The Fractional Calculus Of A Type Of Fractal Function And The Dimension

In recent years, the subject of fractal function attracted manymathematicians and scientists' more and more attention. People discussed thefunction of Weierstrass as a typical function and showed this kind of function'dimension and its fractional calculus functions' dimensions. The paper discussessome definitions of fractional calculus and gets some properties about them.Then we study a kind of more general fractal function and show the function'sfractional calculus functions and their dimensions. The main research work isdescribed in detail in the following:First, we define the fractional integral and fractional derivative of fractalfunction. And we study some properties as to the definition of fractionalcalculus.Secondly, we discuss a kind of fractal function's fractional calculusfunctions and their K-dimensions.Thirdly, we estimate the box dimensions of the kind of fractal function'sfractional calculus functions. The main conclusions as follow:For a fractal function W_θ(t)=sum from n=1 to∞(aⁿcos(bⁿx+θ_n)) (0＜a＜1,b＞1)the fractional integral of order v and the fractional derivative of orderμofW_θ(t) are defined respectively by I_θ(t,v):=D^-v(W_θ(t))=sum from n=1 to∞(aⁿ(cosθ_nC_t(v,bⁿ)-sinθ_nS_t(v,bⁿ))) (0＜ab^-v＜1) g_θ(t,μ):=D^μ(W_θ(t))=sum from n=1 to∞(aⁿcosθ_n/Γ(1-μ)t^-μ-(ab)ⁿconθ_nS_t(1-μ,bⁿ)-(ab)ⁿsinθ_nC_t(1-μ,bⁿ)))(b＞1, 0＜a＜1, 0＜ab^μ＜1)For K-dimensions of the graphs of the functions above, we get theconclusion in the following:Theorem 0.1 Let 0＜a＜1, b＞1, 0＜ab^-v＜1, 0＜ab^μ＜1, I_θ(t, v), g_θ(t,μ) be thefractional integral function and fractional derivative function definedabove.Thenholds for sufficiently small a＜1, sufficiently large b＞1.WhereΓ(f,[a,b]) denotes the graph of the function f on the interval [a,b],δis the positive constant that is arbitrary small.For box dimensions of the graphs of the functions above, we get theconclusion as follow: Theorem 0.2 Let 0＜a＜1,b＞1,0＜ab^-v＜1,0＜ab^μ＜1,I_θ(t,v),g_θ(t,μ) bethe fractional integral function and fractional derivative function definedabove.ThenWhereΓ(f,[a,b]) denotes the graph of the function f on the interval [a,b].δis the positive constant that is arbitrary small.