On the fractional calculus of Besicovitch function

Relationship between fractional calculus and fractal functions has been explored. Based on prior investigations dealing with certain fractal functions, fractal dimensions including Hausdorff dimension, Box dimension, K-dimension and Packing dimension is shown to be a linear function of order of fractional calculus. Both Riemann–Liouville fractional calculus and Weyl–Marchaud fractional derivative of Besicovitch function have been discussed.     

FRACTIONAL INTEGRALS OF THE WEIERSTRASS FUNCTIONS： THE EXACT BOX DIMENSION

The present paper investigates the fractal structure of fractional integrals of Weierstrass functions. The ezact box dimension for such functions many important cases is established. We need to point out that, although the result itself achieved in the present paper is interesting, the new technique and method should be emphasized. These novel ideas might be useful to establish the box dimension or Hausdorff dimension (especially for the lower bounds) for more general groups of functions.     

On the connection between the order of the fractional derivative and the Hausdorff dimension of a fractal function

This paper investigates the fractional derivative of a fractal function. It has been proven that there exists certain linear connection between the order of the Weyl-Marchaud fractional derivatives(WMFD) and the Hausdorff dimension of a fractal function. Graphs and numerical results further show this linear relationship.     

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.     

Vector-valued fractal functions: Fractal dimension and fractional calculus

There are many research available on the study of a real-valued fractal interpolation function and fractal dimension of its graph. In this paper, our main focus is to study the dimensional results for a vector-valued fractal interpolation function and its Riemann–Liouville fractional integral. Here, we give some results which ensure that dimensional results for vector-valued functions are quite different from real-valued functions. We determine interesting bounds for the Hausdorff dimension of the graph of a vector-valued fractal interpolation function. We also obtain bounds for the Hausdorff dimension of the associated invariant measure supported on the graph of a vector-valued fractal interpolation function. Next, we discuss more efficient upper bound for the Hausdorff dimension of measure in terms of probability vector and contraction ratios. Furthermore, we determine some dimensional results for the graph of the Riemann–Liouville fractional integral of a vector-valued fractal interpolation function.     

Fractal Dimension of Graph of Weierstrass Function and Its Derivative of the Fractional Order

In this paper,we obtain the fractal dimension of the graph of the Weierstrass function, its derivative of the fractional order and the relation between the dimension and the order of the fractional derivative.     

一个分形函数的分数阶微积分函数

Based on the combination of fractional calculus with fractal functions, a new type of is introduced； the definition, graph, property and dimension of this function are discussed.     

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.     

The Legendre condition of the fractional calculus of variations

Fractional operators play an important role in modelling nonlocal phenomena and problems involving coarse-grained and fractal spaces. The fractional calculus of variations with functionals depending on derivatives and/or integrals of noninteger order is a rather recent subject that is currently in fast development due to its applications in physics and other sciences. In the last decade, several approaches to fractional variational calculus were proposed by using different notions of fractional derivatives and integrals. Although the literature of the fractional calculus of variations is already vast, much remains to be done in obtaining necessary and sufficient conditions for the optimization of fractional variational functionals, existence and regularity of solutions. Regarding necessary optimality conditions, all works available in the literature concern the derivation of first-order fractional conditions of Euler–Lagrange type. In this work, we obtain a Legendre second-order necessary optimality condition for weak extremizers of a variational functional that depends on fractional derivatives.     

分形插值函数的分数阶微积分的分形维数

证明了线性分形插值函数的Riemann-Liouville分数阶微积分仍然是线性分形插值函数.在基于线性分形插值函数有关讨论的基础上,证明了线性分形插值函数的Box维数与Riemann-.Liouville分数阶微积分的阶之间成立着线性关系.文中给出的例子的图像和数值结果更进一步说明了这个结论.     

Some Remarks on One-dimensional Functions and Their Riemann–Liouville Fractional Calculus

A one-dimensional continuous function of unbounded variation on [0,1] has been constructed.The length of its graph is infnite,while part of this function displays fractal features.The Box dimension of its Riemann–Liouville fractional integral has been calculated.     

一类分形函数的Weyl-Marchaud分数阶导数的Hausdorff维数

首先介绍广义Weierstrass型函数的Weyl-Marchaud分数阶导数,得到了带随机相位的广义Weierstrass型函数的Weyl-Marchaud分数阶导数图像的Hausdorff维数,证明了该分形函数图像的Hausdorff维数与Weyl-Marchaud分数阶导数的阶之间的线性关系.     

Dimension of the Fractal Curve in Plane and Its Derivative of the Fractional Order

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ON A HYPER HILBERT TRANSFORM＊＊＊＊

The authors define the directional hyper Hilbert transform and give ita mixed norm estimate. The similar conclusions for the directional fractional integral of one dimension are also obtained in this paper. As an application of the above results, the authors give the Lp-boundedness for a class of the hyper singular integrals and the fractional integrals with variable kernel. Moreover, as another application of the above results, the authors prove the dimension free estimate for the hyper Riesz transform. This is an extension of the related result obtained by Stein.     

Fractal Dimension of Random Attractors for Non-autonomous Fractional Stochastic Reaction-diffusion Equations

This paper dealswith non-autonomous fractional stochastic reaction-diffusion equations driven by multiplicative noise with s ∈ (0,1). We first present some conditions for estimating the boundedness of fractal dimension of a random invariant set. Then we establish the existence and uniqueness of tempered pullback random attractors. Finally, the finiteness of fractal dimension of the random attractors is proved.     

Box dimension and fractional integral of linear fractal interpolation functions

In this paper, we present a new method to calculate the box dimension of a graph of continuous functions. Using this method, we obtain the box dimension formula for linear fractal interpolation functions (FIFs). Furthermore we prove that the fractional integral of a linear FIF is also a linear FIF and in some cases, there exists a linear relationship between the order of fractional integral and box dimension of two linear FIFs.     

On the fractional derivatives of radial basis functions: Theories and applications

The paper provides the fractional integrals and derivatives of the Riemann‐Liouville and Caputo type for the five kinds of radial basis functions, including the Powers, Gaussian, Multiquadric, Matérn, and Thin‐plate splines, in one dimension. It allows to use high‐order numerical methods for solving fractional differential equations. The results are tested by solving two test problems. The first test case focuses on the discretization of the fractional differential operator while the second considers the solution of a fractional order differential equation.     

Structure function and spectral density of fractal profiles

Spectral density and structure function for fractal profile are analyzed. It is found that the fractal dimension obtained from spectral density is not exactly the same as that obtained from structure function. The fractal dimension of structure function is larger than that of spectral density for small fractal dimension, and is smaller than that of spectral density for larger fractal dimension. The fractal dimension of structure function strongly depends on the spectral density at low and high wave numbers. The spectral density at low wave number affects the structure function at long distance, especially for small fractal dimension. The spectral density at high wave number affects the structure function at short distance, especially for large fractal dimension. This problem is more serious for bifractal profiles. Therefore, in order to obtain a correct fractal dimension, both spectral density and structure function should be checked.     

Electrostatics in fractal geometry: Fractional calculus approach

An electric field in a composite dielectric with a fractal charge distribution is obtained in the spherical symmetry case. The method is based on the splitting of a composite volume into a fractal volume V_d ～ r^d with the fractal dimension d and a complementary host volume V_h = V₃ ? V_d. Integrations over these fractal volumes correspond to the convolution integrals that eventually lead to the employment of the fractional integro-differentiation.     

Analysis of fractal dimension of mixed Riemann-Liouville integral

Numerical Algorithms - In this article, we provide a rigorous study on the fractal dimension of the graph of the mixed Riemann-Liouville fractional integral for various choices of continuous...