A Special Note According to Possible Applications of Fractional-
Order Calculus for Various Special Functions
HÜSEYİN IRMAK
Department of Computer Engineering,
Faculty of Engineering & Architecture,
İstanbul Nişantaşı University,
TR- 34481742, Sarıyer, Istanbul,
TURKEY
Abstract: - The main aim of this special study is to recall certain information about fractional (arbitrary)
order calculus, which has wide and fruitful applications in science and engineering. Then, it aims to consider
various essential definitions related to fractional order integrals and derivatives for stating and proving some
results, as well as to present some of their possible applications to the attention of related researchers.
Key-Words: - Error functions, derivative(s) of fractional-order, differential equations of fractional order,
series expansions, analytic functions, regions in the complex plane, special functions.
Received: March 7, 2024. Revised: September 8, 2024. Accepted: October 11, 2024. Published: November 6, 2024.
1 Introduction and Certain Special
Information
In this first section, specific information about
certain calculus of fractional order and related
applications will be presented.
The concept of operators of fractional order
was introduced almost simultaneously with the
development of the classical ones.
In light of the essential information provided
by written mathematical literature, as special
information, the first known reference can be
found in the correspondence between G. W.
Leibniz and Marquis de l'Hospital in the year
1695, where the question of the meaning of the
semi-derivative was raised. This special question
consequently attracted the interest of many well-
known mathematicians, including Euler,
Liouville, Laplace, Riemann, Grünwald,
Letnikov, and many others. Since the 19th
century, the theory of fractional calculus has
developed rapidly, mostly as a foundation for a
number of applied disciplines, including fractional
geometry, fractional differential equations, and
fractional dynamics.
The extensive applications of fractional order
calculus are very broad nowadays. It is safe to say
that almost no discipline of modern engineering
and science remains untouched by the tools and
techniques of fractional calculus. For example,
wide and fruitful applications can be found in
rheology, viscoelasticity, acoustics, optics,
chemical and statistical physics, robotics, control
theory, electrical and mechanical engineering,
bioengineering, and more.
In fact, one could argue that real-world
processes are generally fractional order systems.
The main reason for the success of its applications
is that these new models of fractional order are
often more accurate than integer-order ones, as
they provide more degrees of freedom than the
corresponding classical models. One of the
intriguing beauties of the subject is that fractional
derivatives (and integrals) are not local (or point)
quantities.
All operators of fractional order type also
consider the entire history of the process, thus
being able to model the non-local and distributed
effects often encountered in natural and technical
phenomena. The calculus of fractional (arbitrary)
order is therefore an excellent set of tools for
describing the memory and hereditary properties
of various materials and processes.
In addition to theoretical studies, in terms of
the various application areas highlighted above,
fractional order calculus is a frequently
encountered research area for both functions of
real independent variables and complex functions
of independent variables. In various applications,
the relevant fractional order calculus expression,
especially fractional order derivatives, gains
importance.
As various references, each of the references
given in [1], [2], [3], [4] and [5] is a main source
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pertaining to fractional-order calculus, and the
references given in [6], [7], [8], [9], [10], [11],
[12], [13], [14], [15], [16], [17], [18] and [19] are
sources for various applications of fractional-
order calculus in different scientific fields.
Especially since our investigation is related to the
theory of complex functions, the studies given in
[20], [21] and [22] are comprehensive references
for those functions. In addition, for various special
functions as well as transformation theories and
their applications, the works in [23], [24], [25],
[26], [27], [28], [29], [30], [31] and [32] can be
provided as different types of references.
2 Definitions, Remarks, Properties
and Special Examples
In this section of the research, we will cover
various types of functions with complex variables.
Now, let us move on to the following section to
provide some relevant definitions.
Firstly, the familiar notations and
denote the set of natural numbers, the set of real
numbers and the set of complex numbers,
respectively.
We now begin by stating and introducing the
fundamental definitions related to fractional-order
integrals and fractional-order derivatives of
functions with complex (or real) variables.
Definition 1. For a (complex) function ,
the fractional integral of (arbitrary) order is
denoted by
and also defined by:
where and the function with complex
variable is analytic in any simply connected
region of the -plane including the origin, and
the multiplicity of is removed by
necessitating to be a real number when
Definition 2. For a (complex) function
the fractional derivative of (arbitrary) order is
denoted by
and, also described as:
where and the analytic function is
constrained, and the multiplicity of is
extinct as in the first definition (just above). For
these definitions above, one may refer to the
essential works given in [1], [5], [10], [22] and
[27].
We specifically note that the value of the
parameter , which relates to the fractional order
of the mentioned integrals and derivatives and
their various possible applications, can also be any
complex number. Due to the related hypotheses, it
is necessary that the real part of the parameter
be greater than zero.
In addition, when the parameter is any
complex number, its real part must be greater than
zero, with where the familiar
notation denotes the greatest integer function
in classical mathematics.
For additional information regarding the
definitions described in Definitions 1 and 2, one
may refer to some of the essential earlier results
presented in [2], [4], [26], [27] and [32].
For our investigation, the following special
information (or assertions), which are directly
related to these main definitions and their possible
applications, is required.
Remark 1. As a result of a straightforward
examination of Definitions 1 and 2, the accuracy
of the well-known properties related to scalar
multiplication and linearity for these definitions is
readily apparent. The details are omitted here.
We note that the value of the parameter,
which represents the fractional-order calculus, can
also be any complex number. Due to the
hypotheses, it is necessary that the real part of
be is greater than zero.
As some special information, we want to
constitute the following assertions as some
remarks by considering the power function with
complex variable given by
for some with (and, of course,
when selecting which are
just below.
Remark 2. In view of the special information
given by (1), (2) and (3), and also by making use
of change of the variable , its fractional
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integral of real order can be easily
determined as the relations given by:
for some value with
Remark 3. By the help of the relevant power
function given by (3), its fractional derivative of
order can be easy determined as the
elementary result consisting of the relationships
given by:
where
Remark 4. In the light of both the definition
given in (2) and the elementary result given by
(5), its fractional derivatives of order can be
also determined as the elementary form given by:
where and
By means of the extensive information
between (1) and (6), the special assertions given
by:
,
and
for all , and also the elementary-special
results given by:
and
can easily be constituted as some more special
examples.
In particular, it is also possible to obtain
several elementary results related to various types
of fractional-order differential equations. In the
simplest term, as a simple example, for an
appropriate function , the fractional-
order equation being of
can easily be designated by the help of a
combining of the elementary results given by (7)
and (8).
For these types of equations, as examples, one
can refer to the essential works [1], [7], [18], [22],
[28] and [32].
3 Final Remarks
As noted, in the previous two sections, some
special information about fractional-order calculus
was first presented, followed by a number of
definitions, fundamental properties, and particular
examples related to fractional-order calculus. In
this final section, we will present special
information consisting of various results and
suggestions directly related to the main goal of
our research, which involves complex-type special
functions
For those results and possible implications,
firstly, we want to center upon only two complex
functions which are called as the complex error
function and the complementary complex error
function, respectively. These special functions
also have important roles in nearly all sciences
and technology.
For example, for the main (complex) error
functions in the familiar forms:
,
are defined by
and
where .
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These two fundamental functions both contain
many general properties and have extensive
relationships with various complex-type special
functions. Especially, by the help of the
equivalent assertions given by
the following main relationship can easily be
achieved:
when one considers the definitions introduced
in (10) and (11).
For additional properties and relationships,
the earlier studies presented in [21], [22], [23],
[24], [25] and [30] can also be examined.
Furthermore, for the (complex) error function
presented in (10), one can consider the definition
given in (2) and make use of the Taylor-Maclaurin
series expansion, which is quite useful in
approximation theory, provided by
the series expansions of the special function
with
can easily be obtained for some and for all
:
Additionally, using the information from the
assertions given in (5) and (13), one can easily
arrive at:
where and denotes the function
defined by (14).
At the same time, along with its undeniable
importance in approximation theory, and in light
of the information presented in (12) and (15), it is
easy to obtain the series expansion of the function
and also its fractional-order derivative(s),
which are:
and also
for some and for all :
where is the mentioned function defined by
(14).
Secondly, inspired by the special fractional-
order differential equation given in (9), we can
both increase the number of such equations and
determine their solutions.
Additionally, in light of information related to
operator theory, various applications of integral
operators and their inverses can be considered for
all possible results. For this extensive
investigation, we will highlight only one example
and present other potential investigations for the
attention of related researchers. For a simple
example, consider the possible solution of the
initial value problem of fractional order in the
special form:
,
where , one can first obtain the explicit
form of the series expansion of
and then focus on the indicated research.
In conclusion, the mathematical literature
contains a wealth of information and applications
related to other special functions and their
implications. For additional information, we
recommend that our readers review the earlier
studies referenced in [4], [9], [21], [23], [24], [30],
[32] and [33].
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Conflict of Interest
The authors have no conflict of interest to declare.
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