This paper discusses further the roughness of Riemann-Liouville fractional integral on an arbitrary fractal continuous functions that follows Rfs. [1]. A novel method is used to reach a similar result for an arbitrary...This paper discusses further the roughness of Riemann-Liouville fractional integral on an arbitrary fractal continuous functions that follows Rfs. [1]. A novel method is used to reach a similar result for an arbitrary fractal function , where is the Riemann-Liouville fractional integral. Furthermore, a general resultis arrived at for 1-dimensional fractal functions such as with unbounded variation and(or) infinite lengths, which can infer all previous studies such as [2] [3]. This paper’s estimation reveals that the fractional integral does not increase the fractal dimension of f(x), i.e. fractional integration does not increase at least the fractal roughness. And the result has partly answered the fractal calculus conjecture and completely answered this conjecture for all 1-dimensional fractal function (Xiao has not answered). It is significant with a comparison to the past researches that the box dimension connection between a fractal function and its Riemann-Liouville integral has been carried out only for Weierstrass type and Besicovitch type functions, and at most Hlder continuous. Here the proof technique for Riemann-Liouville fractional integral is possibly of methodology to other fractional integrals.展开更多
In this paper, the relationship between Riemann-Liouville fractional integral and the box-counting dimension of graphs of fractal functions is discussed.
A one-dimensional continuous function of unbounded variation on [0, 1] has been con- structed. The length of its graph is infinite, while part＂ of this function displays fractal features. The Box dimension of its Rie...A one-dimensional continuous function of unbounded variation on [0, 1] has been con- structed. The length of its graph is infinite, while part＂ of this function displays fractal features. The Box dimension of its Riemann-Liouville fractional integral has been calculated.展开更多
文摘This paper discusses further the roughness of Riemann-Liouville fractional integral on an arbitrary fractal continuous functions that follows Rfs. [1]. A novel method is used to reach a similar result for an arbitrary fractal function , where is the Riemann-Liouville fractional integral. Furthermore, a general resultis arrived at for 1-dimensional fractal functions such as with unbounded variation and(or) infinite lengths, which can infer all previous studies such as [2] [3]. This paper’s estimation reveals that the fractional integral does not increase the fractal dimension of f(x), i.e. fractional integration does not increase at least the fractal roughness. And the result has partly answered the fractal calculus conjecture and completely answered this conjecture for all 1-dimensional fractal function (Xiao has not answered). It is significant with a comparison to the past researches that the box dimension connection between a fractal function and its Riemann-Liouville integral has been carried out only for Weierstrass type and Besicovitch type functions, and at most Hlder continuous. Here the proof technique for Riemann-Liouville fractional integral is possibly of methodology to other fractional integrals.
文摘In this paper, the relationship between Riemann-Liouville fractional integral and the box-counting dimension of graphs of fractal functions is discussed.
基金Supported by National Natural Science Foundation of China(Grant No.11201230)Natural Science Foundation of Jiangsu Province(Grant No.BK2012398)
文摘A one-dimensional continuous function of unbounded variation on [0, 1] has been con- structed. The length of its graph is infinite, while part＂ of this function displays fractal features. The Box dimension of its Riemann-Liouville fractional integral has been calculated.
