Construction of cubic spline hidden variable recurrent fractal interpolation function and its fractional calculus

2021 ◽  
Vol 150 ◽  
pp. 111177
Author(s):  
Mi-Gyong Ri ◽  
Chol-Hui Yun ◽  
Myong-Hun Kim
Keyword(s):  
Fractional Calculus ◽  
Cubic Spline ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Hidden Variable ◽  
Fractal Interpolation Function

FRACTIONAL CALCULUS OF COALESCENCE HIDDEN-VARIABLE FRACTAL INTERPOLATION FUNCTIONS

Fractals ◽  
2017 ◽  
Vol 25 (02) ◽  
pp. 1750019 ◽  
Author(s):  
SRIJANANI ANURAG PRASAD
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Integral ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Hidden Variable ◽  
Fractal Interpolation Function ◽  
Fractal Interpolation Functions ◽  
Free Parameters ◽  
Interpolation Functions

Riemann–Liouville fractional calculus of Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is studied in this paper. It is shown in this paper that fractional integral of order [Formula: see text] of a CHFIF defined on any interval [Formula: see text] is also a CHFIF albeit passing through different interpolation points. Further, conditions for fractional derivative of order [Formula: see text] of a CHFIF is derived in this paper. It is shown that under these conditions on free parameters, fractional derivative of order [Formula: see text] of a CHFIF defined on any interval [Formula: see text] is also a CHFIF.

SOME RESULTS OF CONVERGENCE OF CUBIC SPLINE FRACTAL INTERPOLATION FUNCTIONS

Fractals ◽  
2003 ◽  
Vol 11 (01) ◽  
pp. 1-7 ◽  
Author(s):  
M. ANTONIA NAVASCUÉS ◽  
M. VICTORIA SEBASTIÁN
Keyword(s):  
Experimental Data ◽  
Cubic Spline ◽  
Spline Functions ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Original Function ◽  
New Methods ◽  
Fractal Interpolation Function ◽  
Fractal Interpolation Functions ◽  
Interpolation Functions

Fractal interpolation functions (FIFs) provide new methods of approximation of experimental data. In the present paper, a fractal technique generalizing cubic spline functions is proposed. A FIF f is defined as the fixed point of a map between spaces of functions. The properties of this correspondence allow to deduce some inequalities that express the sensitivity of these functions and their derivatives to those changes in the parameters defining them. Under some hypotheses on the original function, bounds of the interpolation error for f, f′ and f′′ are obtained. As a consequence, the uniform convergence to the original function and its derivative as the interpolation step tends to zero is proved. According to these results, it is possible to approximate, with arbitrary accuracy, a smooth function and its derivatives by using a cubic spline fractal interpolation function (SFIF).

Approximation using hidden variable fractal interpolation function

2015 ◽  
Vol 2 (1) ◽  
pp. 81-114 ◽  
Author(s):  
Arya Chand ◽  
Saurabh Katiyar ◽  
Puthan Viswanathan
Keyword(s):  
Fractal Interpolation ◽  
Interpolation Function ◽  
Hidden Variable ◽  
Fractal Interpolation Function

scholarly journals CUBIC SPLINE SUPER FRACTAL INTERPOLATION FUNCTIONS

Fractals ◽  
2014 ◽  
Vol 22 (01n02) ◽  
pp. 1450005 ◽  
Author(s):  
G. P. KAPOOR ◽  
SRIJANANI ANURAG PRASAD
Keyword(s):  
Generating Function ◽  
Cubic Spline ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Approximation Properties ◽  
Fractal Interpolation Function ◽  
Convergence Results ◽  
Fractal Interpolation Functions ◽  
Interpolation Functions

In the present work, the notion of Cubic Spline Super Fractal Interpolation Function (SFIF) is introduced to simulate an object that depicts one structure embedded into another and its approximation properties are investigated. It is shown that, for an equidistant partition points of [x0, xN], the interpolating Cubic Spline SFIF[Formula: see text] and their derivatives [Formula: see text] converge respectively to the data generating function y(x) ≡ y(0)(x) and its derivatives y(j)(x) at the rate of h2-j+ϵ(0 < ϵ < 1), j = 0, 1, 2, as the norm h of the partition of [x0, xN] approaches zero. The convergence results for Cubic Spline SFIF found here show that any desired accuracy can be achieved in the approximation of a regular data generating function and its derivatives by a Cubic Spline SFIF and its corresponding derivatives.

HIDDEN VARIABLE RECURRENT FRACTAL INTERPOLATION FUNCTIONS WITH FUNCTION CONTRACTIVITY FACTORS

Fractals ◽  
2019 ◽  
Vol 27 (07) ◽  
pp. 1950113
Author(s):  
CHOL-HUI YUN
Keyword(s):  
Iterated Function System ◽  
Function System ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Hidden Variable ◽  
Fractal Interpolation Function ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Better Than ◽  
Interpolation Functions

In this paper, we introduce a construction of hidden variable recurrent fractal interpolation functions (HVRFIFs) with four function contractivity factors. The HVRFIF is a hidden variable fractal interpolation function (HVFIF) constructed using a recurrent iterated function system (RIFS). In the fractal interpolation theory, it is very important to ensure flexibility and diversity of the construction of interpolation functions. RIFSs produce fractal sets with local self-similarity structure. Therefore, the RIFS can describe the irregular and complicated objects in nature better than the iterated function system (IFS). The HVFIF is neither self-similar nor self-affine one. Hence, the HVFIF is more complicated, diverse and irregular than the fractal interpolation function (FIF). The contractivity factors of IFS are very important one that determines characteristics of FIFs. The IFS and RIFS with function contractivity factors can describe the fractal objects in nature better than one with constant contractivity factors. To ensure higher flexibility and diversity of the construction of the FIFs, we present constructions of one variable HVRFIFs and bivariable HVRFIFs using RIFS with four function contractivity factors.

scholarly journals Fractional Calculus of Fractal Interpolation Function on[0,b](b>0)

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
XueZai Pan
Keyword(s):  
Continuous Function ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Iterated Function System ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Fractional Order Calculus ◽  
Fractal Interpolation Function ◽  
Iterated Function ◽  
Fractional Order Integral

The paper researches the continuity of fractal interpolation function’s fractional order integral on[0,+∞)and judges whether fractional order integral of fractal interpolation function is still a fractal interpolation function on[0,b](b>0)or not. Relevant theorems of iterated function system and Riemann-Liouville fractional order calculus are used to prove the above researched content. The conclusion indicates that fractional order integral of fractal interpolation function is a continuous function on[0,+∞)and fractional order integral of fractal interpolation is still a fractal interpolation function on the interval[0,b].

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