Analytical properties of bivariate fractal interpolation functions with vertical scaling factor functions

Ji, Jiabing; Peng, Jigen

doi:10.1080/00207160.2012.722624

Cited by 20 publications

(4 citation statements)

References 22 publications

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“…The transformations like rotations and reflections have been used in [ 23 ] to create continuous fractal surfaces and volumes over rectangular lattices. The stability of the fractal interpolation functions have been studied in [ 14 ]. Instead of using Banach contraction mapping theorem, [ 24 ] employs Rakotch fixed point theorem and Matkowski fixed point theorem for the construction.…”

Section: Introductionmentioning

confidence: 99%

Numerical integration of bivariate fractal interpolation functions on rectangular domains

Aparna

Paramanathan

2023

Eur. Phys. J. Spec. Top.

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This paper primarily focuses on the derivation of fractal numerical integration for the data sets corresponding to two variable signals defined over a rectangular region. Evaluating numerical integration results through the fractal method helps achieve accurate results with minimum computation effort. The formulation of the fractal numerical integration is achieved by considering the recursive relation satisfied by the bivariate fractal interpolation functions for the given data set. The points in the data set have been used to evaluate the coefficients of the iterated function systems. The derivation of these coefficients considering the index of the subrectangles, and the integration formula has been proposed using these coefficients. The bivariate fractal interpolation functions constructed using these coefficients are then correlated with the bilinear interpolation functions. Also, this paper derives a formula for the freely chosen vertical scaling factor that has been used in reducing the approximation error. The obtained formula of the vertical scaling factor is then used in establishing the convergence of the proposed method of integration to the traditional double integration technique through a collection of lemmas and theorems. Finally, the paper concludes with an illustration of the proposed method of integration and the analysis of the numerical integral results obtained for the data sets corresponding to four benchmark functions.

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Section: Introductionmentioning

confidence: 99%

Numerical integration of bivariate fractal interpolation functions on rectangular domains

Aparna

Paramanathan

2023

Eur. Phys. J. Spec. Top.

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“…In 1986, M. F. Barnsley [1] introduced a concept of FIF to model better natural phenomena which are irregular and complicated and the FIFs have been widely studied ever since in many papers. [2][3][4][5][6][7][8][9][10][11][12][13][14][15][17][18][19][20][21][22] In general, to get the FIF, we construct an iterated function system (IFS) on the basis of a given data set and then define a Read-Bajraktarevic operator on some space of continuous functions. A fixed point of the operator is an interpolation function of the data set and its graph is an attractor of the constructed IFS.…”

Section: Introductionmentioning

confidence: 99%

Box-counting dimension and analytic properties of hidden variable fractal interpolation functions with function contractivity factors

Yun¹,

Ri²

2020

Chaos, Solitons & Fractals

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We estimate the bounds of box-counting dimension of hidden variable fractal interpolation functions (HVFIFs) and hidden variable bivariate fractal interpolation functions (HVBFIFs) with four function contractivity factors and present analytic properties of HVFIFs which are constructed to ensure more flexibility and diversity in modeling natural phenomena. Firstly, we construct the HVFIFs and analyze their smoothness and stability. Secondly, we obtain the lower and upper bounds of box-counting dimension of the HVFIFs. Finally, in the similar way, we get the lower and upper bounds of box-counting dimension of HVBFIFs constructed in [21].

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“…Construction, derivative, integral, dimension, smoothness and stability of the FIFs have been widely studied. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] Since the vertical scaling factors, which the contraction transformations of IFS have, determine the charac teristics of FIFs, they are very important. To obtain FIFs with high flexibility, construction of FIFs with fun ction vertical scaling factors and their analytic properties have been studied in many papers.…”

Section: Introductionmentioning

confidence: 99%

Hidden Variable Recurrent Fractal Interpolation Functions With Function Contractivity Factors

Yun¹

2019

Fractals

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In this paper, we introduce a construction of hidden variable recurrent fractal interpolation functions (HVRFIF) with four function contractivity factors. In the fractal interpolation theory, it is very import ant to ensure flexibility and diversity of the construction of interpolation function. Recurrent iterated fu nction system (RIFS) produce fractal sets with local self-similarity structure. Therefore the RIFS can d escribe the irregular and complicated objects in nature better than the iterated function system (IFS). Hidden variable fractal interpolation function (HVFIF) is neither self-similar nor self-affine one. The HVFIF is more complicated, diverse and irregular than the fractal interpolation function (FIF). The co ntractivity factor is important one that determins characteristics of FIFs. We present a constructions of one variable HVRFIFs and bivariable HVRFIFs using RIFS with four function contractivity factors.

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Analytical properties of bivariate fractal interpolation functions with vertical scaling factor functions

Abstract: Based on the construction of bivariate fractal interpolation functions (FIFs), a class of FIFs with vertical scaling factor functions are presented and the analytical properties of smoothness and stability are proved.

Cited by 20 publications

References 22 publications

Numerical integration of bivariate fractal interpolation functions on rectangular domains

Numerical integration of bivariate fractal interpolation functions on rectangular domains

Box-counting dimension and analytic properties of hidden variable fractal interpolation functions with function contractivity factors

Hidden Variable Recurrent Fractal Interpolation Functions With Function Contractivity Factors

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