The Fractional Calculus Of A Class Of Weierstrass-Type Function And The Fractal Dimensions Of Their Graphs

In recent years the Weierstrass-type function as the classical fractal function has attracted people's more and more attention,and the development of fractional calculus for the study of Weierstrass-type function has injected new vitality. Based on the Riemann-Liouville fractional calculus this article will discuss a class of more general fractal function. The main research works described in this article are as follows:First of all, this paper represents the current status of the research on fractal and fractal dimension, gives the concepts of several fractal dimensions that will be used in this article, and introduces the mathematical basis on the Riemann-Liouville fractional calculus.Secondly, on conditional that the basic functionφt kis theβ-order H?lder continuous periodic function, the continuity and nowhere differentiability of a class of generalized Weierstrass-type function are discussed, by the estimation for the bounds of dimension get the box dimension of its graph. In the sense of Riemann-Liouville fractional calculus, derive the form of this function's fractional differential and integral functions, and discuss their continuity, give the K-dimension and box dimension of their graphs.Thirdly, the basic function of the classic Weierstrass-type function is expanded into Lipschitz function, in the case ofα≥1, discuss the function and its Riemann-Liouville fractional differential and integral functions, derive the Hausdorff dimension, K-dimension and box dimension of the three graphs, by the continuity of the differential function obtain the upper and lower bounds estimation. Meanwhile, by using the Matlab, the graphs of this function and its fractional differential function with differentαvalues are drawn to make the results more intuitive.