On Approximation Properties of Fractional Integral for A-Fractal Function

Priyanka, T. M. C.; Valarmathi, R.; Bingi, Kishore; Gowrisankar, A.

doi:https://doi.org/10.1155/2022/6409656

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Volume 2022 | Article ID 6409656 | https://doi.org/10.1155/2022/6409656

On Approximation Properties of Fractional Integral for A-Fractal Function

T. M. C. Priyanka,¹R. Valarmathi,¹Kishore Bingi,²and A. Gowrisankar¹

Academic Editor: Waleed Adel

Received23 Jun 2022

Revised17 Jul 2022

Accepted28 Jul 2022

Published24 Aug 2022

Abstract

In this paper, the Riemann–Liouville fractional integral of an A-fractal function is explored by taking its vertical scaling factors in the block matrix as continuous functions from to . As the scaling factors play a significant role in the generation of fractal functions, the necessary condition for the scaling factors in the block matrix is outlined for the newly obtained function. The resultant function of the fractional integral is demonstrated as an A-fractal function if the scaling factors obey the necessary conditions. Furthermore, this article proposes a fractional operator which defines the Riemann–Liouville fractional integral of an A-fractal function for each continuous function on , where is the space of all continuous functions from closed interval to . In addition, the approximation properties such as linearity, boundedness, and semigroup property of the proposed fractional operator are investigated.

1. Introduction

Simple figures such as circle, triangle, rectangle, and square can be easily described in Euclidean geometry. However, naturally existing intricate geometric shapes such as tree branches, snow flake, Romanesco broccoli, and pine cones could not be described using Euclidean geometry. So to address this issue, Barnsley introduced the “fractal geometry.” Fractal objects are statistically the same regardless of the scale at which they are seen. The complexity of the objects can be measured using fractal dimension which is not required to being an integer, and it is congruent with the idea of spatial dimension. Fractal geometry offers a powerful tool in approximation theory, called the fractal interpolation, which is introduced by Barnsley using iterated function system as a class of functions to interpolate the given data set [1]. A unique attractor of a certain, special type of iterated function system produces the graph of the fractal interpolation function. Initially, Barnsley worked on affine fractal interpolation functions using affine transformations. Later on, more general transformations apart from affine ones have been explored to construct quadratic fractal interpolation function, -fractal interpolation function, hidden variable fractal interpolation function, and so on, for more details see [2–15]. Beyond the approximation theory, fractals are also compiled with various fields such as graph theory [16, 17] and fractional calculus [18, 19]. One of the trending applications of fractals is the fractal robotics, and it has the potential to revolutionize the technologies in an entirely different way which has never been witnessed earlier. The usage of fractal robotics is not limited to the defence technology, medical application, and space application but also includes several domains of science, see for more details [20].

Recently, a novel fractal function known as hidden variable A-fractal function has been introduced by blending -fractal function and hidden variable fractal interpolation function in [21]. The hidden variable fractal interpolation functions are developed with an aim to extend the class of one-dimensional fractal functions to higher dimensions by introducing the “hidden variables.” The class of functions generated is more diverse since their values continuously influence on all of the hidden variables, namely, the coefficients of the possibly high-dimensional affine maps that determine the function. Chand and Kapoor in [22] have presented the coalescence hidden variable fractal interpolation function to approximate both self-affine and nonself-affine functions. The -fractal functions are constructed to approximate the given one-dimensional continuous function. The difference between the -fractal function and the A-fractal function is that the former depends on the scaling factor and approximates the given one-dimensional continuous function whereas the later depends on the block matrix containing both free variables and hidden variables, and it approximates the given two-dimensional continuous function by generating a family of -valued continuous functions. Therefore, to approximate the given valued continuous function , the hidden variable A-fractal function is used, and for the proper choice of parameters in the block matrix , the fractal function preserves certain characteristics such as regularity and monotonicity of the original function . It is a better fractal interpolation function for getting more flexible and appealing curves which may be self-referential or nonself-referential. To this end, the following questions have mainly motivated us to investigate the fractional integral of an A-fractal function. Question 1: Can the fractional integral be implemented on the A-fractal function? If so, what is the resultant function? A fractal operator based on the A-fractal function is investigated in [15]. A natural question that arises from [15] is as follows. Question 2: Is it possible to propose a new operator based on the Riemann–Liouville fractional integral of an A-fractal function?

In literature, many authors have applied various kinds of fractional calculus to different types of fractal functions and have presented novel results. In [23], the box dimension of Weyl–Marchaud fractional derivative of linear fractal interpolation functions is explored. In [18], the authors have demonstrated that there is some linear connection between the order of the fractional derivatives and the dimension of the graph of Weierstrass function. Liang and Zhang have explored the relationship between the order of the Riemann–Liouville fractional calculus and the box dimension of the fractal interpolation function in [19]. Different fractal interpolation functions (FIFs), such as linear FIF, quadratic FIF, hidden variable FIF, and -FIF, and their Weyl–Marchaud fractional derivatives are examined in [24]. In [25], a new fractal-fractional (FF) operational matrix for orthonormal normalized ultraspherical polynomials has been examined. In [26], the rogue wave solutions and the dark wave solutions of Ivancevic option pricing model are constructed. The rogue wave solutions of the Ivancevic option pricing model are obtained via trial function method, and the dark wave solutions of Ivancevic option pricing model are obtained via tanh method. The fourth-order nonlinear Boussinesq water wave equation, which illustrates the spread of long waves in shallow water, is explored in [27]. In [28], the Riemann–Liouville fractional integral for the -fractal function is examined by choosing the vertical scaling factors as variables along with the investigation of the boundedness and linearity of the fractional operator of the -fractal function. For more information, the reader is recommended to see [29–33]. Hence, the fractional integral can be evaluated over the A-fractal function by choosing the scaling factors as continuous functions. The main contribution of the present study is to investigate the Riemann–Liouville fractional integral (simply denoted as R-L fractional integral) of an A-fractal function. Furthermore, it is showed that the R-L fractional integral of an A-fractal function with variable scaling factors is the generalization of the R-L fractional integral of an A-fractal function with constant scaling factors. In order to answer question 2, a fractional operator has been proposed based on the R-L fractional integral of an A-fractal function. Besides, some of its approximation properties are explored.

This paper encompasses the following. Section 2 summarizes the basic definitions required for the paper. Section 3 is devoted to investigate the R-L fractional integral of an A-fractal function with variable scaling factors. In Section 4, a fractional operator is proposed based on the R-L fractional integral of an A-fractal function, and its analytical properties such as linearity and boundedness and its algebraic property such as semigroup property have been investigated. The final remarks of this paper are provided in Section 5.

2. Preliminaries

The generation of affine fractal interpolation function, -fractal function, hidden variable fractal interpolation function, and the hidden variable A-fractal function is described in this section. In addition, the R-L fractional integral of a continuous function is defined in brief.

Consider the set of interpolation datasuch that determines a partition of the closed interval . In order to find the continuous function satisfying , an iterated function system is constructed by considering the following special type of affine transformations:with the constraints

The parameters in the system are all reals, and is the free parameter, called the vertical scaling factor such that .

Now, using the above affine maps, the affine fractal interpolation function is constructed as follows: for the given data set (1), the affine maps are defined by where are the N bicontinuous maps defined by satisfying the Lipschitz condition of order 1. The continuous maps are defined by , and for , are contractions in the second argument,where . The systemdetermines an iterated function system (IFS). Let be the metric space of all nonempty compact subsets of with respect to Hausdorff metric. The Hutchinson map is defined by , for any . Banach fixed point theorem guarantees the existence of the unique invariant set which is called the attractor of the IFS. Let and be the sup norm on . Now, an operator is defined on bythe fixed point of the operator, say satisfying the equationis called the affine fractal interpolation function (AFIF) associated with the IFS (5).

2.1. Hidden Variable Fractal Interpolation Function

Generalize the data set (1) as by including a set of real parameters (see [9]). The hidden variable FIF is constructed by defining the IFS aswhere

Here, and are continuous functions, the parameters and are the free parameters, and is the constrained parameter which are chosen such that , and . The functions and satisfy the following join-up conditions:

By the IFS theory, (8) has a unique invariant set , called the attractor of the IFS (8) and which is the graph of the function satisfying , for all . The first component of the -valued function is called the hidden variable fractal interpolation function which is not self-affine, and the second component is called the fractal function which is self-affine.

For the sample set of interpolation data , choose the scaling parameters , for . Figure 1 represents the hidden variable fractal interpolation function , and Figure 2 illustrates the graph of the fractal function .

2.2. -Fractal Function

For approximating the given continuous function , its corresponding -fractal function is defined as follows. When the continuous functions are chosen aswhere is a continuous function such that and , , and is the scale vector. Then, the function satisfying the following functional equation,is called the -fractal function, simply known as -FIF. For more details of -fractal function, the interested reader can refer to [10, 11, 14].

2.3. Hidden Variable A-Fractal Function

This subsection presents the construction of an A-fractal function corresponding to the continuous -valued function [15, 21]. A continuous function is uttered in order to elicit a whole family of fractal functions parametrized by a specific block matrix with , take a note of the fact when . Consider the partition of and the data set . In the IFS defined in (8), consider the special case,where fulfills and . In this approach, the fixed point of the IFS shown in (8) is the graph of the continuous vector-valued function that obeys the following functional equation:

This function is known as the (hidden variable) A-fractal function of with respect to the partition and the base function . Then, the components of fulfill

The functions are nonself-referential and self-referential functions, respectively. For any choice of the block matrix and for any choice of the base function satisfying the above-mentioned conditions, it is observed that . Hence, the function can be called as the fractal generalization of the continuous function .

When the parameters in the block matrix are taken as continuous functions from to , then the (15) becomes

Definition 1. (see[34]). The R-L fractional integral operator of order with for a continuous function on is defined by the following expression:where . Consider the hidden variable A-fractal function corresponding to the data set . The following is valid for any Equations (18) and (19) are called the R-L fractional integral of and defined at the initial point with .
The R-L fractional integral of and defined at the end point , with , is given as follows:

3. Fractional Calculus on A-Fractal Function

In this section, by choosing the scaling factors as variables, the R-L fractional integral of an A-fractal function is demonstrated as an A-fractal function under certain conditions.

In the following theorem, the scaling factors are chosen as continuous functions for the A-fractal function, and its R-L fractional integral is examined with the presumed initial conditions .

Theorem 1. Let be the A-fractal function with variable scaling factors corresponding to the interpolation points . If , then and generates , wherewith , , and , for ,

Proof. The R-L fractional integral of is given byHere, take and . By using the variable transformation in the third term and the functional (16), the following equation is generatedNow, using the Leibniz rule for fractional integral, the following is obtained:Denote . Now, consider the R-L fractional integral of of order ,Here, take and . By using variable transformation in the third term, one can obtainApplying the Leibniz rule for fractional integral,Denote .
As a result, an A-fractal function’s R-L fractional integral is also an A-fractal function. Take and in the following equation:Hence,The system of equations yieldsBy substituting , the previous equation provides as follows:Take and in the below equation to determine the new data set .Therefore,The equation yieldsBy substituting , the previous equation provides the end point as follows:It is important to check the join-up conditions of . First,when is compared with (31), one can obtainIn the same way, the endpoint conditions of can be verified.
Hence, Theorem 1 has demonstrated that the R-L fractional integral of an A-fractal function with the predetermined initial conditions and is also an A-fractal function for the new data set .

Remark 1. Theorem 1 generalizes the R-L fractional integral of an A-fractal function with constant scaling factors when its integral is defined at the initial point. That is, suppose the vertical scaling factors in Theorem 1 are taken as constants and if where with , , and , , then the following result is obtained for ,The following theorem explores the R-L fractional integral of an A-fractal function with the prescribed condition and .

Theorem 2. Let be the A-fractal function with variable scaling factors corresponding to the interpolation points . If , then and generate where with , with , and , for

Proof. From (18), consider the Riemann–Liouville fractional integral of ,Here, take and . By using variable transformation in the third term as , the above equation is modified asUsing the functional (16), the following equation is generated:Now, using the Leibniz rule for fractional integralHere, denote . The R-L fractional integral of of order is given byHere, take and . By using variable transformation in the third term as , the above equation is modified asUsing the functional (16), the following equation is generated:Now, using the Leibniz rule for fractional integral,Here, denote . As a result, an A-fractal function’s R-L fractional integral is also an A-fractal function. Put and in the following equation:Therefore,The equation yieldsBy substituting , the previous equation provides the initial point as follows:take and in the following equation to determine the new interpolation data .Thus,The equation yieldsBy substituting , the previous equation provides the initial point as follows:It is important to check the join-up conditions of . First,When we compare with (51), we can obtainSimilarly, the endpoint conditions of can be verified.
Theorem 2 clearly shows that the function fulfills the endpoint conditions and . Hence, the R-L fractional integral of an A-fractal function with the predetermined conditions and is also an A-fractal function corresponding to the new data set .

Remark 2. Theorem 2 generalizes the R-L fractional integral of an A-fractal function with constant scaling factors when its integral is defined at the terminal point. That is, suppose the vertical scaling factors in Theorem 2 are taken as constants and if where with , , and , , then the following result is obtained for :

4. Formulation of Fractional Operator

A fractional operator for the A-fractal function on the space of valued continuous functions is formulated in this section. In addition, some properties such as linearity, boundedness, and semigroup property of the fractional operator are discussed.

Definition 2. From Remark 1, the function denotes the R-L fractional integral of -fractal function of order . The fractional operator of -fractal functionis defined byfor all . In a similar way, the fractional operator of -fractal function can be defined such that it provides the function for any given two-dimensional continuous function .
The linearity of in (62) is verified as follows:This easily gives . Hence, is a linear operator.
In the following theorem, by taking the base function as the composition of two functions and , the bound of the fractional operator is evaluated.

Theorem 3. If , where , then the fractional operator is bounded. Moreover,where is the identity operator and .

Proof. For the operator , the sup norm is given byHere, . The R-L fractional integral of a continuous function is bounded by , where (refer to Lemma 2.1 in [34]). Furthermore, , similarly . Thus, the above inequality is simplified asSince , refer [15], thenHence, the bound of has been computed.
Now, for the collection of fractional operators, the composition operation with the fractional orders for on is given byNote that, if and , then (refer, [35]). The succeeding theorem illustrates that the collection of fractional operators forms a semigroup for some fractional orders .

Theorem 4. The collection of fractional operators of order forms a semigroup with respect to composition operation; that is,

Proof. For the valued continuous function defined on the closed interval and for some fractional orders ,Substituting , one can get the following:It is clear that . Hence, the collection of fractional operators forms a semigroup. Furthermore,Therefore, the fractional operator of an A-fractal function defined using the R-L fractional operator is clearly a bounded linear operator. Furthermore, the collection of fractional operators configures a semigroup.

5. Final Remarks

This paper discusses the fractional integral of an A-fractal function, and its vertical scaling factors are the parameters in the block matrix and have been taken as continuous functions in order to broaden the diversity of the fractal function. The questions discussed earlier in this paper have been interpreted apparently. The R-L fractional integral of an A-fractal function is examined to address the first query. It is shown that the resultant function is again an A-fractal function when the norm of the block matrix satisfies a certain condition. Furthermore, the second query has been clarified by proposing a new fractional operator which associates each valued continuous function to its R-L fractional integral of an A-fractal function. The linearity and the bound of its norm are discussed. Besides, the semigroup property is illustrated for the collection of fractional operators.

Many of the real-world experimental functions such as electrical signals, EEG signals, weather patterns, and stock indices are complex and rarely exhibit a sensation of smoothness in their patterns. To model such intricate signals, nondifferentiable interpolants are needed. Owing to the irregular and self-similar nature of the fractal functions, the aforementioned signals can be reconstructed, and their properties can be studied efficiently. Fractal functions are presently being received considerable attention in various areas such as Earth sciences, medical sciences, and other sciences. Among the various fractal functions, A-fractal functions are very unique as they are a blend of hidden variable fractal interpolation functions and -fractal functions; hence, they offer more flexibility in the approximation of several nonlinear signal patterns appearing in the science and engineering fields.

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2022 T. M. C. Priyanka et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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