Fokker–Planck equation on fractal curves

A Fokker–Planck equation on fractal curves is obtained, starting from Chapmann–Kolmogorov equation on fractal curves. This is done using the recently developed calculus on fractals, which allows one to write differential equations on fractal curves. As an important special case, the diffusion and drift coefficients are obtained, for a suitable transition probability to get the diffusion equation on fractal curves. This equation is of first order in time, and, in space variable it involves derivatives of order α, α being the dimension of the curve. An exact solution of this equation with localized initial condition shows departure from ordinary diffusive behavior due to underlying fractal space in which diffusion is taking place and manifests a subdiffusive behavior. We further point out that the dimension of the fractal path can be estimated from the distribution function.     

Logistic混沌系统分形特性研究

对Logistic序列进行研究,利用Matlab数值模拟,通过计算不同初值、不同参数对应的混沌序列的计盒维数,得出结论:只要在数据充分的情况下,Logistic系统的分形维数基本由参数λ决定,与系统初值无关;同时计盒维数并非像熵一样随Logistic系统的参数λ增大而增大.     

Global Existence and Blowup for a Parabolic Equation with a Non-Local Source and Absorption

In this paper we consider a double fronts free boundary problem for a parabolic equation with a non-local source and absorption. The long time behaviors of the solutions are given and the properties of the free boundaries are discussed. Our results show that if the initial value is sufficiently large, then the solution blows up in finite time, while the global fast solution exists for sufficiently small initial data, and the intermediate case with suitably large initial data gives the existence of the global slow solution.     

Fractal dimensions of fractional integral of continuous functions

In this paper,we mainly explore fractal dimensions of fractional calculus of continuous functions defined on closed intervals.Riemann–Liouville integral of a continuous function f(x) of order v(v0) which is written as D~(-v) f(x) has been proved to still be continuous and bounded.Furthermore,upper box dimension of D~(-v) f(x) is no more than 2 and lower box dimension of D~(-v) f(x) is no less than 1.If f(x) is a Lipshciz function,D~(-v) f(x) also is a Lipshciz function.While f(x) is differentiable on [0,1],D~(-v) f(x) is differentiable on [0,1] too.With definition of upper box dimension and further calculation,we get upper bound of upper box dimension of Riemann–Liouville fractional integral of any continuous functions including fractal functions.If a continuous function f(x) satisfying H?lder condition,upper box dimension of Riemann–Liouville fractional integral of f(x) seems no more than upper box dimension of f(x).Appeal to auxiliary functions,we have proved an important conclusion that upper box dimension of Riemann–Liouville integral of a continuous function satisfying H?lder condition of order v(v0) is strictly less than 2-v.Riemann–Liouville fractional derivative of certain continuous functions have been discussed elementary.Fractional dimensions of Weyl–Marchaud fractional derivative of certain continuous functions have been estimated.     

First boundary-value problem in the half-strip for a parabolic-type equation with bessel operator and Riemann–Liouville derivative

The first boundary-value problem in the half-strip for a parabolic-type equation with Bessel operator and Riemann–Liouville derivative is studied. In the case of the zero initial condition, the representation of the solution in terms of the Fox H-function is obtained. The uniqueness of the solution for a class of functions vanishing at infinity is proved. It is shown that when the equation under consideration coincides with the Fourier equation, the obtained representation of the solution becomes the known representation of the solution of the corresponding problem.     

On the local fractional wave equation in fractal strings

The key aim of the present study is to attain nondifferentiable solutions of extended wave equation by making use of a local fractional derivative describing fractal strings by applying local fractional homotopy perturbation Laplace transform scheme. The convergence and uniqueness of the obtained solution by using suggested scheme is also examined. To determine the computational efficiency of offered scheme, some numerical examples are discussed. The results extracted with the aid of this technique verify that the suggested algorithm is suitable to execute, and numerical computational work is very interesting.     

Finite dimensionality of global attractors for a non-classical reaction-diffusion equation with memory

In this paper, we consider a periodic boundary value problem for a non-classical reaction-diffusion equation with memory. In other paper, we use the ω-limit compactness of the solution semigroup {S(t)}_t≥0 to get the existence of a global attractor. The main goal here is to give an estimate of the fractal dimension of the global attractor. By the fractal dimension theorem given by A.O. Celebi et al., we obtain that the fractal dimension of the global attractor for the problem is finite; this makes the results for the non-classical reaction-diffusion equations more substantial and perfect.     

The Second Boundary-Value Problem in a Half-Strip for a Parabolic-Type Equation with Bessel Operator and Riemann–Liouville Partial Derivative

We study the second boundary-value problem in a half-strip for a differential equation with Bessel operator and the Riemann–Liouville partial derivative. In the case of a zero initial condition, a representation of the solution is obtained in terms of the Fox H-function. The uniqueness of the solution is proved for the class of functions satisfying an analog of the Tikhonov condition.     

Quasi-invariance of Lebesgue measure under the homeomorphic flow generated by SDE with non-Lipschitz coefficient

We consider the stochastic flow generated by Stratonovich stochastic differential equations with non-Lipschitz drift coefficients. Based on the author's previous works, we show that if the generalized divergence of the drift is bounded, then the Lebesgue measure on Rd is quasi-invariant under the action of the stochastic flow, and the explicit expression of the Radon-Nikodym derivative is also presented. Finally we show in a special case that the unique solution of the corresponding Fokker-Planck equation is given by the density of the stochastic flow.     

Koch曲线及其分数阶微积分

给出了Koch曲线的一个复值表达式,并且估计了该表达式的分数阶微积分的分形维数,同时给出了此表达式的Weyl-Marchaud分数阶导数的图像.进一步讨论了Koch曲线的图像与某类自仿分形函数图像的联系.最后证明了这类自仿分形函数的分形维数与其分数阶微积分的分形维数成立着线性关系,一个特殊例子的图像和数值结果在文中给出.     

广义Kuramoto—Sivashinsky方程吸引子的分形结构

首先给出广义Ｋｕｒａｍｏｔｏ＿Ｓｉｖａｓｈｉｎｓｋｙ（ＧＫＳ）方程周期初边值问题在Ｈ２空间惯性集的构造，进而给出并证明ＧＫＳ方程吸引子的分形结构，同时发现吸引子的一个分形局部化指数型逼近序列·上述结果精细和推进了［１，３，５，７］关于惯性集和吸引子的结论，刻划了吸引子的一种几何结构     

Fundamental solution and discrete random walk model for a time-space fractional diffusion equation of distributed order

In this paper, we consider a time-space fractional diffusion equation of distributed order (TSFDEDO). The TSFDEDO is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α∈(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of orders β ₁∈(0,1) and β ₂∈(1,2], respectively. We derive the fundamental solution for the TSFDEDO with an initial condition (TSFDEDO-IC). The fundamental solution can be interpreted as a spatial probability density function evolving in time. We also investigate a discrete random walk model based on an explicit finite difference approximation for the TSFDEDO-IC.     

一类四阶差分方程边值问题及分形曲面的生成

本文研究一个二变元四阶差分方程边值问题，证明了此问题的适定性，揭示了解的结构。经过证明和数值模拟，可以作为分形曲面生成和插值的一种新方法。     

Determining surface heat flux in the steady state for the Cauchy problem for the Laplace equation

In this paper, we consider the Cauchy problem for the Laplace equation, in a strip where the Cauchy data is given at x = 0 and the flux is sought in the interval 0<x?1. This problem is typical ill-posed: the solution (if it exists) does not depend continuously on the data. We study a modification of the equation, where a fourth-order mixed derivative term is added. Some error stability estimates for the flux are given, which show that the solution of the modified equation is approximate to the solution of the Cauchy problem for the Laplace equation. Furthermore, numerical examples show that the modified method works effectively.     

A Diffusion Equation on Fractals in Random Media

ＡＤｉｆｕｓｉｏｎＥｑｕａｔｉｏｎｏｎＦｒａｃｔａｌｓｉｎＲａｎｄｏｍＭｅｄｉａ１１ＴｈｅｐａｐｅｒｗａｓｒｅｃｅｉｖｅｄｏｎＪｕｌｙ．３０ｔｈ，１９９７ＤｅＬＩＵ，ＨｏｕｑｉａｎｇＬＩ，ＦｕｘｕａｎＣＨＡＮＧ＆ＨｏｎｇｍｅｉＺＨＡＮＧＤｅｐａｒｔｍ...     

分形统计测度构建及其在投资组合中的应用研究

准确测量证券的风险和收益无论是对投资管理,还是对金融理论研究,甚至对理论成果向实践应用转化都至关重要。本文在证券价格具有分形特征的现实背景下,基于分形理论构建了分形期望和分形方差两个分形统计测度,以克服非分形统计测度在风险收益方面测不准或不可测的缺陷。在此基础上,应用分形统计测度构建了投资组合模型,给出了分形组合模型的解析解;随后,利用实证分析验证了分形统计测度在投资组合应用中的有效性。本文创新之处在于针对证券价格具有分形特征的现实背景构建了分形期望和分形方差两个分形统计测度;并基于分形统计测度构建了投资组合模型,将证券价格普遍存在的分形特征纳入投资组合的研究框架。     

一类分形曲面的精细计盒维数

研究一类分形曲面的精细计盒维数,得到了星积分形曲面与其生成元的精细计盒维数的关系.     

On a time nonlocal problem for inhomogeneous evolution equations

Under consideration is some problem for inhomogeneous differential evolution equation in Banach space with an operator that generates a C ₀-continuous semigroup and a nonlocal integral condition in the sense of Stieltjes. In case the operator has continuous inhomogeneity in the graph norm. We give the necessary and sufficient conditions for existence of a generalized solution for the problem of whether the nonlocal data belong to the generator domain. Estimates on solution stability are given, and some conditions are obtained for existence of the classical solution of the nonlocal problem. All results are extended to a Sobolev-type linear equation, the equation in Banach space with a degenerate operator at the derivative. The time nonlocal problem for the partial differential equation, modeling a filtrating liquid free surface, illustrates the general statements.     

Truncated fractal basin boundaries in the pendulum with nonperiodic forcing

Summary It is well known that oscillators such as the pendulum can have fractal basin boundaries when they are periodically forced with the consequence that the long term behavior of the system may be unpredictable. In engineering and physical applications, the forcing is often nonperiodic and eventually decays to zero, and simulation of the pendulum with decaying forcing (M. Varghese, J. S. Thorp, Physical Review Letters, vol. 60, no. 8, pp. 665–668, Feb. 1988) exhibits truncated fractal basin boundaries which also limit the system predictability. We develop a coordinate change for the pendulum with decaying forcing that allows us to apply standard qualitative methods to study the basin boundaries. We prove that the basin boundaries cannot be fractal and show by example how the extreme stretching and folding leading to a truncated fractal basin boundary may arise.