Time Fractional Diffusion: A Discrete Random Walk Approach

The time fractional diffusion equation is obtained from the standarddiffusion equation by replacing the first-order time derivative with afractional derivative of order (0, 1). From a physicalview-point this generalized diffusion equation is obtained from afractional Fick law which describes transport processes with longmemory. The fundamental solution for the Cauchy problem is interpretedas a probability density of a self-similar non-Markovian stochasticprocess related to a phenomenon of slow anomalous diffusion. By adoptinga suitable finite-difference scheme of solution, we generate discretemodels of random walk suitable for simulating random variables whosespatial probability density evolves in time according to this fractionaldiffusion equation.     

Letter: Self-Similar Bianchi Type VIII and IX Models

It is shown that in transitively self-similar spatially homogeneous tilted perfect fluid models the symmetry vector is not normal to the surfaces of spatial homogeneity. A direct consequence of this result is that there are no self-similar Bianchi VIII and IX tilted perfect fluid models. Furthermore the most general Bianchi VIII and IX spacetime which admits a four dimensional group of homotheties is given.     

Oblique entry of a wedge into an ideal incompressible fluid

A new approach to the solution of the self-similar problem of the entry of a wedge into an ideal fluid at an arbitrary angle to the free surface is proposed. The method is based on the construction of the expressions for the complex velocity and the derivative of the complex potential in a parametric variable domain. An integral and an integro-differential equation are obtained for determining the absolute magnitude and the angle of the velocity vector at the free boundary. The calculated results for the free surface shape, the angles of contact between the free surface and the wedge, and the coefficients of the hydrodynamic forces are presented.     

Self-similar solutions of the problem of displacement of one gas by another in an axisymmetric case with a quadratic drag law

A problem of piston-induced displacement of one gas by another in cracks (porous media) in an axisymmetric case with a quadratic drag law is studied. Self-similar solutions for determining the dynamic characteristics (velocity and pressure) of the displacing and displaced gases are constructed in quadratures. The velocity and pressure are studied as functions of a self-similar variable for several initial conditions and parameters. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 87–92, September–October, 2008.     

Wavelets, generalized white noise and fractional integration: The synthesis of fractional Brownian motion

We provide an almost sure convergent expansion of fractional Brownian motion in wavelets which decorrelates the high frequencies. Our approach generalizes Lévy's midpoint displacement technique which is used to generate Brownian motion. The low-frequency terms in the expansion involve an independent fractional Brownian motion evaluated at discrete times or, alternatively, partial sums of a stationary fractional ARIMA time series. The wavelets fill in the gaps and provide the necessary high frequency corrections. We also obtain a way of constructing an arbitrary number of non-Gaussian continuous time processes whose second order properties are the same as those of fractional Brownian motion.     

壁湍流扩展的自相似标度律的实验研究

对风洞中零压力梯度平板湍流边界层进行了实验研究，用热线风速仪测量了不同法向位置的脉动速度，研究了湍流边界层不同法向位置速度结构函数的扩展的自相似标度律。     

Queueing at large resources driven by long-tailed M/G/∞-modulated processes

We analyze the queue at a buffer with input comprising sessions whose arrival is Poissonian, whose duration is long-tailed, and for which individual session detail is modeled as a stochastic fluid process. We obtain a large deviation result for the buffer occupation in an asymptotic regime in which the arrival rate nr, service rate ns, and buffer level nb are scaled to infinity with a parameter n. This can be used to approximate resources which multiplex many sources, each of which only uses a small proportion of the whole capacity, albeit for long-tailed durations. We show that the probability of overflow in such systems is exponentially small in n, although the decay in b is slower, reflecting the long tailed session durations. The requirements on the session detail process are, roughly speaking, that it self-averages faster than the cumulative session duration. This does not preclude the possibility that the session detail itself has a long-range dependent behavior, such as fractional Brownian motion, or another long-tailed M/G/∞ process. We show how the method can be used to determine the multiplexing gain available under the constraint of small delays (and hence short buffers) for multiplexers of large aggregates, and to compare the differential performance impact of increased buffering as opposed to load reduction. This revised version was published online in June 2006 with corrections to the Cover Date.     

中空圆柱形边界条件下非线性介质中的晶格孤子

本文采用自相似方法求解中空圆柱形边界贝塞尔晶格中变系数非线性薛定谔方程, 得到了与数值模拟解一致的解析解, 表明由非衍射贝塞尔光束诱导的光子晶格可支持稳定的自相似孤子簇. 精确解ψ_mn是n+2层, 2m极的多极孤立波, 其形状及大小在传播过程中保持不变. 关键词： 空间光孤子 贝塞尔晶格 边界 自相似     

Fractal dynamics in the ionization of helium Rydberg atoms

We study the ionization of helium Rydberg atoms in an electric field above the classical ionization threshold within the semiclassical theory.By introducing a fractal approach to describe the chaotic dynamical behavior of the ionization,we identify the fractal self-similarity structure of the escape time versus the distribution of the initial launch angles of electrons,and find that the self-similarity region shifts toward larger initial launch angles with a decrease in the scaled energy.We connect the fractal structure of the escape time plot to the escape dynamics of ionized electrons.Of particular note is that the fractal dimensions are sensitively controlled by the scaled energy and magnetic field,and exhibit excellent agreement with the chaotic extent of the ionization systems for both helium and hydrogen Rydberg atoms.It is shown that,besides the electric and magnetic fields,core scattering is a primary factor in the fractal dynamics.     

基于自相似模型的二维X射线激光等离子体流体力学

从自相似方法出发,建立了简化的二维X射线激光等离子体流体动力学模型. 利用该模型研究了平板靶X射线激光的二维流体力学行为. 研究表明,当激光作用结束后,经过一段时间的演化电子密度和温度沿线聚焦方向逐渐达到平衡,该结果对X射线激光的优化设计具有重要意义. 关键词： X射线激光 二维流体力学计算 自相似方法