The Research On Fractal Interpolation Surfaces And Fractional Calculus Of Fractal Interpolate Functions

The fractal interpolation surface(FIS)is a fractal interpolation function(FIF)which is generated by iterated function system(IFS)or recurrent iterated function system(RIFS).A lot of papers give the construction of FIS and study its dimension,smoothness and so on and obtain many results.In this article,we discuss the FIS on the set of data {(i/n,j/n.xi,j);i,j= 0,1,…,N}.For the bivariate fractal inter-polation surface(BFIS),which is the attractor of the IFS([0,1]2×,?i,j),where?i,j =(Li,j,Fi,j),and Li,j:[0,1]2?[i-1/n,i/n]×[j-1/n,j/n]and Fi,j is defined by Fi,j(x,y,z)= ai,jx+ bi,jy + ci,jxy + dz + fi,j.The Minkowski dimension of the BFIS is given by the formula dim M Gr f = 3 +lgg|d|/log n in paper[1].We improve this method by using the cover of ? columns with scaling appropriately and give a more accurate formula of box dimension of FIS.And the properties of the fractional calculus of fractal interpolation function is also studied and some results are obtained.This article is divided into five chapters and the structure is as follows:Chapter 1 Introduction.We introduce the background and current situation of FIF and FIF fractional calculus.Chapter 2 Preparatory knowledge where gives the relevant preparatory knowledge and concepts of related FIS and fractional calculus according to the problems that we studied.Chapter 3 considers the box dimension of fractal interpolation surface.Taking the dimension of FIF's image as the research foundation,the box dimension of FIS is further discussed and the main results are obtained:Theorem 3.3 Let G is the attractor of IFS(2.3)with interpolation points xi,yj are distributed on I2 uniformaly,that is for all i,j ? {0,1,2,…,N},xi=i/N,yj=j/N.If N for all p ? {0,1,2.,… N},v=(?)|sip|>1 and {(xi,yp,zip)|i = 0,1,2,…,N} are non-collinear,and for all q?{0,1,2,…,N},v=(?)|sqj|>1 and {(xq,yj,zqj)|j=0,1,2,…,N}are non-collinear,then dimB ?(G)= max{2 + log v/N,2 + log v/N}.Otherwise dimB?(G)= 2.We give a better method of estimating the box dimension of FIS,and seek a method of calculation which is suitable for estimating the dimension of fractal inter-polation function surfaces.Chapter 4 mainly considers FIF and explores the fractional calculus of it and gets some results:Theorem 4.2 Let f(x)is the FIF defined by(4.1)and we have that f(x)is the FIF defined by {(Li(x),Fi(x,y))}Ni=1.And for i=1,2,…,N,we get that Fi,v(x,y)= aivciy +qi,v(x),Chapter 5 is the summary and prospect.Summarizing up the work of this article,we get the formula to estimate the dimension of FIS.We also get two important conclusions in the fractional calculus of FIF.In addition,we put forward some questions that needed further exploration.