# On Approximation Properties of Fractional Integral for A-Fractal Function

Authors: T. M. C. Priyanka; R. Valarmathi; Kishore Bingi; A. Gowrisankar
Journal: Mathematical Problems in Engineering
 (2022)

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## 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 from0,1 to ℝ. 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 CI,ℝ2, where CI,ℝ2 is the space of all continuous functions from closed interval I⊂ℝ to ℝ2. In addition, the approximation properties such as linearity, boundedness, and semigroup property of the proposed fractional operator are investigated.


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## Body

## 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, α-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 blendingα-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 α-fractal functions are constructed to approximate the given one-dimensional continuous function. The difference between the α-fractal function and the A-fractal function is that the former depends on the scaling factor α and approximates the given one-dimensional continuous function whereas the later depends on the block matrix A containing both free variables and hidden variables, and it approximates the given two-dimensional continuous function by generating a family of ℝ2-valued continuous functions. Therefore, to approximate the given ℝ2 valued continuous function f, the hidden variable A-fractal function fA is used, and for the proper choice of parameters in the block matrix A, the fractal function preserves certain characteristics such as regularity and monotonicity of the original function f. 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 α-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 α-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 α-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. Section2 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,α-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 data(1)xm,ym∈I×ℝ:m=0,1,…,N,such that x0<x1<⋯<xN determines a partition of the closed interval I=x0,xN. In order to find the continuous function f:I⟶ℝ satisfying fxm=ym,∀m=0,1,…,N, an iterated function system is constructed by considering the following special type of affine transformations:(2)wmxy=am0cmdmxy+emfm,with the constraints(3)wmx0y0=xm−1ym−1,wmxNyN=xmym,m=1,2,…,N.The parametersam,cm,dm,em,andfm in the system are all reals, and dm is the free parameter, called the vertical scaling factor such that dm<1.Now, using the above affine maps, the affine fractal interpolation function is constructed as follows: for the given data set (1), the affine maps wm are defined by wmx,y=Lmx,Fmx,y where Lm:I⟶Im=xm,xm−1,∀m=1,2,…,N are the N bicontinuous maps defined by Lmx=amx+bm satisfying the Lipschitz condition of order 1. The continuous maps Fm:X⊂I×ℝ⟶ℝ are defined by Fmx,y=dmy+qmx, and for sm∈0,1, Fm are contractions in the second argument,(4)Fmx,t1−Fmx,t2≤smt1−t2,where x∈I,t1,t2∈ℝ. The system(5)X;wm:m=1,2,…,N,determines an iterated function system (IFS). Let ℋX be the metric space of all nonempty compact subsets of X with respect to Hausdorff metric. The Hutchinson map W:ℋX⟶ℋX is defined by W=∪m=1NwmB∗, for any B∗∈ℋX. Banach fixed point theorem guarantees the existence of the unique invariant set A∗=WA∗ which is called the attractor of the IFS. Let CI=g:I⟶ℝ|g is continuous,gx0=y0,gxN=yN and ρ be the sup norm on CI. Now, an operator T is defined on CI,ρ by(6)Tgx=FmLm−1x,gLm−1x,the fixed point of the operator, say f satisfying the equation(7)fx=FmLm−1x,fLm−1x,is 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 xm,ym,zm∈I×ℝ2:m=0,1,…,N by including a set of real parameters zm (see [9]). The hidden variable FIF is constructed by defining the IFS as(8)I×ℝ2;wm:m=1,2,…,N,where(9)wmx,y,z=Lmx,Fmx,y,z,Lmx=amx+bm,Fmx,y,z=αmy+βmz+pmx,γmz+qmx.Here,pmx and qmx are continuous functions, the parameters αm and γm are the free parameters, and βm is the constrained parameter which are chosen such that αm<1, γm<1 and βm+γm<1. The functions Lm and Fm satisfy the following join-up conditions:(10)Lmx0=xm−1,LmxN=xm,Fmx0,y0,z0=ym−1,zm−1,FmxN,yN,zN=ym,zm.By the IFS theory, (8) has a unique invariant set G, called the attractor of the IFS (8) and which is the graph of the function f:I⟶ℝ2 satisfying fxm=ym.zm, for all m=0,1,…,N. The first component f1 of the ℝ2-valued function f=f1,f2 is called the hidden variable fractal interpolation function which is not self-affine, and the second component f2 is called the fractal function which is self-affine.For the sample set of interpolation data0,10,10,20,40,20,70,80,90,90,10,40, choose the scaling parameters αm=0⋅2,βm=0⋅4,γm=0⋅5, for m=1,2,3. Figure 1 represents the hidden variable fractal interpolation function f1, and Figure 2 illustrates the graph of the fractal function f2.Figure 1
First component off=f1,f2, the hidden variable fractal interpolation function f1 with scaling parameters αm=0⋅2,βm=0⋅4.Figure 2
Second component off=f1,f2, the fractal function f2 with scaling parameters γm=0⋅5.


### 2.2.α-Fractal Function

For approximating the given continuous functionf:I⟶ℝA2, its corresponding α-fractal function is defined as follows. When the continuous functions qm are chosen as(11)qmx=f°Lmx−αmbx,where b:I⟶ℝ is a continuous function such that bx0=fx0 and bxN=fxN, b≠f, and αm=α1,α2,…,αN is the scale vector. Then, the function satisfying the following functional equation,(12)fαx=fx+αmfα−b°Lm−1x,∀x∈I,m=1,2,…,N,is called the α-fractal function, simply known as α-FIF. For more details of α-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 continuousℝ2-valued function [15, 21]. A continuous function f:I⟶ℝ2 is uttered in order to elicit a whole family of fractal functions fA parametrized by a specific block matrix A=Am,m=1,2,…,N with Am=αmβm0γm, take a note of the fact fA=f when A=0. Consider the partition x0<x1<⋯<xN of I and the data set D=xm,f1xm,f2xm:m=1,2,…,N. In the IFS I×ℝ2;wm:m=1,2,…,N defined in (8), consider the special case,(13)pmx=f1°Lmx−αmb1x−βmb2x,qmx=f2°Lmx−γmb2x,where b=b1,b2∈CI,ℝ2 fulfills bx0=fx0 and bxN=fxN. In this approach, the fixed point of the IFS shown in (8) is the graph of the continuous vector-valued function fA=f1A,f2A that obeys the following functional equation:(14)fAx=fx+AmfA−bLm−1x,x∈I,m=1,2,…,N.This functionfA is known as the (hidden variable) A-fractal function of f with respect to the partition x0<x1<⋯<xN and the base function b. Then, the components f1Aandf2A of fA fulfill(15)f1Ax=αmf1ALm−1x+βmf2ALm−1x+pmLm−1x,f2Ax=γmf2ALm−1x+qmLm−1x.The functionsf1Aandf2A are nonself-referential and self-referential functions, respectively. For any choice of the block matrix A and for any choice of the base function b satisfying the above-mentioned conditions, it is observed that fAxm=fxm,∀m=0,1,…,N. Hence, the function fA can be called as the fractal generalization of the continuous function f.When the parameters in the block matrix are taken as continuous functions from0,1 to ℝ, then the (15) becomes(16)f1Ax=αmLm−1xf1ALm−1x+βmLm−1xf2ALm−1x+pmLm−1x,f2Ax=γmLm−1xf2ALm−1x+qmLm−1x.Definition 1.
(see[34]). The R-L fractional integral operator of order μ>0 with a≥0 for a continuous function on a,∞ is defined by the following expression:(17)Iaμfx=1Γμ∫axx−yμ−1fydy,where Γμ=∫0∞e−uuμ−1du. Consider the hidden variable A-fractal function corresponding to the data set xm,ym,zm:m=0,1,…,N. The following is valid for any μ∈0,1(18)Ix0μf1x=1Γμ∫x0xx−tμ−1f1tdt,(19)Ix0μf2x=1Γμ∫x0xx−tμ−1f2tdt.
Equations (18) and (19) are called the R-L fractional integral of f1 and f2b2−4acdefined at the initial point x0 with Ix0μf1x0=Ix0μf2x0=0.
The R-L fractional integral off1 and f2 defined at the end point xN, with IxNμf1xN=IxNμf2xN=0, is given as follows:(20)IxNμf1x=−1Γμ∫xNxx−tμ−1f1tdt,(21)IxNμf2x=−1Γμ∫xNxx−tμ−1f2tdt.


## 2.1. Hidden Variable Fractal Interpolation Function

Generalize the data set (1) as xm,ym,zm∈I×ℝ2:m=0,1,…,N by including a set of real parameters zm (see [9]). The hidden variable FIF is constructed by defining the IFS as(8)I×ℝ2;wm:m=1,2,…,N,where(9)wmx,y,z=Lmx,Fmx,y,z,Lmx=amx+bm,Fmx,y,z=αmy+βmz+pmx,γmz+qmx.Here,pmx and qmx are continuous functions, the parameters αm and γm are the free parameters, and βm is the constrained parameter which are chosen such that αm<1, γm<1 and βm+γm<1. The functions Lm and Fm satisfy the following join-up conditions:(10)Lmx0=xm−1,LmxN=xm,Fmx0,y0,z0=ym−1,zm−1,FmxN,yN,zN=ym,zm.By the IFS theory, (8) has a unique invariant set G, called the attractor of the IFS (8) and which is the graph of the function f:I⟶ℝ2 satisfying fxm=ym.zm, for all m=0,1,…,N. The first component f1 of the ℝ2-valued function f=f1,f2 is called the hidden variable fractal interpolation function which is not self-affine, and the second component f2 is called the fractal function which is self-affine.For the sample set of interpolation data0,10,10,20,40,20,70,80,90,90,10,40, choose the scaling parameters αm=0⋅2,βm=0⋅4,γm=0⋅5, for m=1,2,3. Figure 1 represents the hidden variable fractal interpolation function f1, and Figure 2 illustrates the graph of the fractal function f2.Figure 1
First component off=f1,f2, the hidden variable fractal interpolation function f1 with scaling parameters αm=0⋅2,βm=0⋅4.Figure 2
Second component off=f1,f2, the fractal function f2 with scaling parameters γm=0⋅5.


## 2.2.α-Fractal Function

For approximating the given continuous functionf:I⟶ℝA2, its corresponding α-fractal function is defined as follows. When the continuous functions qm are chosen as(11)qmx=f°Lmx−αmbx,where b:I⟶ℝ is a continuous function such that bx0=fx0 and bxN=fxN, b≠f, and αm=α1,α2,…,αN is the scale vector. Then, the function satisfying the following functional equation,(12)fαx=fx+αmfα−b°Lm−1x,∀x∈I,m=1,2,…,N,is called the α-fractal function, simply known as α-FIF. For more details of α-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 continuousℝ2-valued function [15, 21]. A continuous function f:I⟶ℝ2 is uttered in order to elicit a whole family of fractal functions fA parametrized by a specific block matrix A=Am,m=1,2,…,N with Am=αmβm0γm, take a note of the fact fA=f when A=0. Consider the partition x0<x1<⋯<xN of I and the data set D=xm,f1xm,f2xm:m=1,2,…,N. In the IFS I×ℝ2;wm:m=1,2,…,N defined in (8), consider the special case,(13)pmx=f1°Lmx−αmb1x−βmb2x,qmx=f2°Lmx−γmb2x,where b=b1,b2∈CI,ℝ2 fulfills bx0=fx0 and bxN=fxN. In this approach, the fixed point of the IFS shown in (8) is the graph of the continuous vector-valued function fA=f1A,f2A that obeys the following functional equation:(14)fAx=fx+AmfA−bLm−1x,x∈I,m=1,2,…,N.This functionfA is known as the (hidden variable) A-fractal function of f with respect to the partition x0<x1<⋯<xN and the base function b. Then, the components f1Aandf2A of fA fulfill(15)f1Ax=αmf1ALm−1x+βmf2ALm−1x+pmLm−1x,f2Ax=γmf2ALm−1x+qmLm−1x.The functionsf1Aandf2A are nonself-referential and self-referential functions, respectively. For any choice of the block matrix A and for any choice of the base function b satisfying the above-mentioned conditions, it is observed that fAxm=fxm,∀m=0,1,…,N. Hence, the function fA can be called as the fractal generalization of the continuous function f.When the parameters in the block matrix are taken as continuous functions from0,1 to ℝ, then the (15) becomes(16)f1Ax=αmLm−1xf1ALm−1x+βmLm−1xf2ALm−1x+pmLm−1x,f2Ax=γmLm−1xf2ALm−1x+qmLm−1x.Definition 1.
(see[34]). The R-L fractional integral operator of order μ>0 with a≥0 for a continuous function on a,∞ is defined by the following expression:(17)Iaμfx=1Γμ∫axx−yμ−1fydy,where Γμ=∫0∞e−uuμ−1du. Consider the hidden variable A-fractal function corresponding to the data set xm,ym,zm:m=0,1,…,N. The following is valid for any μ∈0,1(18)Ix0μf1x=1Γμ∫x0xx−tμ−1f1tdt,(19)Ix0μf2x=1Γμ∫x0xx−tμ−1f2tdt.
Equations (18) and (19) are called the R-L fractional integral of f1 and f2b2−4acdefined at the initial point x0 with Ix0μf1x0=Ix0μf2x0=0.
The R-L fractional integral off1 and f2 defined at the end point xN, with IxNμf1xN=IxNμf2xN=0, is given as follows:(20)IxNμf1x=−1Γμ∫xNxx−tμ−1f1tdt,(21)IxNμf2x=−1Γμ∫xNxx−tμ−1f2tdt.


## 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 conditionsy^0=0andz^0=0.Theorem 1.
LetfA be the A-fractal function with variable scaling factors corresponding to the interpolation points xm,ym,zm∈I×ℝ2:m=0,1,…,N. If maxamμαm∞,amμβm+γm∞:m=1,2,…,N<1, then Lmx,F^1mAx,y^,z^m=1N and Lmx,F^2mAx,z^m=1N generates Ix0μfAx=Ix0μf1Ax,Ix0μf2Ax, where(22)F^1mAx,y^,z^=amμαmxy^+amμβmxz^+p^mx,F^2mAx,z^=amμγmxz^+q^mx,with ∑m=1NamμαmxN≠1, ∑m=1NamμγmxN≠1, and y^0=0,z^0=0, for m=1,2,…,Nb2−4ac,(23)p^mx=y^m−1+f1Am,μx+amμIx0μf1°Lmx−amμαmxIx0μb1x−amμβmxIx0μb2x,q^mx=z^m−1+f2Am,μx+amμIx0μf2°Lmx−amμγmxIx0μb2x,y^m=∑i=1mf1Ai,μxN+aiμαixNy^N+aiμβixNz^N+aiμIx0μf1°LixN−aiμαixNIx0μb1xN−aiμβixNIx0μb2xNy^N=∑i=1Nf1Ai,μxN+aiμβixNz^i+aiμIx0μf1°LixN−aiμαixNIx0μb1xN−aiμβixNIx0μb2xN1−∑i=1NaiμαixN,z^m=∑i=1mf2Ai,μxN+aiμγixNz^N+aiμIx0μf2°LixN−aiμγixNIx0μb2xN,z^N=∑i=1Nf2Ai,μxN+aiμIx0μf2°LixN−aiμγixNIx0μb2xN1−∑i=1NaiμγixN.Proof.
The R-L fractional integral off1A is given by(24)Ix0μf1ALmx=1Γμ∫x0LmxLmx−tμ−1f1Atdt=1Γμ∫x0xm−1xm−1−tμ−1f1Atdt+1Γμ∫x0xm−1Lmx−tμ−1−xm−1−tμ−1f1Atdt+1Γμ∫xm−1LmxLmx−tμ−1f1Atdt.
Here, takey^m−1=1/Γμ∫x0xm−1xm−1−tμ−1f1Atdt and f1Am,μx=1/Γμ∫x0xm−1Lmx−tμ−1−xm−1−tμ−1f1Atdt. By using the variable transformation t=Lmu in the third term and the functional (16), the following equation is generated(25)Ix0μf1ALmx=y^m−1+f1Am,μx+amΓμ∫x0xamμ−1x−uμ−1αmuf1Au+βmuf1Au+pmudu=y^m−1+f1Am,μx+amμΓμ∫x0xx−uμ−1αmuf1Audu+amμΓμ∫x0xx−uμ−1βmμ×f1Aμdu+amμΓμ∫x0xx−uμ−1f1°Lmu−αmub1−βmub2udu.
Now, using the Leibniz rule for fractional integral, the following is obtained:(26)Ix0μf1ALmx=y^m−1+f1Am,μx+amμ∑k=0∞vkIx0μ+kf1AxDx0kαmx+amμ∑k=0∞vkIx0μ+kf2AxDx0kβmx+amμIx0μf1°Lmx−amμ∑k=0∞vkIx0μ+kb1xDx0kαmx−amμ∑k=0∞vkIx0μ+kb2xDx0kβmx=y^m−1+f1Am,μx+amμαmxIx0μf1Ax+amμβmxIx0μf2Ax+amμIx0μf1°Lmx−amμαmxIx0μb1x−amμβmxIx0μb2x=amμαmxIx0μf1Ax+amμβmxIx0μf2Ax+p^mx=F^1mx,Ix0μf1Ax,Ix0μf2Ax.
Denotep^mx=y^m−1+f1Am,μx+amμIx0μf1°Lmx−amμαmxIx0μb1x−amμβmxIx0μb2x. Now, consider the R-L fractional integral of f2A of order v,(27)Ix0μf2ALmx=1Γμ∫x0LmxLmx−tμ−1f2Atdt=1Γμ∫x0xm−1xm−1−tμ−1f2Atdt+1Γμ∫x0xm−1Lmx−tμ−1−xm−1−tμ−1f2Atdt+1Γμ∫xm−1LmxLmx−tμ−1f2Atdt.
Here, takez^m−1=1/Γμ∫x0xm−1xm−1−tμ−1f2Atdt and f2Am,μx=1/Γμ∫x0xm−1Lmx−tμ−1−xm−1−tμ−1f2Atdt. By using variable transformation t=Lmu in the third term, one can obtain(28)Ix0μf2ALmx=z^m−1+f2Am,μx+amΓμ∫x0xLmx−Lmuμ−1f2ALmudu=z^m−1+f2Am,μx+amμΓμ∫x0xx−uμ−1γmuf2Audu+amμΓμ∫x0xx−uμ−1f2°Lmu−γmub2udu=z^m−1+f2Am,μx+amμΓμ∫x0xx−uμ−1γmuf2Audu+amμΓμ∫x0xx−uμ−1f2°Lmudu−amμΓμ∫x0xx−uμ−1γmub2udu.
Applying the Leibniz rule for fractional integral,(29)Ix0μf2ALmx=z^m−1+f2Am,μx+amμ∑k=0∞vkIx0μ+kf2AxDx0kγmx+amμIx0μf2°Lmx−amμ∑k=0∞vkIx0μ+kb2xDx0kγmx=z^m−1+f2Am,μx+amμγmxIx0μf2Ax+amμIx0μf2°Lmx−amμγmxIx0μb2x=amμγmxIx0μf2Ax+q^mx=F^2mx,Ix0μf2Ax.
Denoteq^mx=z^m−1+f2Am,μx+amμIx0μf2°Lmx−amμγmxIx0μb2x.
As a result, an A-fractal function’s R-L fractional integral is also an A-fractal function. Takex=xN and LmxN=xm in the following equation:(30)Ix0μf1ALmx=y^m−1+f1Am,μx+amμαmxIx0μf1Ax+amμβmxIx0μf2Ax+amμIx0μf1°Lmx−amμαmxIx0μb1x−amμβmxIx0μb2x.
Hence,(31)y^m−y^m−1=f1Am,μxN+amμαmxNy^N+amμβmxNz^N+amμIx0μf1°LmxN−amμαmxNIx0μb1xN−amμβmxNIx0μb2xN.
The system of equationsy^m=y^0+∑i=1my^i−y^i−1 yields(32)y^m=∑i=1mf1Ai,μxN+aiμαixNy^N+aiμβixNz^N+aiμIx0μf1°LixN−aiμαixNIx0μb1xN−aiμβixNIx0μb2xN.
By substitutingm=N, the previous equation provides y^N as follows:(33)y^N=∑i=1Nf1Ai,μxN+aiμβixNz^i+aiμIx0μf1°LixN−aiμαixNIx0μb1xN−aiμβixNIx0μb2xN1−∑i=1NaiμαixN.
Takex=xN and LmxN=xm in the below equation to determine the new data set xm,z^m:m=0,1,…,N.(34)Ix0μf2ALmx=z^m−1+f2Am,μx+amμγmxIx0μf2Ax+amμIx0μf2°Lmx−amμγmxIx0μb2x.
Therefore,(35)z^m−z^m−1=f2Am,μxN+amμγmxNz^N+amμIx0μf2°LmxN−amμγmxNIx0μb2xN.
The equationz^m=z^0+∑i=1mz^i−z^i−1 yields(36)z^m=∑i=1mf2Ai,μxN+aiμγixNz^N+aiμIx0μf2°LixN−aiμγixNIx0μb2xN.
By substitutingm=N, the previous equation provides the end point z^N as follows:(37)z^N=∑i=1Nf2Ai,μxN+aiμIx0μf2°LixN−aiμγixNIx0μb2xN1−∑i=1NaiμγixN.
It is important to check the join-up conditions ofF^mx,y^,z^. First,(38)p^mx0=y^m−1+f1Am,μx0+amμIx0μf1°Lmx0−amμαmx0Ix0μb1x0−amμβmx0Ix0μb2x0,p^mx0=y^m−1,p^mxN=y^m−1+f1Am,μxN+amμIx0μf1°LmxN−amμαmxNIx0μb1xN−amμβmxNIx0μb2xN,when p^mxN is compared with (31), one can obtain(39)p^mxN=y^m−amμαmxNy^N−amμβmxNz^N.
In the same way, the endpoint conditions ofq^m can be verified.
Hence, Theorem1 has demonstrated that the R-L fractional integral of an A-fractal function with the predetermined initial conditions y^0=0 and z^0=0 is also an A-fractal function for the new data set xm,y^m,z^m:m=0,1,…,N.Remark 1.
Theorem1 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 Am′<1 where Am′=amμαmβm0γm with ∑m=1Namμαm≠1, ∑m=1Namμγm≠1, and y^0=0, z^0=0, then the following result is obtained for m=1,2,…,N,(40)p^mx=y^m−1+f1i,μAx+amμIx0μf1°Lmx−amμαmIx0μb1x−amμβmIx0μb2x,q^mx=z^m−1+f2i,μANx+amμIx0μf2°Lmx−amμγmIx0μb2x,f1Am,μx=1Γμ∫x0xm−1Lmx−tμ−1−xm−1−tμ−1f1Atdt,f2Am,μx=1Γμ∫x0xm−1Lmx−tμ−1−xm−1−tμ−1f2Atdt,y^m=∑i=1mf1Ai,μxN+aiμαiy^N+aiμβiz^N+aiμIx0μf1°LixN−aiμαiIx0μb1xN−aiμβiIx0acμb2xN,−aiμαiIx0acμb1xN−aiμβiIx0acμb2xN,y^N=∑i=1Nf1Ai,μxN+aiμβiz^N+aiμIx0μf1°LixN−aiμαiIx0μb1xN−aiμβiIx0μb2xN1−∑i=1Naiμαi,z^m=∑i=1mf2Ai,μxN+aiμγiz^N+aiμIx0μf2°LixN−aiμγiIx0μb2xN,z^N=∑i=1Nf2Ai,μxN+aiμIx0μf2°LixN−aiμγiIx0μb2xN1−∑i=1Naiμγi.
The following theorem explores the R-L fractional integral of an A-fractal function with the prescribed conditiony^N=0 and z^N=0.Theorem 2.
LetfA be the A-fractal function with variable scaling factors corresponding to the interpolation points xm,ym,zm∈I×ℝ2:m=0,1,2,…,N. If maxamμαm∞,amμβm+γm∞:m=1,2,…,N<1, then Lmx,F^1mAx,y^,z^m=1N and Lmx,F^2mAx,z^m=1N generate IxNμfAx=IxNμf1Ax,IxNμf2Ax where F^1mAx,y^,z^=amμαmxy^+amμβmxz^+p^mx with F^2mAx,z^=am−vγmxz^+q^mx, with ∑m=1Namμαmx0≠1, ∑m=1Namμγmx0≠1 and y^N=0, z^N=0 for m=1,2,…,N(41)p^mx=y^m−f1Am,μx+amμIxNμf1°Lmx−amμαmxIxNμb1x−amμβmxIxNμb2x,q^mx=z^m−f2Am,μx+amμIxNμf2°Lmx−amμγmxIxNμb2x,y^m−1=−∑i=mNf1Ai,μx0−aiμαix0y^0−aiμβix0z^0−aiμIxNμf1°Lix0+aiμαix0IxNμb1x0+aiμβix0IxNμb2x0,y^0=−∑i=1Nf1Ai,μx0−aiμβix0z^0−aiμIxNμf1°Lix0+aiμαix0IxNμb1x0+aiμβix0IxNμb2x01−∑i=1Naiμαix0,z^m−1=−∑i=mNf2Ai,μx0−aiμγix0z^0−aiμIxNμf2°Lix0+aiμγix0IxNμb2x0,z^0=−∑i=1Nf2Ai,μx0−aiμIxNμf2°Lix0+aiμγix0IxNμb2x01−∑i=1Naiμγix0.Proof.
From (18), consider the Riemann–Liouville fractional integral of f1A,(42)IxNμf1ALmx=−1Γμ∫LmxxNLmx−tμ−1f1Atdt=−1Γμ∫xmxNxm−tμ−1f1Atdt−1Γμ∫xmxNLmx−tμ−1−xm−tμ−1f1Atdt−1Γμ∫LmxxmLmx−tμ−1f1Atdt.
Here, takey^m=1/Γμ−∫xmxNxm−tμ−1f1Atdt and f2Am,μx=1/Γμ∫xmxNLmx−tμ−1−xm−tμ−1f1Atdt. By using variable transformation in the third term as t=Lmu, the above equation is modified as(43)IxNμf1ALmx=y^m−f1Am,μx−amΓμ∫xxNLmx−Lmuμ−1f1ALmudu.
Using the functional (16), the following equation is generated:(44)IxNμf1ALmx=y^m−f1Am,μx−amΓμ∫xxNamμ−1x−uμ−1αmuf1Au+βmuf2Au+pmudu=y^m−f1Am,μx−amμΓμ∫xxNx−uμ−1αmuf1Audu−amμΓμ∫xxNx−uμ−1βmuf2Audu−amμΓμ∫xxNx−uμ−1f1°Lmu−αmub1u−βmub2udu.
Now, using the Leibniz rule for fractional integral(45)IxNμf1ALmx=y^m−f1Am,μx+amμ∑k=0∞vkIxNμ+kf1AxDxNkαmx+amμ∑k=0∞vkIxNμ+kf2AxDxNkβmx+amμIxNμf1°Lmx−amμ∑k=0∞vkIxNμ+kb1xDxNkαmx−amμ∑k=0∞vkIxNμ+kb2xDxNkβmx=y^m−f1Am,μx+amμαmxIxNμf1Ax+amμβmxIxNμf2Ax+amμIxNμf1°Lmx−amμαmxIxNμb1x−amμβmxIxNμb2x=amμαmxIxNμf1Ax+amμβmxIxNμf2Ax+p^mx=F^1mx,IxNμf1Ax,IxNμf2Ax.
Here, denotep^mx=y^m−f1Am,μx+amμIxNμf1°Lmx−amμαmxIxNμb1x−amμβmxIxNμb2x. The R-L fractional integral of f2A of order v is given by(46)IxNμf2ALmx=−1Γμ∫LmxxNLmx−tμ−1f2Atdt=−1Γμ∫xmxNxm−tμ−1f2Atdt−1Γμ∫xmxNLmx−tμ−1−xm−tμ−1f2Atdt−1Γμ∫LmxxmLmx−tμ−1f2Atdt.
Here, takez^m=1/Γμ−∫xmxNxm−tμ−1f2Atdt and f2Am,μx=1/Γμ∫xmxNLmx−tμ−1−xm−tμ−1f2Atdt. By using variable transformation in the third term as t=Lmu, the above equation is modified as(47)IxNμf2ALmx=z^m−f2Am,μx−amΓμ∫xxNLmx−Lmuμ−1f2ALmudu.
Using the functional (16), the following equation is generated:(48)IxNμf2ALmx=z^m−f2Am,μx−amΓμ∫xxNamμ−1x−uμ−1γmuf2Au+qmudu=z^m−f2Am,μx−amμΓμ∫xxNx−uμ−1γmuf2Audu−amμΓμ∫xxNx−uμ−1f2°Lmudu+amμΓμ∫xxNx−uμ−1γmub2udu.
Now, using the Leibniz rule for fractional integral,(49)IxNμf2ALmx=z^m−f2Am,μx+amμ∑k=0∞vkIxNμ+kf2AxDxNkγmx+amμIxNμf2°Lmx−amμIxNμ∑k=0∞vkIxNμ+kb2xDxNkγmx=z^m−f2Am,μx+amμγmxIxNμf2Ax+amμIxNμf2°Lmx−amμγmxIxNμb2x=amμγmxIxNμf2Ax+q^mx=F^2mx,IxNμf2Ax.
Here, denoteq^mx=z^m−f2Am,μx+amμIxNμf2°Lmx−amμγmxIxNμb2x. As a result, an A-fractal function’s R-L fractional integral is also an A-fractal function. Put x=x0 and Lmx0=xm−1 in the following equation:(50)IxNμf1ALmx=y^m−f1Am,μx+amμαmxIxNμf1Ax+amμβmxIxNμf2Ax+amμIxNμf1°Lmx−amμαmxIxNμb1x−amμβmxIxNμb2x.
Therefore,(51)ym^−y^m−1=f1Am,μx0−amμαmx0y^0−amμβmx0z^0−amμIxNμf1°Lmx0+amμαmx0IxNμb1x0+amμβmx0IxNμb2x0.
The equationy^m−1=y^N−∑i=mNy^i−y^i−1 yields(52)y^m−1=−∑i=mNf1Ai,μx0−aiμαix0y^0−aiμβix0z^0−aiμIxNμf1°Lix0+aiμαix0IxNμb1x0+aiμβix0IxNμb2x0.
By substitutingm=1, the previous equation provides the initial point y^0 as follows:(53)y^0=−∑i=1Nf1Ai,μx0−aiμβix0z^0−aiμIxNμf1°Lix0+aiμαix0IxNμb1x0+aiμβix0IxNμb2x01−∑i=1Naiμαix0,take x=x0 and Lmx0=xm−1 in the following equation to determine the new interpolation data xm,z^m:m=0,1,…,N.(54)IxNμf2ALmx=z^m−f2Am,μx+amμγmxIxNμf2Ax+amμIxNμf2°Lmx−amμγmxIxNμb2x.
Thus,(55)z^m−z^m−1=f2Am,μx0−amμγmx0z^0−amμIxNμf2°Lmx0+amμγmx0IxNμb2x0.
The equationz^m−1=z^N−∑i=mNz^i−z^i−1 yields(56)z^m−1=−∑i=mNf2Ai,μx0−aiμγix0z^0−aiμIxNμf2°Lix0+aiμγix0IxNμb2x0.
By substitutingm=1, the previous equation provides the initial point z^0 as follows:(57)z^0=−∑i=1Nf2Ai,μx0−aiμIxNμf2°Lix0+aiμγix0IxNμb2x01−∑i=1Naiμγix0.
It is important to check the join-up conditions ofF^mx,y^,z^. First,(58)p^mx0=y^m−f1Am,μx0+amμIxNμf1°Lmx0−amμαmx0IxNμb1x0−amμβmx0IxNμb2x0.
When we comparep^mx0 with (51), we can obtain(59)p^mx0=y^m−1−amμαmx0y^0−amμβmx0z^0,p^mxN=y^m−f1Am,μxN+amμIxNμf1°LmxN−amμαmxNIxNμb1xN−amμβmxNIxNμb2xN=y^m.
Similarly, the endpoint conditions ofq^m can be verified.
Theorem2 clearly shows that the function IxNμfA fulfills the endpoint conditions F^mx0,y^0,z^0=y^m−1,z^m−1 and F^mxN,y^N,z^N=y^m,z^m. Hence, the R-L fractional integral of an A-fractal function with the predetermined conditions y^N=0 and z^N=0 is also an A-fractal function corresponding to the new data set xm,y^m,z^m:m=0,1,…,N.Remark 2.
Theorem2 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 Am′<1 where Am′=amμαmβm0γm with ∑m=1Namμαm≠1, ∑m=1Namμγm≠1, and y^N=0, z^N=0, then the following result is obtained for m=1,2,…,N:(60)p^mx=ym^−f1m,μAx+amμIxNμf1°Lmx−amμαmIxNμb1x−βmamμIxNμb2x,q^mx=z^m−f2m,μAx+amμIxNμf2°Lmx−amμγmIxNμb2x,f1Am,μx=1Γμ∫xmxNLmx−tμ−1−xm−tμ−1f1Atdt,f2Am,μx=1Γμ∫xmxNLmx−tμ−1−xm−tμ−1f2Atdt,y^m−1=−∑i=1NfAi,μx0−aiμαiy^0−aiμβiz^0−aiμIxNμf1°Lix0+aiμαiIxNμb1x0+aiμβiIxNμb2x01,y^0=−∑i=1Nf1Ai,μx0−aiμβiz^0−aiμIx0μf1°Lix0+aiμαiIx0μb1x0+aiμβiIx0μb2x01−∑i=1Naiμαi,z^m−1=−∑i=1Nf2Ai,μx0−aiμγiz^0−aiμIxNμf2°Lix0+aiμγiIxNμb2x0,z^0=−∑i=1Nf2Ai,μx0−aiμIx0μf2°Lix0+aiμγiIx0μb2x01−∑i=1Naiμγi.


## 4. Formulation of Fractional Operator

A fractional operator for the A-fractal function on the space ofℝ2 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 Remark1, the function Ix0μfA denotes the R-L fractional integral of A-fractal function of order v. The fractional operator of A-fractal function(61)ℱμA:CI,ℝ2⟶CI,ℝ2f↦Ix0μfA,is defined by(62)ℱμAfx=y^m−1+f1Am,μLm−1x+amμαmIx0μf1ALm−1x+amμβmIx0μf2ALm−1x+amμIx0μf1x−amμαmIx0μb1Lm−1x−amμβmIx0μb2Lm−1x+z^m−1+f2Am,μLm−1x+amμγmIx0μf2ALm−1x+amμIx0μf2x−amμγmIx0μb2Lm−1x,for all x∈Im,m=1,2,…,N. In a similar way, the fractional operator of A-fractal function can be defined such that it provides the function IxNμfA for any given two-dimensional continuous function f.
The linearity ofℱμA in (62) is verified as follows:(63)ℱμArf=ry^m−1+rf1Am,μLm−1x+amμαmIx0μrf1ALm−1x+amμβmIx0μrf2ALm−1x+amμIx0μrf1x−amμαmIx0μrb1Lm−1x−amμβmIx0μrb2Lm−1x+rz^m−1+rf2Am,μLm−1x+amμγmIx0μrf2ALm−1x+amμIx0μrf2x−amμγmIx0μrb2Lm−1x,ℱμAsg=sy^m−1+sg1Am,μLm−1x+amμαmIx0μsg1ALm−1x+amμβmIx0μsg2ALm−1x+amμIx0μsg1x−amμαmIx0μsb1′Lm−1x−amμβmIx0μsb2′Lm−1x+sz^m−1+sg2Am,μLm−1x+amμγmIx0μsg2ALm−1x+amμIx0μsg2x−amμγmIx0μsb2′Lm−1x.
This easily givesℱμArf+sg=rℱμAf+sℱμAg. Hence, ℱμA is a linear operator.
In the following theorem, by taking the base function as the composition of two functionsL and f, the bound of the fractional operator is evaluated.Theorem 3.
Ifb=Lf, where L:LipCI,ℝ2⟶CI,ℝ2, then the fractional operator ℱμA is bounded. Moreover,(64)ℱμAf∞≤A1J2A11−A11−L+3−Lf∞,where Id is the identity operator and J=xN−x0μ/μΓμ.Proof.
For the operatorℱμAf, the sup norm is given by(65)ℱμAf∞=Ix0μfA∞≤max1≤m≤Nsupy^m−1+f1Am,μLm−1x+z^m−1+f2Am,μLm−1x+amμαmIx0μf1A−Ix0μb1Lm−1x+amμβmIx0μf2A−Ix0μb2Lm−1x+amμγmIx0μf2A−Ix0μb2Lm−1x+amμ‖Ix0μf1+Ix0μf2x≤max1≤m≤NAm1supy^m−1+f1Am,μLm−1x+z^m−1+f2Am,μLm−1x+Ix0μf1A−Ix0μb1Lm−1x+Ix0μf2A−Ix0μb2Lm−1x+Ix0μf1+Ix0f2x=A1supy^m−1+f1Am,μLm−1x+z^m−1+f2Am,μLm−1x+Ix0μfA−Ix0μbLm−1x+Ix0μfx.
Here,A1=max1≤m≤Nαm,βm+γm≤1 . The R-L fractional integral of a continuous function Ix0μfx is bounded by Jfx∞, where J=xN−x0μ/vΓv (refer to Lemma 2.1 in [34]). Furthermore, y^m−1+f1Am,μLm−1x∞≤xm−1−x0μ/vΓμf1A∞, similarly z^m−1+f2Am,μLm−1x∞≤xm−1−x0μ/vΓμf2A∞. Thus, the above inequality is simplified as(66)≤A1xm−1−x0μvΓμfA∞+JfA−b∞+Jf∞≤A2JfA∞+Jf∞−b∞.
SincefA∞≤A1/1−A1Id−L+1f∞, refer [15], then(67)≤A12JA11−A1Id−L+1f∞+Jf∞−Lf∞≤A12JA11−A1f∞−2JA11−A1Lf∞+3Jf∞−Lf∞=A1J2A11−A11−L+3−Lf∞.
Hence, the bound ofℱμA has been computed.
Now, for the collection of fractional operators, the composition operation with the fractional ordersk,l>0 for f on CI,ℝ2 is given by(68)IakIalf=Iak+lf.
Note that, iff∈CI,ℝ2 and k,l>0, then Iak+lf∈CI,ℝ2 (refer, [35]). The succeeding theorem illustrates that the collection of fractional operators forms a semigroup for some fractional orders μ1,μ2>0.Theorem 4.
The collection of fractional operators of orderμ>0,ℱμA:CI,ℝ2⟶CI,ℝ2 forms a semigroup with respect to composition operation; that is,(69)ℱμ1ℱμ2Af=ℱμ2ℱμ1Af.Proof.
For theℝ2 valued continuous function f defined on the closed interval I and for some fractional orders μ1,μ2>0,(70)ℱμ1ℱμ2Af=amμ1+μ2αmΓμ1Γμ2∫x0xx−tμ1−1∫x0tt−uμ2−1f1ALm−1ududt+amμ1+μ2βmΓμ1Γμ2∫x0xx−tμ1−1∫x0tt−uμ2−1f2ALm−1ududt+amμ1+μ2Γμ1Γμ2∫x0xx−tμ1−1∫x0tt−uμ2−1pmLm−1ududt+y^m−1,μ1+μ2+f1Am,μ1+μ2Lm−1x+amμ1+μ2γmΓμ1Γμ2∫x0xx−tμ1−1∫x0tt−uμ2−1f2ALm−1ududt+amμ1+μ2Γμ1Γμ2∫x0xx−tμ1−1∫x0tt−uμ2−1qmLm−1ududt+z^m−1,μ1+μ2+f2Am,μ1+μ2Lm−1x,ℱμ1ℱμ2Af=amμ1+μ2αmΓμ1Γμ2∫x0xf1ALm−1x∫uxx−tμ1−1t−uμ2−1dtdu+amμ1+μ2βmΓμ1Γμ2∫x0xf2ALm−1u∫uxx−tμ1−1t−uμ2−1dtdu+amμ1+μ2Γμ1Γμ2∫x0xpmLm−1x∫uxx−tμ1−1t−uμ2−1dtdu+y^m−1,μ1+μ2+f1Am,μ1+μ2Lm−1x+amμ1+μ2γmΓμ1Γμ2∫x0xf2ALm−1x∫uxx−tμ1−1t−uμ2−1dtdu+amμ1+μ2Γμ1Γμ2∫x0xqmLm−1x∫uxx−tμ1−1t−uμ2−1dtdu+z^m−1,μ1+μ2+f2Am,μ1+μ2Lm−1x.
Substitutingt=u+vx−u, one can get the following:(71)ℱμ1ℱμ2Af=amμ1+μ2αmΓμ1Γμ2∫x0xf1ALm−1ux−uμ1+μ2−1∫011−vμ1−1vμ2−1dvdu+amμ1+μ2βmΓμ1Γμ2∫x0xf2ALm−1ux−uμ1+μ2−1∫011−vμ1−1vμ2−1dvdu+amμ1+μ2Γμ1Γμ2∫x0xpmLm−1ux−uμ1+μ2−1∫011−vμ1−1vμ2−1dvdu+y^m−1,μ1+μ2+f1Am,μ1+μ2Lm−1x+amμ1+μ2γmΓμ1Γμ2∫x0xf2ALm−1ux−uμ1+μ2−1∫011−vμ1−1vμ2−1dvdu+amμ1+μ2Γμ1Γμ2∫x0xqmLm−1ux−uμ1+μ2−1∫011−vμ1−1vμ2−1dvdu+z^m−1,μ1+μ2+f2Am,μ1+μ2Lm−1x=amμ1+μ2αmIx0μ1+μ2f1ALm−1x+amμ1+μ2βmIx0μ1+μ2f2ALm−1x+amμ1+μ2Ix0μ1+μ2pmLm−1x+y^m−1,μ1+μ2+f1Am,μ1+μ2Lm−1x+amμ1+μ2γmIx0μ1+μ2f2ALm−1x+amμ1+μ2Ix0μ1+μ2qmLm−1x+z^m−1,μ1+μ2+f2Am,μ1+μ2Lm−1x=ℱμ1+μ2Af.
It is clear thatℱμ1+μ2Af∈CI,ℝ2. Hence, the collection of fractional operators ℱμA:CI,ℝ2⟶CI,ℝ2 forms a semigroup. Furthermore,(72)ℱμ1ℱμ2Af=ℱμ1+μ2Af=ℱμ2+μ1Af=ℱμ2ℱμ1Af.
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 eachℝ2 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 andα-fractal functions; hence, they offer more flexibility in the approximation of several nonlinear signal patterns appearing in the science and engineering fields.



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