The Fractional Calculus Of A Type Of Function With Unbounded Variation And The Fractal Dimension

The exploration of fractal theory has been attracted widely attention in recent years.The main work of fractal research lies in the estimation of fractal dimension.The estimation of fractal dimension of various fractal functions and the estimation of fractal dimension of fractional calculus are systematically studied.In this paper,we mainly discuss unbounded variation functions and get some basic properties.Furthermore,we go deep into them.The properties of unbounded variation functions have been discussed in a more general case.The fractal dimension of these unbounded variation functions and the fractal dimension of fractional integral have been studied.In this paper,four fractal dimensions and two fractional calculus have been de-fined.Secondly,on the basis of the definition of bounded variation,the definition of unbounded variation has been investigated through the dichotomy.Secondly,the correlative properties of continuous functions with bounded varia-tion are discussed.It is proved that the Hadamard fractional integral of continuous functions with bounded variation is still bounded variation.Accordingly,The fractal dimensions of the Hadamard fractional integral of these functions have been given.Then,two 1-dimensional continuous functions have been constructed,which con-tain only one unbounded variation point,and the connection and difference between them are described.Based on the structure,some relative properties of these functions are analyzed,and the fractal dimensions of this function have been given.Furthermore,1-dimensional continuous functions with countable and uncountable unbounded variation points have been constructed.The Box dimension of the continu-ous function with uncountable unbounded variation points has been studied.And the relative theorems and inferences have also been given.Finally,the conclusion of our work is drawn and the outlook for future work is presented.