intechopen.com

Sets of Fractional Operators and Some of Their Applications

️Wed Oct 26 2022

Open access peer-reviewed chapter

Reviewed: 19 August 2022 Published: 26 October 2022

DOI: 10.5772/intechopen.107263

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

A. Torres-Hernandez*
- Faculty of Science, Department of Physics, Universidad Nacional Autónoma de México, Mexico City, Mexico
- Department of Information and Communication Technologies, Music and Machine Learning Lab, Universitat Pompeu Fabra, Barcelona, Spain
F. Brambila-Paz
- Faculty of Science, Department of Mathematics, Universidad Nacional Autónoma de México, Mexico City, Mexico
R. Ramirez-Melendez
- Department of Information and Communication Technologies, Music and Machine Learning Lab, Universitat Pompeu Fabra, Barcelona, Spain

*Address all correspondence to: anthony.torres@ciencias.unam.mx

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

On the other hand, a fractional differential equation is an equation that involves at least one differential operator of order α, with n−1<α≤n 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 α→n. For example, let f:Ω⊂R→R be a function such that f∈Lloc1ab, where Lloc1ab denotes the space of locally integrable functions on the open interval ab⊂Ω. One of the fundamental operators of fractional calculus is the operator Riemann−Liouville fractional integral, which is defined as follows [4, 5]:

aIxαfx≔1Γα∫axx−tα−1ftdt,

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]:

aDxαfx≔aIx−αfx,ifα<0dndxnaIxn−αfx,ifα≥0,

where n=α and aIx0fx≔fx. On the other hand, let f:Ω⊂R→R be a function n-times differentiable such that f,fn∈Lloc1ab. Then, the Riemann−Liouville fractional integral also allows constructing the operator Caputo fractional derivative, which is defined as follows [4, 6]:

aCDxαfx≔aIx−αfx,ifα<0aIxn−αfnx,ifα≥0,

where n=α and aIx0fnx≔fnx. Furthermore, if the function f fulfills that fka=0∀k∈01⋯n−1, 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+1Γμ−α+1xμ−α,α∈R\Z,

where if 1≤α≤μ 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:Rm→R and the canonical basis of Rm denoted by êkk≥1, it is possible to define the following fractional operator of order α using Einstein notation

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,αnh≔oxα:∃okαhxandlimα→nokαhx=∂knhx∀k≥1,

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

O0,x,αnh≔oxα:∃okαhx=∂kn+μα∂kαhxandlimα→nμα∂kαhx=0∀k≥1,

with which it is possible to obtain the following result:

Ifoi,xα,oj,xα∈Ox,αnhwithi≠j⇒∃ok,xα=12oi,xα+oj,xα∈Ox,αnh.

E10

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

Ox,αn,ch≔{oxα:∃okαhx∀k≥1andlimα→nokαhx≠∂knhxinatleastonevaluek≥1},

E11

with which it is possible to obtain the following result:

Ifoi,xα=êkoi,kα∈Ox,αnh⇒∃oj,xα=êkoi,σjkα∈Ox,αn,ch,

E12

where σj:12⋯m→12⋯m 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,n∈Rd with α=êkαk and n=êknk, it is possible to define the following set of fractional tensor operators

Ox,αnh≔oxα:∃oxαhxandoxα∈Ox,α1n1h×Ox,α2n2h×⋯×Ox,αdndh.

E13

3. Groups of fractional operators

Considering a function h:Ω⊂Rm→Rm, it is possible to define the following sets

mOx,αnh≔oxα:oxα∈Ox,αnhk∀k≤m,

E14

mOx,αn,ch≔oxα:oxα∈Ox,αn,chk∀k≤m,

E15

mOx,αn,uh≔mOx,αnh∪mOx,αn,ch,

E16

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

mMOx,α∞,uh≔⋂k∈ZmOx,αk,uh,

E17

which under the classical Hadamard product it is fulfilled that

ox0∘hx≔hx∀oxα∈mMOx,α∞,uh.

E18

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

mMx,α∞h≔{Ah,α=Ah,αoxα:oxα∈mMOx,α∞,uhandAh,αx=Ah,αjkx≔okαhjx},

E19

and therefore, considering that when using the classical Hadamard product in general oxpα∘oxqα≠oxp+qα. Assuming the existence of a fixed set of matrices mMx,α∞h, joined with a modified Hadamard product that fulfills the following property

oi,xpα∘oj,xqα≔oi,xpα∘oj,xqα,ifi≠jHadamardproductoftypehorizontaloi,xp+qα,ifi=jHadamardproductoftypevertical,

E20

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,xpα∘Aαoj,xqα≔Aαoi,xpα∘oj,xqα,ifi≠jAαoi,xp+qα,ifi=j,

E21

through the following set [24, 26]:

mGFIMα≔Aα∘r=Aαoxrα:∃Aα∘r∈mMx,α∞⋅∀r∈ZandAα∘r=Aα∘rjk≔okrα.

E22

Where ∀Ai,α∘p,Aj,α∘q,Aj,α∘r∈mGFNRα, with i≠j, the following property is defined

Ai,α∘p∘Aj,α∘q∘Aj,α∘r=Ai,α∘p∘Aj,α∘q∘Aj,α∘r=Ak,α∘1≔Ak,αoi,xpα∘oj,xq+rα,p,q,r∈Z\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,α∘1∈mGFIMαsuchthatAk,αok,xα=Ak,αoi,xpα∘oj,xqα∃Ak,α∘r=Ak,α∘r−1∘Ak,α∘1=Ak,αoi,xrpα∘oj,xrqα.

E24

It is necessary to mention that for each operator oxα∈mMOx,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 oxα∈mMOx,α∞,uh. So, considering the modified Hadamard product given by (20), it is possible to define the following set of fractional matrix operators

mGAαoxα≔Aα∘r=Aαoxrα:r∈ZandAα∘r=Aα∘rjk≔okrα,

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, ∀Aα∘p,Aα∘q∈mGAαoxα it is fulfilled that

Aα∘p∘Aα∘q=Aα∘pjk∘Aα∘qjk=okp+qα=Aα∘p+qjk=Aα∘p+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:

∀Aα∘p,Aα∘q,Aα∘r∈mGAαoxαitisfulfilledthatAα∘p∘Aα∘q∘Aα∘r=Aα∘p∘Aα∘q∘Aα∘r0.1cm∃Aα∘0∈mGAαoxαsuchthat∀Aα∘p∈mGAαoxαitisfulfilledthatAα∘0∘Aα∘p=Aα∘p0.1cm∀Aα∘p∈mGAαoxα∃Aα∘−p∈mGAαoxαsuchthatAα∘p∘Aα∘−p=Aα∘00.1cm∀Aα∘p,Aα∘q∈mGAαoxαitisfulfilledthatAα∘p∘Aα∘q=Aα∘q∘Aα∘p.

E27

Theorem 1.2 Let oxα be a fractional operator such that oxα∈mMOx,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,

Proof: To prove the theorem it is enough to define a bijective homomorphism between the sets mGAαoxα and Z+. Let ψ:mGAαoxα→Z+ be a function with inverse function ψ−1:Z+→mGAαoxα. So, the functions ψ and ψ−1 may be defined as follows

with which it is possible to obtain the following results:

∀Aα∘p,Aα∘q∈mGAαoxαitisfulfilledthatψAα∘p∘Aα∘q=ψAα∘p+q=p+q=ψAα∘p+ψAα∘q∀p,q∈Z+itisfulfilledthatψ−1p+q=Aα∘p+q=Aα∘p∘Aα∘q=ψ−1p∘ψ−1q.

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 oxα∈mMOx,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 ℍ of the group Z+, it is possible to define the following set of fractional matrix operators

mGAαoxαℍ≔Aα∘r=Aαoxrα:r∈ℍandAα∘r=Aα∘rjk≔okrα,

E31

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

Example 1 Let Zn be the set of residual classes less than n. So, considering a fractional operator oxα∈mMOx,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:Rm→Rm be a function such that ∃mMOx,α∞,uh. So, if it is fulfilled the following condition

∀oxα∈mMOx,α∞,uh∃mGAαoxα⊂mGFIMα,

E33

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

mGFIMα=⋃oxα∈mMOx,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 oxα∈mMOx,α∞,uh, it is possible to define the following set of fractional matrix operators

mG∗AαoxαZp+≔Aα∘r=Aαoxrα:r∈Zp+andAα∘r=Aα∘rjk≔okrα,

E35

which corresponds to an Abelian group under the following operation

Example 2 Let oxα be a fractional operator such that oxα∈mMOx,α∞,uh. So, considering the set Z13+, it is possible to define the Abelian group of fractional matrix operators mG∗AαoxαZ13+. Furthermore, all possible combinations of the elements of the group under the operation (36) are summarized below:

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

∀Aα∘r∈mGFIMα∃Ah,rα∈mMx,α∞hsuchthatAh,rα≔Aαoxrα∘AαTh.

E37

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

Letα0∈R\Z⇒∀Ah,α0∈mMx,α∞h∃ΦFIM=ΦFIMAh,α0∴∀Ah,α0∃ΦFIMAh,α:α∈R\Z,

E38

Letα0∈R\Z⇒∀Aα0∘1∈mGFIMα∃ΦFIM=ΦFIMAα0∴∀Aα0∃ΦFIMAα:α∈R\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:Ω⊂Rn→Rn, that is,

where ⋅:Rn→R denotes any vector norm, or equivalently

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:Rn→Rn be a function. It is possible to build a sequence xii≥1 by defining the following iterative method

So, if it is fulfilled that xi→ξ∈Rn and the function Φ is continuous around ξ, we obtain that

ξ=limi→∞xi+1=limi→∞Φxi=Φlimi→∞xi=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

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:Rn→Rn be an iteration function. If Φ defines a sequence xii≥1 such that xi→ξ∈Rn. So, Φ has an order of convergence of order (at least) p in Bξδ, where it is fulfilled that:

p≔1,iflimx→ξΦ1x≠02,iflimx→ξΦ1x=0.

E45

5. Fractional fixed-point method

Let N0 be the set N∪0, if γ∈N0m and x∈Rm, then it is possible to define the following multi-index notation

γ!≔∏k=1mγk!,γ≔∑k=1mγk0.1cm,xγ≔∏k=1mxkγk∂γ∂xγ≔∂γ1∂x1γ1∂γ2∂x2γ2⋯∂γm∂xmγm.

E46

So, considering a function h:Ω⊂Rm→R and the fractional operator

sxαγoxα≔o1αγ1o2αγ2⋯omαγm,

E47

it is possible to define the following set of fractional operators

Sx,αn,γh≔{sxαγ=sxαγoxα:∃sxαγhxwithoxα∈Ox,αsh∀s≤n2

andlimα→ksxαγhx=∂kγ∂xkγhx∀α,γ≤n},

E48

from which it is possible to obtain the following results:

Ifsxαγ∈Sx,αn,γh⇒limα→0sxαγhx=o10o20⋯om0hx=hxlimα→1sxαγhx=o1γ1o2γ2⋯omγmhx=∂γ∂xγhx∀γ≤nlimα→qsxαγhx=o1qγ1o2qγ2⋯omqγmhx=∂qγ∂xqγhx∀qγ≤qnlimα→nsxαγhx=o1nγ1o2nγ2⋯omnγmhx=∂nγ∂xnγhx∀nγ≤n2,

E49

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

mSx,αn,γh≔sxαγ:sxαγ∈Sx,αn,γhk∀k≤m.

E50

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

Ifx∈Baδ⇒limx→aox−aγx−aγ→0∀γ≥1,

E51

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

Rαγna≔rαγn:limx→arαγnx=0∀γ≥nandrαγnx≤ox−an∀x∈Baδ,

E52

where rαγn:Baδ⊂Ω→Rm. So, considering the previous set and some Baδ⊂Ω, it is possible to define the following sets of fractional operators

mTx,α,pn,q,γah≔{txα,p=txα,psxαγ:sxαγ∈mSx,αM,γhand

txα,phx≔∑γ=0p1γ!êjsxαγhjax−aγ+rαγpx∀α≤n∀p≤q},

E53

mTx,α∞,γah≔{txα,∞=txα,∞sxαγ:sxαγ∈mSx,α∞,γhand

txα,∞hx≔∑γ=0∞1γ!êjsxαγhjax−aγ},

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,p∈mTx,α,p1,q,γahandα→1⇒tx1,phx=ha+∑γ=1p1γ!êj∂γ∂xγhjax−aγ+rγpx,

E55

Iftxα,p∈mTx,α,pn,1,γahandp→1⇒txα,1hx=ha+∑k=1mêjokαhjax−ak+rαγ1x.

E56

Let f:Ω⊂Rn→Rn be a function with a point ξ∈Ω such that fξ=0. So, for some xi∈Bξδ⊂Ω 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,∞fx≈fxi+∑k=1nêjokαfjxix−xik,

E57

which may be rewritten more compactly as follows

txα,∞fx≈fxi+okαfjxix−xi.

E58

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

0≈fxi+okαfjxiξ−xi⇒ξ≈xi−okαfjxi−1fxi.

E59

So, defining the following matrix

Af,αx=Af,αjkx≔okαfjx−1,

E60

it is possible to define the following fractional iterative method

xi+1≔Φαxi=xi−Af,αxifxi,i=0,1,2,⋯,

E61

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

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

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:Rn→Rn, whose particular case occurs when g=f. So, it is possible to define in a general way a fractional fixed-point method as follows

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 xii≥1 to the value ξ∈Ω. 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

Convδξ≔Φ:limx→ξΦαx=ξα∈B(ξδ)

E64

which may be interpreted as the set of fractional fixed-point methods that define a convergent sequence xii≥1 to some value ξα∈Bξδ. 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):

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×Rn→Rn be an iteration function such that Φ∈Convδξ in a region Ω. So, if Φ is given by the equation (62) and fulfills the following condition

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=xk−∑j=1nAg,αkjxfjx,

E67

and considering that f1x=f1jlx≔∂lfjx, it is possible to obtain the following result

Φ1klαx=∂lΦkαx=δkl−∑j=1nAg,αkjxf1jlx+∂lAg,αkjxfjx,

E68

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

Φ1klαξ=δkl−∑j=1nAg,akjξf1jlξ.

E69

Then, if Φ∈Convδξ 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,l≤n,

E70

which may be rewritten more compactly as follows

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

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:

Ord1ξ≔Φ∈Convδξ:limx→ξΦ1ax≠0,

E73

Ord2ξ≔Φ∈Convδξ:limx→ξΦ1ax=0,

E74

ord1ξ≔Φ∈Convδξ:limx→ξAg,ax≠f1ξ−1orlimα→1Ag,aξ≠f1ξ−1,

E75

ord2ξ≔Φ∈Convδξ:limx→ξAg,ax≠f1ξ−1orlimα→1Ag,aξ≠f1ξ−1,

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

ord1ξ⊂Ord1ξandord2ξ⊂Ord1ξ∪Ord2ξ,

E77

with which it is possible to define the following sets

Ord21ξ≔ord2ξ∩Ord1ξandOrd22ξ≔ord2ξ∩Ord2ξ.

E78

5.1 Acceleration of the order of convergence of the set Ord21ξ

Let f:Ω⊂Rn→Rn be a function with a point ξ∈Ω 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

OrdNR2ξ≔f:limx→ξΦNR1x=0.

E79

So, it is possible to define the following corollary:

Corollary 1.8 Let f:Ω⊂Rn→Rn be a function such that f∈OrdNR2ξ, and let Φ:R\Z×Rn→Rn be an iteration function given by the equation (62) such that Φ∈ord1ξ. So, if Φ also fulfills the following condition

Then, Φ∈Ord21ξ. 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

αfxkx≔α,if∣xk∣≠0andfx≥δ01if∣xk∣=0orfx≥δ0

E81

obtaining a new matrix that may be denoted as follows

Ag,αfx=Ag,αfjkx,α∈R\Z,

E82

and with which it is fulfilled that Φ∈Ord22ξ.

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,

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,αxi≔Ag,αxx−→xi,x∈Rn,xi∈Cn.

E84

Therefore, it is possible to obtain the following corollaries:

Corollary 1.9 Let f:Ω⊂Cn→Cn be a function such that f∈OrdNR2ξ, let g:Cn→Cn be a function such that g1x=f1x∀x∈Bξδ, and let Φ:R\Z×Cn→Cn be an iteration function given by the equation (62). So, for each operator oxα∈nOx,α1g such that there exists the matrix Ag,α−1=Aαoxα∘AαTg, it follows that the matrix fulfills the following condition

limα→1Ag,αx=f1x−1∀x∈Bξδ.

E85

As a consequence, by the Corollary 1.8, if ΦAg,α∈Ord21ξ⇒ΦAg,αf∈Ord22ξ.

Corollary 1.10 Let f:Ω⊂Cn→Cn be a function such that f∈OrdNR2ξ, let gkk=1N be a finite sequence of functions gk:Cn→Cn such that it defines a finite sequence of operators ok,xαk=1N through the following condition

ok,xα∈nMOx,α∞,ugk∀k≥1,

E86

and let Φ:R\Z×Cn→Cn 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,α−1=∑k=1NAαok,xα∘AαTgkandlimα→1AN,αx=f1x−1∀x∈Bξδ.

E87

As a consequence, by the Corollary 1.8, if ΦAN,α∈Ord21ξ⇒ΦAN,αf∈Ord22ξ.

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

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

The pair GAαoxα,∘m is an Abelian group.
The pair GAαoxα,∗m is an Abelian monoid.
∀Aα∘p,Aα∘q,Aα∘r∈RAαoxαm, the operation ∗ is distributive with respect to the operation ∘, that is,

Aα∘p∗Aα∘q∘Aα∘r=Aα∘p∗Aα∘q∘Aα∘p∗Aα∘rAα∘p∘Aα∘q∗Aα∘r=Aα∘p∗Aα∘r∘Aα∘q∗Aα∘r.

E89

References

1. De Oliveira EC, Machado JAT. A review of definitions for fractional derivatives and integral. Mathematical Problems in Engineering. 2014;2014:1-6
2. Teodoro GS, Machado JAT, De Oliveira EC. A review of definitions of fractional derivatives and other operators. Journal of Computational Physics. 2019;388:195-208
3. Valério D, Ortigueira MD, Lopes AM. How many fractional derivatives are there? Mathematics. 2022;10:737
4. Hilfer R. Applications of Fractional Calculus in Physics. Singapore: World Scientific Publishing Co. Pte. Ltd.; 2000. pp. 3-73
5. Oldham K, Spanier J. The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order. London: Academic Press, Inc.; 1974. pp. 25-121
6. Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier; 2006. pp. 69-132
7. Torres-Hernandez A, Brambila-Paz F, Iturrarán-Viveros U, Caballero-Cruz R. Fractional newton-raphson method accelerated with aitken’s method. Axioms. 2021;10(2):1-25. DOI: 10.3390/axioms10020047
8. Thomas J. Leibniz rule for fractional derivatives generalized and an application to infinite series. SIAM Journal on Applied Mathematics. 1970;18(3):658-674
9. Almeida R. A caputo fractional derivative of a function with respect to another function. Communications in Nonlinear Science and Numerical Simulation. 2017;44:460-481
10. Hui F, Wu G-C, Yang G, Huang L-L. Continuous time random walk to a general fractional fokker–planck equation on fractal media. The European Physical Journal Special Topics. 2021;2021:1-7
11. Fan Q, Wu G-C, Hui F. A note on function space and boundedness of the general fractional integral in continuous time random walk. Journal of Nonlinear Mathematical Physics. 2022;29(1):95-102
12. Abu-Shady M, Kaabar MKA. A generalized definition of the fractional derivative with applications. Mathematical Problems in Engineering. 2021;2021:1-9
13. Khaled M. New fractional derivative with non-singular kernel for deriving legendre spectral collocation method. Alexandria Engineering Journal. 2020;2020:1909-1917
14. Rahmat MRS. A new definition of conformable fractional derivative on arbitrary time scales. Advances in Difference Equations. 2019;2019(1):1-16
15. Vanterler J et al. On the hilfer fractional derivative. Communications in Nonlinear Science and Numerical Simulation. 2018;60:72-91
16. Jarad F, Uğurlu E, Abdeljawad T, Baleanu D. On a new class of fractional operators. Advances in Difference Equations. 2017;2017(1):1-16
17. Atangana A, Gómez-Aguilar JF. A new derivative with normal distribution kernel: Theory, methods and applications. Physica A: Statistical Mechanics and Its Applications. 2017;476:1-14
18. Yavuz M, Özdemir N. Comparing the new fractional derivative operators involving exponential and mittag-leffler kernel. Discrete & Continuous Dynamical Systems-S. 2020;13:995
19. Liu J-G, Yang X-J, Feng Y-Y, Cui P. New fractional derivative with sigmoid function as the kernel and its models. Chinese Journal of Physics. 2020;68:533-541
20. Yang X-J, Machado JAT. A new fractional operator of variable order: Application in the description of anomalous diffusion. Physica A: Statistical Mechanics and its Applications. 2017;481:276-283
21. Atangana A. On the new fractional derivative and application to nonlinear fisher’s reaction–diffusion equation. Applied Mathematics and Computation. 2016;273:948-956
22. He J-H, Li Z-B, Wang Q-l. A new fractional derivative and its application to explanation of polar bear hairs. Journal of King Saud University-Science. 2016;28(2):190-192
23. Sene N. Fractional diffusion equation with new fractional operator. Alexandria Engineering Journal. 2020;59(5):2921-2926
24. Torres-Hernandez A, Brambila-Paz F. Sets of fractional operators and numerical estimation of the order of convergence of a family of fractional fixed-point methods. Fractal and Fractional. 2021;5(4):240
25. Torres-Hernandez A, Brambila-Paz F, Montufar-Chaveznava R. Acceleration of the order of convergence of a family of fractional fixed point methods and its implementation in the solution of a nonlinear algebraic system related to hybrid solar receivers. Applied Mathematics and Computation. 2022;429:127231. DOI: 10.1016/j.amc.2022.127231
26. Torres-Hernandez A. Code of a multidimensional fractional quasi-newton method with an order of convergence at least quadratic using recursive programming. Applied Mathematics and Sciences: An International Journal (MathSJ). 2022;9:17-24. DOI: 10.5121/mathsj.2022.9103
27. Safdari-Vaighani A, Heryudono A, Larsson E. A radial basis function partition of unity collocation method for convection–diffusion equations arising in financial applications. Journal of Scientific Computing. 2015;64(2):341-367
28. Torres-Hernandez A, Brambila-Paz F, Torres-Martínez C. Numerical solution using radial basis functions for multidimensional fractional partial differential equations of type black–scholes. Computational and Applied Mathematics. 2021;40(2):248
29. Traore A, Sene N. Model of economic growth in the context of fractional derivative. Alexandria Engineering Journal. 2020;59(6):4843-4850
30. Guariglia E. Fractional calculus, zeta functions and shannon entropy. Open Mathematics. 2021;19(1):87-100
31. Torres-Henandez A, Brambila-Paz F. An approximation to zeros of the riemann zeta function using fractional calculus. Mathematics and Statistics. 2021;9(3):309-318
32. Vega E D-l, Torres-Hernandez A, Rodrigo PM, Brambila-Paz F. Fractional derivative-based performance analysis of hybrid thermoelectric generator-concentrator photovoltaic system. Applied Thermal Engineering. 2021;193:1169
33. Torres-Hernandez A, Brambila-Paz F, Rodrigo PM, De-la-Vega E. Reduction of a nonlinear system and its numerical solution using a fractional iterative method. Journal of Mathematics and Statistical Science. 2020;6:285-299
34. Erfanifar R, Sayevand K, Esmaeili H. On modified two-step iterative method in the fractional sense: Some applications in real world phenomena. International Journal of Computer Mathematics. 2020;97(10):2109-2141
35. Cordero A, Girona I, Torregrosa JR. A variant of chebyshev’s method with 3th-order of convergence by using fractional derivatives. Symmetry. 2019;11(8):1017
36. Torres-Hernandez A, Brambila-Paz F. Fractional newton-raphson method. Applied Mathematics and Sciences: An International Journal (MathSJ). 2021;8:1-13. DOI: 10.5121/mathsj.2021.8101
37. Gdawiec K, Kotarski W, Lisowska A. Newton’s method with fractional derivatives and various iteration processes via visual analysis. Numerical Algorithms. 2021;86(3):953-1010
38. Gdawiec K, Kotarski W, Lisowska A. Visual analysis of the newton’s method with fractional order derivatives. Symmetry. 2019;11(9):1143
39. Akgül A, Cordero A, Torregrosa JR. A fractional newton method with 2th-order of convergence and its stability. Applied Mathematics Letters. 2019;98:344-351
40. Candelario G, Cordero A, Torregrosa JR. Multipoint fractional iterative methods with (2+ 1) th-order of convergence for solving nonlinear problems. Mathematics. 2020;8:452
41. Candelario G, Cordero A, Torregrosa JR, Vassileva MP. An optimal and low computational cost fractional newton-type method for solving nonlinear equations. Applied Mathematics Letters. 2022;124:107650

Written By

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