Researching On Fractional Calculus And Fractal Interpolation Function

Fractional calculus is related to fractal, in particular is closely linked with the fractal function,which is a powerful tool to study the fractal function. Fractal interpolation function (FIF for short), generated by the a special class of iterated function systems, is the fractal function. It is of great significance to use the fractional calculus to study it. For Riemann-Liouville fractional calculus and fractal interpolation function, this paper does the research as follows:First of all, the paper represents the development and the current status of the research on fractal, fractal dimension, fractional calculus and fractal interpolation, introduces the basic concepts of the Box dimension, Riemann- Liouville fractional calculus and basic theory of fractal interpolation function.Secondly, the paper introduces the concept ofαorder continuous function, studies the continuity and H?lder continuity of fractional integral, gives the norm of fractional integral operator and discusses its compactness.Thirdly, based on FIF's iterated function systems, this paper gives iterated function systems for generating FIF's fractional calculus , and then discusses the H?lder continuity of FIF's fractional integral and gives one sufficient and necessary condition for existence and continutity of FIF's fractional differential. Next, this paper gives dimension formulas of FIF function graph and FIF's fractional calculus functions graph, and relationship of their dimensions. The results show that fractional integration can reduce the dimension of FIF's graph and fractional differential can increase the dimension of FIF's graph.