Sub-shape Function And Fractional Calculus: The Application Of The Constructive Method

As a new branch in mathematics, The subject of fractals and their applications have attracted many mathematicians and scientists's more and more attentions in recent years. Futhermore, fractal analysis has achieved some important development.Several fractal functions, such as the Weierstrass functions, the Besicovitch functions, the Rademacher functions, the Takagi functions, etc, and their graph dimensions and fractal properties are investigated extensively due to the special fractal structures.Fractional calculus, as an important tool of investigating fractal sets, also attracts much interest recently. The combination of fractional calculus with fractal analysis, in other words, appling fractional calculus to analyze fractal structures of graphs of some special functions, is a crucial problem in this direction.The present paper mainly investigates the relationship between the fractional calculus and the fractal functions. First, we show the existence of the fractional calculus of some fractal functions, thus, we define the fractional integrals and derivatives of some fractal functions. Secondly, we sketch graphes of the fractal functions, their fractional integrals and derivatives and make some comparisions of these functions. Thirdly, by using some constructive means, we establish the exact fractal dimensions of fractional integrals and derivatives of graphs of some important fractal functions. This paper contains the following results(i) The Riemann-Liouville operators are defined as follow:For a fractal functionthe fractional integral of order v and the fractional derivative of order of W(t) are defined respectively byWe first have the following initial result.fractional integral defined above. ThenWhere F(f, I) denotes the graph of the function f on the interval I. In Chapter 3 we finally establish the following theorem:theorem 0.1. Thenholds for sufficiently large > 1.(ii) Consider the following Weiertrass functionthe fractional integral of order v and fractional derivatives of W(t) of order are defined respectively byandFor the K-dimension of l(t) and m(t), we haveTheorem 0.3 Let 0 < a + v < 1, I = [0,1], l(t) be defined above. ThenFurthermore, in Chapter 4, by a series of calculations, we prove that(iii) Consider the following Besicovitch functionthe fractional integrals and derivatives of B(t) can be denned asrespectively. We haveTheorem 0.5 Let b(t) satiesfy 2, 0 < v < l.Then(iv) For the fractal functions defined in (ii), consider the Weyl-Marchaud fractional derivatives of W(t) of order v. ThenTheorem 0.6 Let 0 < v < a < 1, then for > 0, we have dimH (p, I) dimK (p, I) = dimP (p, I) = dimB (p, I) = 2 - + v. Finally, we remark some recently developments in this direction. Some problems for further investigation are also listed in Chapter 5.