对该目的，采用理论研究与试验研究相结合的方法，首先研究了胶泥缓冲器的耗能机理，进而设计开发了较理想的缓冲器产品，具体研究内容和成果如下：研究了胶泥的本构模型——分数微分Maxwell模型。

Finally it offers the analysis of the fault-tolerance system as well as its test verification.

这样，运行于空间环境中的系统的设计目标就是在保证系统实时性

However, it is still difficult to find out the real reason of the increase of the keratinized gingiva during the orthodontic extrusion whether it is due to the proliferation of the gingival tissue or to its elastic nature, that is because the proliferation of the fibroblasts as well as the connective tissue changes in volume and the intra-fibers spaces were all immeasurable in most of the studies.

然而，仍然难以找到真正的原因增加的角化牙龈在正畸挤压它是否是由于扩散的牙龈组织或其弹性性质，这是因为增殖的成纤维细胞以及作为结缔组织的变化量和内部纤维空格都是不可估量的，大多数的研究。