A Geometric Based Connection between Fractional Calculus and Fractal Functions

Establishing the accurate relationship between fractional calculus and fractals is an important research content of fractional calculus theory.In the present paper,we investigate the relationship between fractional calculus and fractal functions,based only on fractal dimension considerations.Fractal dimension of the Riemann-Liouville fractional integral of continuous functions seems no more than fractal dimension of functions themselves.Meanwhile fractal dimension of the Riemann-Liouville fractional differential of continuous functions seems no less than fractal dimension of functions themselves when they exist.After further discussion,fractal dimension of the Riemann-Liouville fractional integral is at least linearly decreasing and fractal dimension of the Riemann-Liouville fractional differential is at most linearly increasing for the Holder continuous functions.Investigation about other fractional calculus,such as the Weyl-Marchaud fractional derivative and the Weyl fractional integral has also been given elementary.This work is helpful to reveal the mechanism of fractional calculus on continuous functions.At the same time,it provides some theoretical basis for the rationality of the definition of fractional calculus.This is also helpful to reveal and explain the internal relationship between fractional calculus and fractals from the perspective of geometry.

Hausdorff Measure of Homogeneous Cantor Set

This paper gives the Hausdorff measure of a class of homogeneous Cantor sets.

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（v 〉 0） which is written as D-Vf（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 HSlder 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 HSlder condition of order v（v 〉 0） 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 Fractional Derivatives of a Fractal Function

The present paper investigates the fractional derivatives of Weierstrass function, proves that there exists some linear connection between the order of the fractional derivatives and the dimension of the graphs of Weierstrass function.

Two-dimensional Wave Equations with Fractal Boundaries

This paper focuses on two cases of two-dimensional wave equations with fractal boundaries. The first case is the equation with classical derivative. The formal solution is obtained. And a definition of the solution is given. Then we prove that under certain conditions, the solution is a kind of fractal function, which is continuous, differentiable nowhere in its domain. Next, for specific given initial position and 3 different initial velocities, the graphs of solutions are sketched. By computing the box dimensions of boundaries of cross-sections for solution surfaces, we evaluate the range of box dimension of the vibrating membrane. The second case is the equation with p-type derivative. The corresponding solution is shown and numerical example is given.