On Fractional - and bi-calculi
AMER H. DARWEESH, ABDELAZIZ M.D. MAGHRABI
Department of Mathemtics, Jordan University of Science and Technology, Irbid 22110, JORDAN
Department of Mathematics, Eastern Mediterranean University, Gazimagusa, Mersin 10, TURKEY
Abstract: - In this paper we introduce fractional - and bi-calculi using Riemann-Liouville approach
and Caputo approach as well. An effort is put into explaining the basic principles of these calculi since
they are not as common as classical calculus. This was also done for tanh-, bi-tanh- multiplicative, and
bigeometric calculi and in the general case as well. Generalizations are also investigated where the
homeomorphisms are arbitrary.
Key-Words: - Derivative, integral, non-Newtonian calculus, fractional derivative, homeomorphism.
Received: April 19, 2022. Revised: January 14, 2023. Accepted: February 9, 2023. Published: March 7, 2023.
1 Introduction
In the seventeenth century, Isaac Newton and
Gottfried Leibniz laid the foundations for the
classical -or sometimes called- Newtonian
calculus. That particular calculus has proved its
mathematical strength. Indeed, it is the most
applicable theory used in sciences. Fractional
calculus, even though it is usually thought that
it is a relatively new subject, it has dated back
to 1695 when L’Hoptial wrote to Leibnitz
asking about the interpretation of
when
, see [1]. In the previous century, many
mathematicians have given different
perspectives and approaches in an attempt to
answer this question. The same question arises
when one considers
or
. These are the
multiplicative and bigeometric derivatives
respectively. In the period 1967 to 1970,
Michael Grossman and Robert Katz initiated
many calculi considering different operations
and viewing classical calculus as an additive
type that depend on addition and subtraction as
their foundation [2]. Using that view, they came
up with what we call multiplicative and
bigeometric calculi [1-6], that which depends
on multiplication and division. More precisely,
defining - to represent the main
algebraic operations performed on . The
function is a bijection from onto an interval
that induced the field and metric structures
from onto . Letting , we see that
on , the exponential-operations give
rise to two pairs of calculi on functions
. This will be further explained later on in the
second section. This paper is organized in the
following way. In Section 2, we explain briefly
the principles of - and bi- calculi, and give
examples regarding multiplicative and
bigeometric calculi. Moreover, we mention the
Newtonian versions of Caputo and Riemann-
Liouville approaches to this subject. Then, we
introduce some theorems for - and bi-calculi
and we also mention theorems from [6] as well
which are the stepping stones used in this paper.
In Section 3, we define - and bi- fractional
calculi, and based on them we also define it
with respect to non-Newtonian calculi, which
are the bigeometric, tanh-, and bi-tanh-
fractional calculi. The multiplicative case is
discussed in [6]. Moreover, we mention some
results which are the relations between - and
bi- fractional calculi and the Newtonian
fractional calculi considering the mentioned
approaches. The notation is rather different than
the one that was introduced in [2], which
denotes the bijection as instead of . This
was done because the letter is more
convenient when discussing fractional calculi
since is commonly used for denoting the
order.
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.10
Amer H. Darweesh, Abdelaziz M. D. Maghrabi
E-ISSN: 2224-2678
87
Volume 22, 2023
2 Elements of - and bi-
Differentiation
Consider an increasing homeomorphism
. For , we define the following
operations:
1.
2.
3.
4.
5. if and only if
It is easy to check that under the above
operations becomes an ordered field. We call
this field the -non-Newtonian interval.
Moreover, the following real-valued function
defines a metric on :
6.
Moreover, for any , we define the -
power of by
It is easy to check that the following properties
are true:
1. For , we have
2. For , we have
3. For , we have
Remark 1 This metric is compatible with the
operations on the field , that is, the above
operations are continuous.
With this metric we can define, as usual, the
limit of a function that is defined on a -non-
Newtonian interval .
Definition 2 Let be -interval, and .
For , we define
to be the limit from the metric to itself.
That is, if as .
In the next proposition, we see the relation
between the usual limit and the -limit.
Proposition 3 Let be a -interval, and
. Then,
Proof. Let
. By the
definition of bi-limit, we have
In other words,
Hence,
That is,
Therefore,
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.10
Amer H. Darweesh, Abdelaziz M. D. Maghrabi
E-ISSN: 2224-2678
88
Volume 22, 2023
Definition 4 Let be a -interval, and
. For , we define
to be the limit from the usual metric on to the
metric . That is, as
.
Proposition 5 Let be a -interval, and
. Then,
Proof. Let
. By the
definition of -limit, we have
Therefore,
Or equivalently,
Based on these types of limits, we can
develop - and bi-calculi, that is to define a
derivative and an integral with respect to -
operations.
Remark 6 From now on, for the sake of
convenience and brevity, we will use the
operations , , , and instead of ,
, , and .
Definition 7 The bi-derivative of a function
, where with is denoted
and given by
(1)
Consider Equation 1. Using Proposition 3, one
has
This yields the following results.
Proposition 8 The bi-derivative of a function
, where is a -interval,is given by
or in the other notations,
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.10
Amer H. Darweesh, Abdelaziz M. D. Maghrabi
E-ISSN: 2224-2678
89
Volume 22, 2023
If we denote the bi-derivative of by
, we can easily obtain
the following result.
Proposition 9 Let be a -interval, and
. Then,
(2)
or in the other notations,
Conversely, we can write the ordinary
derivative in terms of bi-derivative as follows.
Proposition 10 Let , be a -
interval. Then, and
or in the other notations,
Example 11 Take , with
, then for functions , we can use
Proposition 8 to define the bigeometric
derivative as follows
as it is expected in the bigeometric calculus.
If we define from the Newtonian field into
the -non-Newtonian interval , we can
introduce a weaker version of differentiablity
and integrability.
Definition 12 The -derivative of a function
, where with is
denoted and given by
Proposition 13 The -derivative of a function
, where is a -interval, is given by
Proposition 14 Let . Then, is -
differentiable at if and only if is
bi-differentiable at . In this case,
.
Proof. The proof follows from Propositions 8
and 13.
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.10
Amer H. Darweesh, Abdelaziz M. D. Maghrabi
E-ISSN: 2224-2678
90
Volume 22, 2023
Theorem 15 Let be differentiable
with on , and let . Then
is differentiable if and only if is -
differentiable.
Proof. Let be differentiable at . By the
inverse function theorem, the function
is differentiable on . Hence, the function
is differentiable at . That is,
exists. Therefore,
By Proposition 13, we have .
Hence, is -differentiable at . Now, suppose
that is -differentiable. Then,
and hence
It follows that the function is
differentiable at . By the chain rule and the
fact that is differentiable everywhere, we
conclude that is
differentiable at .
Theorem 16 Let be differentiable
with on , and let . Then
is differentiable if and only if is bi-
differentiable. Moreover,
Proof. Let be differentiable at . By the
inverse function theorem, the function
is differentiable on . Hence, the function
is differentiable at . Let , then
is differentiable at . By the chain rule,
is differentiable at , and
Therefore,
By Proposition 8, exists and
Now, suppose that is bi-differentiable.
Then,
Therefore,
Since
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.10
Amer H. Darweesh, Abdelaziz M. D. Maghrabi
E-ISSN: 2224-2678
91
Volume 22, 2023
we have
It follows that the function is
differentiable at . By the chain rule, we
conclude that is
differentiable at .
We denote the -derivative by .
With this notation, one can obtain the following
result.
Theorem 17 Let , then
Proof. The proof will be done using
mathematical induction and Proposition 13.
Example 18 For and ,
we obtain the geometric derivative (which is
also called derivative or multiplicative
derivative) of .
(3)
By Theorem , we have
(4)
Moreover, one immediately realizes the
relation
(5)
The multiplicative derivative and the additive
derivative can be used to express each other.
Indeed, we have the following equation
(6)
Using Faà di Bruno formula on equation (4),
one also arrives at the following
(7)
For a simpler variant of Faà di Bruno formula,
refer to [7]. This gives a brief overview of
multiplicative and bigeometric calculi.
Example 19 The tanh-derivative for
, is denoted and given by
(8)
Moreover, the order tanh-derivative is
given by
(9)
Theorem 20 Let be -times
differentiable, then
(10)
Thus, the bi- derivative is a Gauss vector of
the -derivative. Equivalently,
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.10
Amer H. Darweesh, Abdelaziz M. D. Maghrabi
E-ISSN: 2224-2678
92
Volume 22, 2023
(11)
(12)
Proof. This proof will be done using
mathematical induction. Let then
we have . This implies,
.
Hence, the theorem is true at . Assume
that it is true for , by the induction
hypothesis we have,
This concludes the proof. The other forms are
obtained by manipulating the substitution
Remark 21 The first form which includes the
variables and were introduced to obtain a
simple proof.
Example 22 Let be -times differentiable,
then
(13)
Which has equivalent forms,
(14)
(15)
Example 23 Let be -times
. Then,
(16)
Which are equivalent to the forms,
(17)
. (18)
Remark 24 By Theorem (20), we can
comprehend the relation between - and bi-
calculi. Indeed, -calculus is not only a
weakened version of bi-calculus, rather bi-
calculus is the change in -calculus with
respect to , which is equivalent to
stating that -calculus is the change in bi-
calculus with respect to
3 Elements of - and bi-Riemann
integration
Using the structure of the metric , one can
define the boundedness of . Precisely,
is -bounded if , for all
. That is, if , for all
.
Definition 25 Let be -bounded. Let
in , and be a
partition on . The function is called bi-
Riemann integrable if there is such that
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.10
Amer H. Darweesh, Abdelaziz M. D. Maghrabi
E-ISSN: 2224-2678
93
Volume 22, 2023
for any and any choice of ,
there is a satisfies:
whenever In this case we
write,
Remark 26 The definition above is independent
of the choice of . That is, if the
above limit exists, then for any choice of ,
the limit is the same.
It is worth mentioning that if is
Riemann integrable, and is piecewise
continuously differentiable on , then
(19)
Example 27 The bigeometric integral, denoted
is given by
Definition 28 Let be -bounded. Let
in , and be partition
on . The function is called -Riemann
integrable if there is such that for any
and any choice of , there is a
satisfies:
whenever
In this case we write
It is clear that from the definition above if
is Riemann integrable, and hence is
Riemann integrable, then is -Riemann
integrable and
Example 29 The geometric integral, denoted
is given by
Theorem 30 Let
be the -integral
of . Then,
(20)
Proof. The proof will be done using
mathematical induction. For it is clear.
Assume that it holds true for , then
we get
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.10
Amer H. Darweesh, Abdelaziz M. D. Maghrabi
E-ISSN: 2224-2678
94
Volume 22, 2023
Example 31 It is clear from the definition of the
multiplicative integral that
(21)
Additionally, we have the tanh-integral
(22)
4 Definitions of fractional - and bi-
calculi
For , the Riemann-Liouville
integral is given by,
Moreover, for , , by
analytic continuation of the RL-integral to
, the Riemann-Liouville fractional
derivative is given by
Whereas the Caputo derivative is given by,
Definition 32 For , define the -
Gamma function by
The definitions that will be discussed in this
section are dealt with in a similar fashion to that
logic used in the definitions of Riemann-
Liouville and Caputo. Indeed, we have the
following definitions:
Definition 33 Let , where is a -
interval. The - fractional Riemann-Liouville
integral of order is denoted and
given by
It is easy to see that
(23)
As an example, the fractional multiplicative
Riemann-Liouville integral of order
is defined as follows:
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.10
Amer H. Darweesh, Abdelaziz M. D. Maghrabi
E-ISSN: 2224-2678
95
Volume 22, 2023
(Check) Moreover, the tanh- fractional integral
is denoted and given by
Proposition 34 The - fractional Riemann-
Liouville integral operator satisfies the property
Proof. For
Similarly,
This proves the assertion.
Definition 35 Let , where is a -
interval. The bi- fractional Riemann-Liouville
integral of order is denoted and
given by
It is easy to see that
(24)
Proposition 36 The bi- fractional Riemann-
Liouville integral operator satisfies the property
Proof. For , one has
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.10
Amer H. Darweesh, Abdelaziz M. D. Maghrabi
E-ISSN: 2224-2678
96
Volume 22, 2023
The other part is similar.
Definition 37 Let and
, where is a -interval. The - fractional
Riemann-Liouville derivative of order is
denoted and given by
It is easy to see that
(25)
As examples we have the multiplicative and
tanh- versions, respectively, which are denoted
and given by
(26)
(27)
Definition 38 Let and ,
where is a -interval. The bi- fractional
Riemann-Liouville derivative of order is
denoted and given by
[
It is easy to see that
(28)
Definition 39 The - fractional Caputo
derivative of order is denoted and given by
(29)
As examples,
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.10
Amer H. Darweesh, Abdelaziz M. D. Maghrabi
E-ISSN: 2224-2678
97
Volume 22, 2023
(30)
(31)
Theorem (20) paves the way for the bi-
fractional calculi without the use of heavy
machinery. It is an immediate out-growth of it
in some sense. In the following definitions, the
subscript is to clarify that the operations
are carried out with respect to .
Definition 40 The bi- fractional integral of
order is denoted and given by
(32)
As examples,
(33)
(34)
This notation in (34) is just an abbreviation,
since equation (32) is rather tedious. It means
that all the arguments would change from
everywhere except
at the boundaries of integration.
Definition 41 The bi- fractional Riemann-
Liouville derivative of order is denoted and
given by
(35)
As examples,
(36)
(37)
Definition 42 The bi- fractional Caputo
derivative of order is denoted and given by
(38)
(39)
Remark 43 One can see that the Hadamard
fractional derivative is the logarithm of the
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.10
Amer H. Darweesh, Abdelaziz M. D. Maghrabi
E-ISSN: 2224-2678
98
Volume 22, 2023
bigeometric RL derivative of . That is, it is a
RL derivative on the manifold under the
diffeomorphism . This hints that many of the
fractional derivatives that are defined may
indeed be a RL derivative on a given manifold,
from a differential geometric point of view.
(40)
(41)
These definitions also allows one to calculate
various - and bi-fractional derivatives, and
integrals. Hence, all the results that holds true
for Caputo and Riemann-Liouville
differintegrals are also true under the influence
of the homeomorphism For a more general
case, consider a function , where
are ordered fields equipped with the
usual metric and their field structure are based
on the algebraic operations similar to those in
the beginning of the second section under the
influence of the homeomorphisms
. Then, we can define the bi()-
derivative in a similar fashion, where
(42)
And the bi()-integral,
(43)
Remark 44 The domains of the
homeomorphisms may be a subset of the real
numbers.
Example 45 Consider
,
defined by , .
Then we have,
and,
Where the tangent function is on domain
.
Remark 46 Many other fractional calculi may
be defined under the influence of a
diffeomorphism defined as a composition of
finitely many diffeomorphisms.
5 Conclusion
In this paper, the very basic definitions of
fractional calculus are established in the
relatively new -calculi and bi-calculi, which
are promising to be of great use. Indeed, they
give an interpretation of the so-called
fractional calculus under the scope of the
discussed subject, as Remark 43 mentions. This
paper also reveals a new form of the discussed
calculi as seen in the first section which is
useful in proofs. We have also arrived at an
important link which in future papers will make
establish relations between -calculi and bi-
calculi in a smooth and practical way as well as
a relation to the Newtonian versions, where
various analogs such as the gamma
function.
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.10
Amer H. Darweesh, Abdelaziz M. D. Maghrabi
E-ISSN: 2224-2678
99
Volume 22, 2023
References:
[1] Kochubei A, Luchko Y, Handbook of
Fractional Calculus with Applications, de
Gruyter, Berlin, 2019.
[2] Agamirza Bashirov, Emine Mısırlı, Y ucel
Tando gdu, and Ali Ozyapıcı. On modeling
with multiplicative differ-
ential equations. Applied Mathematics-A
Journal of Chinese Universities, 26(4):425–
438, 2011.
Michael Grossman and Robert Katz. Non-
Newtonian Calculus. Non-Newtonian
Calculus, 1972.
[3] Dick Stanley. A multiplicative calculus.
Problems, Resources, and Issues in
Mathematics Undergraduate Studies,
9(4):310–326, 1999.
[4] Agamirza Bashirov, Emine Kurpınar, and Ali
Ozyapıcı. Multiplicative calculus and its
applications. Journal of
mathematical analysis and applications,
337(1):36–48, 2008.
[5] Agamirza E Bashirov and Sajedeh Norozpour.
On an alternative view to complex calculus.
Mathematical Methods
in the Applied Sciences, 41(17):7313–7324,
2018.
[6] Svetlin G Georgiev. Focus on Calculus. Nova
Science, Hauppauge, NY, May 2020.
[7] Raymond Mortini. The f`aa di bruno formula
revisited. Elemente der Mathematik,
68(1):33–38, 2013.
15
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Both authors contributed equally.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding is granted.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.10
Amer H. Darweesh, Abdelaziz M. D. Maghrabi
E-ISSN: 2224-2678
100
Volume 22, 2023
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
