Open access peer-reviewed chapter

# Sets of Fractional Operators and Some of Their Applications

Written By

A. Torres-Hernandez, F. Brambila-Paz and R. Ramirez-Melendez

Reviewed: 19 August 2022 Published: 26 October 2022

DOI: 10.5772/intechopen.107263

From the Edited Volume

## Operator Theory - Recent Advances, New Perspectives and Applications

Edited by Abdo Abou Jaoudé

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## Abstract

This chapter presents one way to define Abelian groups of fractional operators isomorphic to the group of integers under addition through a family of sets of fractional operators and a modified Hadamard product, as well as one way to define finite Abelian groups of fractional operators through sets of positive residual classes less than a prime number. Furthermore, it is presented one way to define sets of fractional operators which allow generalizing the Taylor series expansion of a vector-valued function in multi-index notation, as well as one way to define a family of fractional fixed-point methods and determine their order of convergence analytically through sets.

### Keywords

• fractional operators
• set theory
• group theory
• fractional iterative methods
• fractional calculus of sets

## 1. Introduction

In one dimension, a fractional derivative may be considered in a general way as a parametric operator of order α, such that it coincides with conventional derivatives when α is a positive integer n. So, when it is not necessary to explicitly specify the form of a fractional derivative, it is usually denoted as follows

dαdxα.E1

On the other hand, a fractional differential equation is an equation that involves at least one differential operator of order α, with n1<αn for some positive integer n, and it is said to be a differential equation of order α if this operator is the highest order in the equation. The fractional operators have many representations [1, 2, 3], but one of their fundamental properties is that they allow retrieving the results of conventional calculus when αn. For example, let f:ΩRR be a function such that fLloc1ab, where Lloc1ab denotes the space of locally integrable functions on the open interval abΩ. One of the fundamental operators of fractional calculus is the operator Riemann−Liouville fractional integral, which is defined as follows [4, 5]:

aIxαfx1Γαaxxtα1ftdt,E2

where Γ denotes the Gamma function. It is worth mentioning that the above operator is a fundamental piece to construct the operator Riemann−Liouville fractional derivative, which is defined as follows [4, 6]:

where n=α and aIx0fxfx. On the other hand, let f:ΩRR be a function n-times differentiable such that f,fnLloc1ab. Then, the Riemann−Liouville fractional integral also allows constructing the operator Caputo fractional derivative, which is defined as follows [4, 6]:

aCDxαfxaIxαfx,ifα<0aIxnαfnx,ifα0,E4

where n=α and aIx0fnxfnx. Furthermore, if the function f fulfills that fka=0k01n1, the Riemann−Liouville fractional derivative coincides with the Caputo fractional derivative, that is,

So, applying the operator (3) with a=0 to the function xμ, with μ>1, we obtain the following result [7]:

0Dxαxμ=Γμ+1Γμα+1xμα,αR\Z,E6

where if 1αμ it is fulfilled that 0Dxαxμ=0CDxαxμ.

## 2. Sets of fractional operators

Before continuing, it is necessary to mention that due to the large number of fractional operators that may exist [1, 2, 3, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], some sets must be defined to fully characterize elements of fractional calculus. It is worth mentioning that characterizing elements of fractional calculus through sets is the main idea behind of the methodology known as fractional calculus of sets [24, 25]. So, considering a scalar function h:RmR and the canonical basis of Rm denoted by êkk1, it is possible to define the following fractional operator of order α using Einstein notation

oxαhxêkokαhx.E7

Therefore, denoting by kn the partial derivative of order n applied with respect to the k-th component of the vector x, using the previous operator it is possible to define the following set of fractional operators

Ox,αnhoxα:okαhxandlimαnokαhx=knhxk1,E8

which may be proved to be a nonempty set through the following set of fractional operators

O0,x,αnhoxα:okαhx=kn+μαkαhxandlimαnμαkαhx=0k1,E9

with which it is possible to obtain the following result:

Ifoi,xα,oj,xαOx,αnhwithijok,xα=12oi,xα+oj,xαOx,αnh.E10

So, the complement of the set (8) may be defined as follows

Ox,αn,ch{oxα:okαhxk1andlimαnokαhxknhxinatleastonevaluek1},E11

with which it is possible to obtain the following result:

Ifoi,xα=êkoi,kαOx,αnhoj,xα=êkoi,σjkαOx,αn,ch,E12

where σj:12m12m denotes any permutation different from the identity. On the other hand, the set (8) may be considered as a generating set of sets of fractional tensor operators. For example, considering α,nRd with α=êkαk and n=êknk, it is possible to define the following set of fractional tensor operators

Ox,αnhoxα:oxαhxandoxαOx,α1n1h×Ox,α2n2h××Ox,αdndh.E13

## 3. Groups of fractional operators

Considering a function h:ΩRmRm, it is possible to define the following sets

mOx,αnhoxα:oxαOx,αnhkkm,E14
mOx,αn,choxα:oxαOx,αn,chkkm,E15
mOx,αn,uhmOx,αnhmOx,αn,ch,E16

where hk:ΩRmR denotes the k-th component of the function h. So, it is possible to define the following set of fractional operators

mMOx,α,uhkZmOx,αk,uh,E17

which under the classical Hadamard product it is fulfilled that

ox0hxhxoxαmMOx,α,uh.E18

As a consequence, it is possible to define the following set of matrices

mMx,αh{Ah,α=Ah,αoxα:oxαmMOx,α,uhandAh,αx=Ah,αjkxokαhjx},E19

and therefore, considering that when using the classical Hadamard product in general oxoxoxp+qα. Assuming the existence of a fixed set of matrices mMx,αh, joined with a modified Hadamard product that fulfills the following property

by omitting the function h, the resulting set mMx,α has the ability to generate a group of fractional matrix operators Aα that fulfill the following equation

Aαoi,xAαoj,xAαoi,xoj,x,ifijAαoi,xp+qα,ifi=j,E21

through the following set [24, 26]:

mGFIMαAαr=Aαox:AαrmMx,αrZandAαr=Aαrjkok.E22

Where Ai,αp,Aj,αq,Aj,αrmGFNRα, with ij, the following property is defined

Ai,αpAj,αqAj,αr=Ai,αpAj,αqAj,αr=Ak,α1Ak,αoi,xoj,xq+rα,p,q,rZ\0,E23

since it is considered that through combinations of the Hadamard product of type horizontal and vertical the fractional operators are reduced to their minimal expression. As a consequence, it is fulfilled that

Ak,α1mGFIMαsuchthatAk,αok,xα=Ak,αoi,xoj,xAk,αr=Ak,αr1Ak,α1=Ak,αoi,xrpαoj,xrqα.E24

It is necessary to mention that for each operator oxαmMOx,α,uh it is possible to define a group [26], which is isomorphic to the group of integers under the addition, as shown by the following theorems:

Theorem 1.1 Let oxα be a fractional operator such that oxαmMOx,α,uh. So, considering the modified Hadamard product given by (20), it is possible to define the following set of fractional matrix operators

mGAαoxαAαr=Aαox:rZandAαr=Aαrjkok,E25

which corresponds to the Abelian group generated by the operator Aαoxα.

Proof: It should be noted that due to the way the set (25) is defined, just the Hadamard product of type vertical is applied among its elements. So, Aαp,AαqmGAαoxα it is fulfilled that

AαpAαq=AαpjkAαqjk=okp+qα=Aαp+qjk=Aαp+q,E26

with which it is possible to prove that the set mGAαoxα fulfills the following properties, which correspond to the properties of an Abelian group:

Aαp,Aαq,AαrmGAαoxαitisfulfilledthatAαpAαqAαr=AαpAαqAαr0.1cmAα0mGAαoxαsuchthatAαpmGAαoxαitisfulfilledthatAα0Aαp=Aαp0.1cmAαpmGAαoxαAαpmGAαoxαsuchthatAαpAαp=Aα00.1cmAαp,AαqmGAαoxαitisfulfilledthatAαpAαq=AαqAαp.E27

Theorem 1.2 Let oxα be a fractional operator such that oxαmMOx,α,uh and let Z+ be the group of integers under the addition. So, the group generated by the operator Aαoxα is isomorphic to the group Z+, that is,

mGAαoxαZ+.E28

Proof: To prove the theorem it is enough to define a bijective homomorphism between the sets mGAαoxα and Z+. Let ψ:mGAαoxαZ+ be a function with inverse function ψ1:Z+mGAαoxα. So, the functions ψ and ψ1 may be defined as follows

ψAαr=randψ1r=Aαr,E29

with which it is possible to obtain the following results:

Aαp,AαqmGAαoxαitisfulfilledthatψAαpAαq=ψAαp+q=p+q=ψAαp+ψAαqp,qZ+itisfulfilledthatψ1p+q=Aαp+q=AαpAαq=ψ1pψ1q.E30

Therefore, from the previous results, it follows that the function ψ defines an isomorphism between the sets mGAαoxα and Z+.

Then, from the previous theorems it is possible to obtain the following corollaries:

Corollary 1.3 Let oxα be a fractional operator such that oxαmMOx,α,uh and let Z+ be the group of integers under the addition. So, considering the modified Hadamard product given by (20) and some subgroup of the group Z+, it is possible to define the following set of fractional matrix operators

mGAαoxαAαr=Aαox:randAαr=Aαrjkok,E31

which corresponds to a subgroup of the group generated by the operator Aαoxα, that is,

mGAαoxαmGAαoxα.E32

Example 1 Let Zn be the set of residual classes less than n. So, considering a fractional operator oxαmMOx,α,uh and the set Z14, it is possible to define the Abelian group of fractional matrix operators mGAαoxαZ14. Furthermore, all possible combinations of the elements of the group under the modified Hadamard product given by (20) are summarized below:

Corollary 1.4 Let h:RmRm be a function such that mMOx,α,uh. So, if it is fulfilled the following condition

oxαmMOx,α,uhmGAαoxαmGFIMα,E33

such that mGAαoxα is the group generated by the operator Aαoxα. As a consequence, it is fulfilled that

mGFIMα=oxαmMOx,α,uhmGAαoxα.E34

It is necessary to mention that the Corollary 1.3 allows generating groups of fractional operators under other operations, as shown in the following corollary:

Corollary 1.5 Let Zp+ be the set of positive residual classes less than p, with p a prime number. So, for each fractional operator oxαmMOx,α,uh, it is possible to define the following set of fractional matrix operators

mGAαoxαZp+Aαr=Aαox:rZp+andAαr=Aαrjkok,E35

which corresponds to an Abelian group under the following operation

AαrAαs=Aαrs.E36

Example 2 Let oxα be a fractional operator such that oxαmMOx,α,uh. So, considering the set Z13+, it is possible to define the Abelian group of fractional matrix operators mGAαoxαZ13+. Furthermore, all possible combinations of the elements of the group under the operation (36) are summarized below:

On the other hand, defining Aαh=Aαhjkhk, it is possible to obtain the following result:

AαrmGFIMαAh,mMx,αhsuchthatAh,AαoxAαTh.E37

Therefore, if ΦFIM denotes the iteration function of some fractional iterative method [26], it is possible to obtain the following results:

Letα0R\ZAh,α0mMx,αhΦFIM=ΦFIMAh,α0Ah,α0ΦFIMAh,α:αR\Z,E38
Letα0R\ZAα01mGFIMαΦFIM=ΦFIMAα0Aα0ΦFIMAα:αR\Z.E39

To finish this section, it is necessary to mention that the applications of fractional operators have spread to different fields of science such as finance [27, 28], economics [29], number theory through the Riemann zeta function [30, 31], and in engineering with the study for the manufacture of hybrid solar receivers [32, 33]. It is worth mentioning that there exists also a growing interest in fractional operators and their properties for solving nonlinear algebraic systems [24, 34, 35, 36, 37, 38, 39, 40, 41], which is a classical problem in mathematics, physics and engineering, which consists of finding the set of zeros of a function f:ΩRnRn, that is,

ξΩ:fξ=0,E40

where :RnR denotes any vector norm, or equivalently

ξΩ:fkξ=0k1.E41

Although finding the zeros of a function may seem like a simple problem, it is generally necessary to use numerical methods of the iterative type to solve it.

## 4. Fixed-point method

Let Φ:RnRn be a function. It is possible to build a sequence xii1 by defining the following iterative method

xi+1Φxi,i=0,1,2,.E42

So, if it is fulfilled that xiξRn and the function Φ is continuous around ξ, we obtain that

ξ=limixi+1=limiΦxi=Φlimixi=Φξ,E43

the above result is the reason by which the method (42) is known as the fixed-point method. Furthermore, the function Φ is called an iteration function. On the other hand, considering the following set

Bξδx:xξ<δ,E44

it is possible to define the following corollary, which allows characterizing the order of convergence of an iteration function Φ through its Jacobian matrix Φ1 [7]:

Corollary 1.6 Let Φ:RnRn be an iteration function. If Φ defines a sequence xii1 such that xiξRn. So, Φ has an order of convergence of order (at least) p in Bξδ, where it is fulfilled that:

p1,iflimxξΦ1x02,iflimxξΦ1x=0.E45

## 5. Fractional fixed-point method

Let N0 be the set N0, if γN0m and xRm, then it is possible to define the following multi-index notation

γ!k=1mγk!,γk=1mγk0.1cm,xγk=1mxkγkγxγγ1x1γ1γ2x2γ2γmxmγm.E46

So, considering a function h:ΩRmR and the fractional operator

sxαγoxαo1αγ1o2αγ2omαγm,E47

it is possible to define the following set of fractional operators

Sx,αn,γh{sxαγ=sxαγoxα:sxαγhxwithoxαOx,αshsn2
andlimαksxαγhx=xhxα,γn},E48

from which it is possible to obtain the following results:

IfsxαγSx,αn,γhlimα0sxαγhx=o10o20om0hx=hxlimα1sxαγhx=o1γ1o2γ2omγmhx=γxγhxγnlimαqsxαγhx=o1qγ1o2qγ2omqγmhx=xhxqγqnlimαnsxαγhx=o1nγ1o2nγ2omnγmhx=xhxnγn2,E49

and as a consequence, considering a function h:ΩRmRm, it is possible to define the following set of fractional operators

mSx,αn,γhsxαγ:sxαγSx,αn,γhkkm.E50

On the other hand, using little-o notation it is possible to obtain the following result:

IfxBaδlimxaoxaγxaγ0γ1,E51

with which it is possible to define the following set of functions

Rαγnarαγn:limxarαγnx=0γnandrαγnxoxanxBaδ,E52

where rαγn:BaδΩRm. So, considering the previous set and some BaδΩ, it is possible to define the following sets of fractional operators

mTx,α,pn,q,γah{txα,p=txα,psxαγ:sxαγmSx,αM,γhand
txα,phxγ=0p1γ!êjsxαγhjaxaγ+rαγpxαnpq},E53
mTx,α,γah{txα,=txα,sxαγ:sxαγmSx,α,γhand
txα,hxγ=01γ!êjsxαγhjaxaγ},E54

which allow generalizing the Taylor series expansion of a vector-valued function in multi-index notation [7], where M=maxnq. As a consequence, it is possible to obtain the following results:

Iftxα,pmTx,α,p1,q,γahandα1tx1,phx=ha+γ=1p1γ!êjγxγhjaxaγ+rγpx,E55
Iftxα,pmTx,α,pn,1,γahandp1txα,1hx=ha+k=1mêjokαhjaxak+rαγ1x.E56

Let f:ΩRnRn be a function with a point ξΩ such that fξ=0. So, for some xiBξδΩ and for some fractional operator txα,nTx,α,γxif, it is possible to define a type of linear approximation of the function f around the value xi as follows

txα,fxfxi+k=1nêjokαfjxixxik,E57

which may be rewritten more compactly as follows

txα,fxfxi+okαfjxixxi.E58

where okαfjxi denotes a square matrix. On the other hand, if xξ and since fξ=0, it follows that

0fxi+okαfjxiξxiξxiokαfjxi1fxi.E59

So, defining the following matrix

Af,αx=Af,αjkxokαfjx1,E60

it is possible to define the following fractional iterative method

xi+1Φαxi=xiAf,αxifxi,i=0,1,2,,E61

which corresponds to the more general case of the fractional Newton-Raphson method [25, 36].

Let f:ΩRnRn be a function with a point ξΩ such that fξ=0. So, considering an iteration function Φ:R\Z×RnRn, the iteration function of a fractional iterative method may be written in general form as follows

ΦαxxAg,αxfx,αR\Z,E62

where Ag,α is a matrix that depends, in at least one of its entries, on fractional operators of order α applied to some function g:RnRn, whose particular case occurs when g=f. So, it is possible to define in a general way a fractional fixed-point method as follows

xi+1Φαxi,i=0,1,2,.E63

Before continuing, it is worth mentioning that one of the main advantages of fractional iterative methods is that the initial condition x0 can remain fixed, with which it is enough to vary the order α of the fractional operators involved until generating a sequence convergent xii1 to the value ξΩ. Since the order α of the fractional operators is varied, different values of α can generate different convergent sequences to the same value ξ but with a different number of iterations. So, it is possible to define the following set

ConvδξΦ:limxξΦαx=ξαB(ξδ)E64

which may be interpreted as the set of fractional fixed-point methods that define a convergent sequence xii1 to some value ξαBξδ. So, denoting by card the cardinality of a set, under certain conditions it is possible to prove the following result (see reference [24], proof of Theorem 2):

cardConvδξ=cardR,E65

from which it follows that the set (64) is generated by an uncountable family of fractional fixed-point methods. Before continuing, it is necessary to define the following proposition [7]:

Proposition 1.7 Let Φ:R\Z×RnRn be an iteration function such that ΦConvδξ in a region Ω. So, if Φ is given by the equation (62) and fulfills the following condition

limxξAg,αx=f1ξ1.E66

Then, Φ fulfills a necessary (but not sufficient) condition to be convergent of order (at least) quadratic in Bξδ.

Proof: If Φ is given by the equation (62), the k-th component of the function Φ may be written as follows

Φkαx=xkj=1nAg,αkjxfjx,E67

and considering that f1x=f1jlxlfjx, it is possible to obtain the following result

Φ1klαx=lΦkαx=δklj=1nAg,αkjxf1jlx+lAg,αkjxfjx,E68

where δkl denotes the Kronecker delta. On the other hand, since f has a point ξΩ such that fξ=0, it follows that

Φ1klαξ=δklj=1nAg,akjξf1jlξ.E69

Then, if ΦConvδξ and has an order of convergence (at least) quadratic in Bξδ, by the Corollary 1.6, it is fulfilled the following condition

j=1nAg,akjξf1jlξ=δkl,k,ln,E70

which may be rewritten more compactly as follows

Ag,aξf1ξ=In,E71

where In denotes the identity matrix of n×n. Therefore, any matrix Ag,a that fulfills the following condition

limxξAg,ax=f1ξ1,E72

ensures that the iteration function Φ given by the equation (62), fulfills a necessary (but not sufficient) condition to be convergent of order (at least) quadratic in Bξδ.

Considering the Corollary 1.6 and the Proposition 1.7, it is possible to define the following sets to classify the order of convergence of some fractional iterative methods:

Ord1ξΦConvδξ:limxξΦ1ax0,E73
Ord2ξΦConvδξ:limxξΦ1ax=0,E74
ord1ξΦConvδξ:limxξAg,axf1ξ1orlimα1Ag,aξf1ξ1,E75
ord2ξΦConvδξ:limxξAg,axf1ξ1orlimα1Ag,aξf1ξ1,E76

On the other hand, considering that depending on the nature of the function f, there exist cases in which the Newton-Raphson method can present an order of convergence (at least) linear [7]. So, it is possible to obtain the following relations between the previous sets

ord1ξOrd1ξandord2ξOrd1ξOrd2ξ,E77

with which it is possible to define the following sets

Ord21ξord2ξOrd1ξandOrd22ξord2ξOrd2ξ.E78

### 5.1 Acceleration of the order of convergence of the set Ord21ξ

Let f:ΩRnRn be a function with a point ξΩ such that fξ=0, and denoting by ΦNR to the iteration function of the Newton-Raphson method, it is possible to define the following set of functions

OrdNR2ξf:limxξΦNR1x=0.E79

So, it is possible to define the following corollary:

Corollary 1.8 Let f:ΩRnRn be a function such that fOrdNR2ξ, and let Φ:R\Z×RnRn be an iteration function given by the equation (62) such that Φord1ξ. So, if Φ also fulfills the following condition

limα1Ag,aξ=f1ξ1.E80

Then, ΦOrd21ξ. Therefore, it is possible to assign a positive value δ0, and replace the order α of the fractional operators of the matrix Ag,α by the following function

αfxkxα,ifxk0andfxδ01ifxk=0orfxδ0E81

obtaining a new matrix that may be denoted as follows

Ag,αfx=Ag,αfjkx,αR\Z,E82

and with which it is fulfilled that ΦOrd22ξ.

It is necessary to mention that, for practical purposes, it may be defined that if a fractional iterative method Φ fulfills the properties of the Corollary 1.8 and uses the function (81), it may be called a fractional iterative method accelerated. Finally, it is necessary to mention that fractional iterative methods may be defined in the complex space [24], that is,

Φαx:αR\ZandxCn.E83

However, due to the part of the integral operator that fractional operators usually have, it may be considered that in the matrix Ag,α each fractional operator okα is obtained for a real variable xk, and if the result allows it, this variable is subsequently substituted by a complex variable xik, that is,

Ag,αxiAg,αxxxi,xRn,xiCn.E84

Therefore, it is possible to obtain the following corollaries:

Corollary 1.9 Let f:ΩCnCn be a function such that fOrdNR2ξ, let g:CnCn be a function such that g1x=f1xxBξδ, and let Φ:R\Z×CnCn be an iteration function given by the equation (62). So, for each operator oxαnOx,α1g such that there exists the matrix Ag,α1=AαoxαAαTg, it follows that the matrix fulfills the following condition

limα1Ag,αx=f1x1xBξδ.E85

As a consequence, by the Corollary 1.8, if ΦAg,αOrd21ξΦAg,αfOrd22ξ.

Corollary 1.10 Let f:ΩCnCn be a function such that fOrdNR2ξ, let gkk=1N be a finite sequence of functions gk:CnCn such that it defines a finite sequence of operators ok,xαk=1N through the following condition

ok,xαnMOx,α,ugkk1,E86

and let Φ:R\Z×CnCn be an iteration function given by the Eq. (62). So, if there exists a matrix AN,α such that it fulfills the following conditions

AN,α1=k=1NAαok,xαAαTgkandlimα1AN,αx=f1x1xBξδ.E87

As a consequence, by the Corollary 1.8, if ΦAN,αOrd21ξΦAN,αfOrd22ξ.

## 6. Conclusions

It is worth mentioning that it is feasible to develop more complex algebraic structures of fractional operators using the presented results. For example, without loss of generality, considering the modified Hadamard product (20) and the operation (36), a commutative and unitary ring of fractional operators may be defined as follows

RAαoxαGAαoxα,,mm,E88

in which it is not difficult to verify the following properties:

1. The pair GAαoxα,m is an Abelian group.

2. The pair GAαoxα,m is an Abelian monoid.

3. Aαp,Aαq,AαrRAαoxαm, the operation is distributive with respect to the operation , that is,

AαpAαqAαr=AαpAαqAαpAαrAαpAαqAαr=AαpAαrAαqAαr.E89

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Written By

A. Torres-Hernandez, F. Brambila-Paz and R. Ramirez-Melendez

Reviewed: 19 August 2022 Published: 26 October 2022