Al-Azhar University-Gaza
Deanship of Postgraduate Studies
Faculty of Science
Department of Mathematics
A Study on -Bessel Type Functions with
Applications
by
Noor Nidal Khattab
Supervisors
Prof. Dr. Tariq O. Salim
Dr. Ghazi S. Khammash
A THESIS
Submitted in Partial Fulfillment of the
Requirements of The Degree of
Master of Science in Mathematics
2021-2022
Dedication
This thesis is dedicated to:
The sake of Allah, my Creator and my Master.
My great teacher and messenger, Mohammed (May Allah bless and grant
him), who taught us the purpose of life.
My great parents, who have never failed to give us financial and moral
support, for giving all our needs during the time we developed our system
and for teaching us that even the largest task can be accomplished if it is
done one step as a time.
My beloved brothers and sisters, who stand by me when things look
bleak.
My friends who encourage and support me.
All the people in my life who touch my heart, I dedicate this research.
II
Acknowledgment
All thanks and appreciation are being to The Almighty Allah, who
helped me to accomplish this modest work, which I hope will benefit all
people.
I offer my sincere thanks and gratitude to everyone who helped me,
and was a light that shined the way for me, and here I especially mention
Prof. Dr. Tariq O. Salim and Dr. Ghazi S. Khammash for their kindness
in accepting to supervise this humble work, and for all the advice and
instructions they gave me, which were like a beacon in all my steps.
To them all, I send all my thanks and gratitude.
III
Abstract
In this thesis, we define a new type of -Bessel function using the gamma function. And get the various formulation terms of others special
function.
In chapter one we display the most important definition and theorem
which one used this study.
In chapter two we define new genialized K-Bessel type and provide
some integral transform and also discussed some other transforms as
special cases of our main result.
In chapter three we study monoticity properties and log- convexity of
the function which consists of the ratio of two different orders K-Bessel
function.
Finally in chapter four we develop fractional kinetic equation involving
generalized
-Bessel function via Laplace transforms and Sumudu
transform. Also, the graphical interpretation of the solution by employing
MATLAB is given.
IV
‫الملخص‬
‫في ىذه انذراسة قًنا تتؼزيف نٌع جذيذ ين دانة ‪ -K‬تيسم تاستخذاو دانة ‪ -K‬جايا‪.‬‬
‫في انفصم األًل قًنا تؼزض أىى انتؼزيفات ًاننظزيات انًستخذية في ىذه انذراسة‪.‬‬
‫في انفصم انثاني ػزفنا دانة ‪ -K‬تيسم ًقًنا تذراستيا ػهى تؼط تحٌالت انتكايم ًحصهنا ػهى‬
‫نتائج ين ىذه اننظزيات‪.‬‬
‫في انفصم انثانج قًنا تذراسة انتزايذ ًانتناقص ًانتحذب انهٌغاريتًي نذانة يكٌنو ين نسثو تين‬
‫دانتين ين ‪-K‬تيسم يختهفتين في انزتة‪.‬‬
‫أخيزا في انفصم انزاتغ قًنا تصياغة يؼادالت انحزكة استخذاو دانة ‪-K‬تيسم ًقًنا تحم ىذه‬
‫انًؼادالت تاستخذاو تحٌيم التالس ًسٌيٌدً ً فسزنا ىذه اننتائج تاستخذاو تزنايج انًاث الب‪.‬‬
‫‪V‬‬
Contents
Dedication .................................................................................................. II
Acknowledgment ......................................................................................III
Abstract .................................................................................................... IV
‫ انًهخص‬......................................................................................................... V
Chapter One:Some Basic Theory ...............................................................1
1.1Introduction and Preliminaries…………………………………....2
1.2Some Definitions of Integral Transforms.......................................4
1.3 Special Functions………………………………………………...7
1.4 Riemann-Liouville Fractional Calculus.........................................9
1.5 Fractional Kinetic Equation..........................................................10
1.6 Inequality……………….……………………………………….12
Chapter Two:Integral Transforms Involving a Generalized
-Bessel
Function ....................................................................................................13
2.1 Introduction……………………………………………………..14
2.2 Some Integral Transform of -Bessel Function………………...14
Chapter Three: Monotonicity and Properties of Log Convex - Bessel
Function ....................................................................................................27
3.1 Basic Properties of Convex Functions………………………….28
3.2 Properties of Log Convex - Bessl Function.............................32
Chapter Four:Certain Fractional Kinetic Equation Involving the -Bessel
Function ....................................................................................................43
4.1Introduction……………………………………………………...44
VI
4.2 Fractional Kinetic Equations Involving Generalized -Bessel
Function via Laplace Transform ...............................................................44
4.3 Fractional Kinetic Equations Involving Generalized -Bessel
Function via Sumudu Transform ..............................................................54
References .................................................................................................66
VII
Chapter One
Some Basic Theory
1
1.1 Introduction and Preliminaries
Bessel functions are named for Friedrich Wilhelm Bessel (1784 1846), an investigation which was further developed by Lord Rayleigh in
1878, where he demonstrated that Bessel's functions are particular cases of
Laplace's functions. [21]
Bessel, while receiving named credit for these functions, did not
incorporate them into his work as an astronomer until 1817. The Bessel
function was the result of Bessel's study of a problem of Kepler for
determining the motion of three bodies moving under mutual gravitation.
[21]
The notation
was first used by Hansen [21] (1843) and
subsequently by Schlomilch [52] (1857) and later modified to
by
Watson (1922).
Subsequent studies of Bessel functions included the works of
Mathews [36] in 1895, “A treatise on Bessel functions and their
applications to physics” written in collaboration with Andrew Gray. It was
the first major treatise on Bessel functions in English and covered topics
such as applications of Bessel functions to electricity, hydrodynamics and
diffraction. In 1922, Watson first published his comprehensive examination
of Bessel functions “A Treatise on the Theory of Bessel Functions” [58].
By mentioning some notations with some results and definitions used
in this thesis. The Bessel functions of the first kind
are defined by
their power series representation as [47]
∑
The
( )
-Bessel function of the first kind has been generalized and
discussed in [47] and defined as
2
∑
( )
(
where
)
and
A more general the -Bessel function of the first kind is defined by
the following series [47]
∑
( )
where
and
Here
Pochhammer symbol defined as [17]
the well-known
{ }
,
while
refer to
denotes the
gamma function defined by [32]
∫
The - gamma function
has the relation
( )
where
Here
if
.
refer to the well-known Gamma function defined by Euler's
integral as follows [54]
∫
Also, equivalently, the generalized -Bessel function of the first kind
has been introduced by [40] is in the form
3
∑
( )
(
)
where
and
and
is
-Pochhammer symbol.
We introduce the generalized
Bessel function of the first kind in
the following form
( )
∑
(
)
for
and
and
are the
where
Pochhammer symbol.
Note that
( )
∑
∑
( )
1.2 Some Definitions of Integral Transforms
In this investigation, the following integral transforms are applied to get
the images of generalized - Bessel function under study.
Definition 1.1
Euler transform of the function
{
}
is defined by [54]
∫
where
4
Definition 1.2
is defined by [54]
Laplace transform of the function
{
}
∫
Definition 1.3
The Sumudu transform of the -Bessel function is given by [54]
[
∫
]
∫
( )
∑
(
)
∫
∑
(
[
]
( )
)
∑
(
)
Definition 1.4
The inverse of Sumudu transform is given by [54]
[
]
( )
*∑
(
[
]
+
)
( )
∑
(
)
Definition 1.5
The - transform with
as a complex parameter defined by [58]
5
{
}
where
∫
and
is the Bessel function of the second kind
defined by [55] as
(
where
)
is the Whittaker function.
we needed the following formula [36]
∫
(
)
where
Definition 1.6
We recall here the following results known as Whittaker transform
[64]
(
∫
where
and
) (
)
is the Whittaker function [60] (see e.g.
[36]),
(
where
)
(
)
is the confluent hypergeometric function written as
(
)
Also, we have the transformation formula
∫
6
1.3 Special Functions
Some special functions are used and applied to get some results in
this investigation. We mention these special functions as follows.
Definition 1.7 [54]
The Beta function
is define by
∫
Hint: Let
then
∫
The relation between Gamma and Beta functions:
Theorem 1.8 [54]
If
, then
.
Definition 1.9 [54]
Mittag-Leffler function of two parameters is defined as
∑
and when
we get the Mittag-Leffler function of one parameter
∑
Definition 1.10
In [55] the generalized hypergeometric function is written as
7
∑
∏
∏
provided
and
where
known
Pochhammer symbol.
Definition 1.11
Also, Fox-Wright generalization
of hypergeometric function
[18,57, 59]:
*
+
∑
(
(
)
(
)
(
)
where
∑
)
, and
∑
for suitably bounded value of
In [20] the generalized -Wright function defined for
and
is written as
[
(
)
]
∑
∏
∏
(
)
Definition 1.12
The generalized Lommel-Wright function [41] is defined as
( )
∑
( )
(
)
8
as
Prieto et al. [45] obtained some results related to fractional calculus
operators of generalized Lommel-Wright function. Konovska [41] studied
convergence of series involving generalized Lommel-Wright function.
It is observed that for
, generalized Lommel-Wright function
(4.10) reduces to generalized Bessel function
( )
[42]:
( )
∑
1.4 Riemann-Liouville Fractional Calculus
In this section we presentation the application of Sumudu transform as
well as Laplace transform to solve some fractional order differential using
Riemann Liouville differential operator.
The theory of fractional calculus is mainly based upon the study of the
well-known Riemann Liouville integral and derivative operators defined as
follows.
Definition 1.13 (Fractional integral operator) [42]
The Riemann-Liouville fractional integral order
defined by
∫
Definition 1.14 (Fractional derivative operator) [42]
The Riemann-Liouville fractional derivative of order
is defined by
(
) ∫
9
is
Laplace transform of the basic fractional operator
(a) The Laplace transform of the Riemann-Liouville integral of order
, is given by [33]
{
}
(b) The Laplace transform of Riemann-Liouville fractional differential
operator of order
{
, is given by [33]
}
∑
for
[
]
.
Sumudu transform of the basic fractional operator
a) The Sumudu transform of the Riemann-Liouville fractional
derivative of
{
of order , is given by [28]
}
∑
[
]
b) The Sumudu transform of Riemann-Liouville fractional integral of
of order
{
is given by [44]
}
1.5 Fractional Kinetic Equation
In recent years, fractional kinetic equations gained remarkable interest
due to their applications in astrophysics and mathematical physics. The
extension and generalization of fractional kinetic equations involving many
fractional operators were found [1,6-7, 23-26, 35, 49, 51, 60].
Haubold and Mathai [23] established a fractional differential equation
between rate of change of the reaction, the destruction rate and the
production rate as follows:
10
where
is the rate of reaction,
is the rate of destruction,
is the rate of production and
denotes the function defined
.
A special case of (1.37) when spatial fluctuations or inhomogeneities
in
are neglected, is given by the following differential equation
[23,34]
with the initial condition
species
at time
is the number of densities of the
. If we remove the index
and integrate the
standard kinetic equation (1.38), we have
where
is standard fractional integral operator.
Haubold and Mathai [23] obtained the fractional generalization of the
standard kinetic equation (1.39) as
where
is Riemann-Liouville fractional integral operator.
Solution of equation (1.40) is given by [23]
∑
Further, Saxena and Kalla consider the following fractional kinetic equation
(
where
)
denotes the number density of a given species at ,
is the number density of the species at
11
, is constant and
1.6 Inequality
Young’s inequality [13]
If a,b ≥ 0 , p,q >0 ,
+
= 1 then ab ≤
+
12
(1.41)
Chapter Two
Integral Transforms Involving
a Generalized
-Bessel Function
13
2.1 Introduction
In special function some polynomials and functions have importance
in mathematical field also in engineering, applied sciences, biology,
chemistry, and physical sciences. One of the important functions is the
Bessel function which has been developed to new class called
-Bessel
function which motivate the future research work in special functions and
its applications in different fields. recently, Ali et al. [3], Ghayasuddin and
Khan [20], Khan et al. [29, 30, 31] and Nisar et al. [39] gave certain
interesting new class of integral formulas involving some types of the
generalized Bessel functions, which are expressed in terms of the
generalized (Wright) hypergeometric function.
2.2 Some Integral Transform of -Bessel Function
We introduce the generalized -Bessel function of the first kind in
the following form
( )
∑
(
)
for
and
and
are the -Pochhammer symbol.
Note that:
( )
∑
14
where
The integral transforms for generalized -Bessel functions such as
Euler, Laplace, Whittaker… etc. are listed in the following theorems.
Theorem 2.1
If
and
{
( )
( )
(
( )
∫
)}
then
( )
( )
(
)
[
]
(
where
) (
)
is the Wright hypergeometric function (1.25).
Proof:
∫
(
∫
)
∑
(
)
( )
∑
(
)
(
)
∫
∑
Now by using (1.4) we get
15
∫
( ) ∑
( )
(
)
Now using (1.6) we get
∫
(
( ) ∑
)
( )
⁄
( )
(
)
(
( )
)
( )
( )
(
∑
)
(
(
)
(
)
)
From (1.25) we have
∫
( )
( )
( )
Moreover, if we set
(
)
[
]
(
) (
in above theorem we get the following result
contained in:
16
)
Corollary 2.2
If
and
}
{
then
∫
( )
When
*
(
and
+
)
then we get
Corollary 2.3
If
{
and
( )
(
)}
(
)
the
∫
( )
*
is −Bessel function defined by (1.2) and
where
+
is the Wright
hypergeometric function (1.3).
Lemma 2.4
If
{
and
}
have
∫
Proof:
Let
.
17
then we
Then
∫( )
∫
( )
In view of the definition of Laplace transform, we get
∫( )
Theorem 2.5
(
If
{
( )
)
( )
| |
and
}
, then we
have
∫
( )
( )
( )
(
)
*
+
(
) (
Proof:
∫
∫
⁄
∑
(
)
⁄
∑
(
∫
)
By lemma 2.4
18
)
∫
⁄
∑
(
)
Now by applying (1.4) and (1.6) in above equation, we have
∫
( )
(
∑
( )
( )
( )
( )
(
(
∑
Moreover, if we set
)
)
[
(
( )
(
(
) (
(
( )
)
)
(
( )
⁄
)
]
) (
)
in above theorem we get the following result
contained in:
Corollary 2.6
{
If
and | |
}
then we have
∫
( )
*
(
)
19
+
)
)
Moreover, if we set
in corollary above, we get the following result
contained in:
Corollary 2.7
{
If
and | |
}
then we have
∫
( )
*
(
+
)
Theorem 2.8
(
,
{
( )
( )
)
}
and
then we have
∫
( )
(
( )
) (
)
[
( )
]
(
) (
Proof:
∫
∫
∑
(
(
)
( )
∑
(
∫
)
20
)
)
Using (1.6) in above expression, we get
∫
(
( )
)(
)
∑
(
(
( )
)
) (
)
( )(
)
∑
( )
( )
(
)
(
( )
( )
(
) (
)(
)
∑
(
) (
)
and by using (1.15), we get
∫
( )
( )
( )
(
[
(
) (
)
]
) (
)
which complete the proof of (2.10).
Moreover, if we set
in above theorem we get the following result
contained in:
21
)
Corollary 2.9
,
}
(
)
{
and
then we have
∫
(
( )
)
[
]
(
)
Theorem 2.10
If
,
{
| |
( )
(
,
( )
}
)
,
, then we have
∫
( )( )
( )
(
[
) (
) (
]
(
)
(
Proof:
By setting
in the left side of (2.12) then we
∫
∫( )
)
(
( ) )( )
22
)
( ( ) )
∫( )
( )∑
(
)
( )
∑
(
∫
)
( )
∑
(
)
(
) (
)
( )
(
)
( ) (
) (
)
∑
(
( ) (
) (
)
in view of (2.3), we can obtain our result (2.15).
Moreover, if we set
in above theorem we get the following result
contained in:
Corollary 2.11
If
,
,
{
}
∫
23
,| |
(
)
, then we have
)
( )
(
) (
)
[
]
(
)
Theorem 2.12
(
,
{
| |
( )
( )
)
}
and
, then we have
∫
( )
( )
( ) (
(
[
(
)
) (
)
]
) (
) (
)
Proof:
∫
in the previous integral, suppose that
so
∫
24
∫( )
( ( ) )
∫( )
( )
∑
(
( )
)
( )
∑
(
( )
∫
)
Now from (1.18) we get
∫
( )
( )
∑
(
)
(
)
(
) (
Now from (1.4) and (1.5) we get
)
∫
( )
(
∑
( )
(
(
(
)
25
(
(
)
( )
( ) (
)
)
) (
( )
) ( )
)
)
(
) (
)
∑ (
) (
( )
(
) (
)
)
( )
( ) (
(
[
(
)
) (
)
]
) (
Moreover, if we set
) (
)
in above theorem we get the following result
contained in:
Corollary 2.13
If
(
,
{
)
}
and | |
, then we
have
∫
( )
(
)
(
)
[
]
(
)
(
)
26
Chapter Three
Monotonicity and Properties of
Log Convex - Bessel Function
27
3.1 Basic Properties of Convex Functions
Convex functions play an important role in many areas of
mathematics. They are especially important in the study of optimization
problems where they are distinguished by a number of convenient
properties.
For instance, a strictly convex function on an open set has no more
than one minimum. Even in infinite-dimensional spaces, under suitable
additional hypotheses, convex functions continue to satisfy such properties
and as a result, they are the most well-understood functionals in the calculus
of variations.
In probability theory, a convex function applied to the expected value
of a random variable is always bounded above by the expected value of the
convex function of the random variable. This result, known as Jensen's
inequality, can be used to deduce inequalities such as the arithmetic–
geometric mean inequality and Hölder's inequality.
Definition 3.1 [13]
A function
is called convex, if
i) the domain of the function
ii) for any
is convex set ;
and every
one has
If the above inequality is strict whenever
and
,
is called strictly convex.
Definition 3.2 [61]
A function
i)
is said to be logarithmically convex (log-convex) if
is convex or equivalently;
28
ii) for all
[
and
]
the two following inequalities hold,
(
) (
)
Theorem 3.1 [13]
The sum of two log-convex function is log-convex.
Proof:
Let
for all
is log-convex is equivalent to
and
(
) (
, since
and
)
are log-convex.
We have the following estimate
(
) (
)
(
) (
)
the claim follows by combining the above two inequalities together.
Set
and
By dividing
and
then we need to show
on both sides, we may assume that
.
Thus, it suffices to show
For
by Young's inequality, we see that
Similarly, we have
Combing these two inequalities together we obtain
29
Theorem 3.2 [13]
Suppose
are twice differentiable over an open domain. Then
the following are equivalent
i)
is convex
ii)
For al
́
iii)
For all
Proof:
We prove from (i) to (ii) then from (ii) to (iii)
If
is convex, by definition 2.1
[
for all
] and
.
After rewriting, we have
(
[
for all
]. As
)
, we get
́
Suppose inequality
holds, for all
and let
we have
́
́
30
take any
multiplying
by
and
by
and adding together, we get
́
Now, from
to
(
)
we prove both of these claims first in dimension 1
and generalized,
Let
and
, we have
́
́
́
by using
́
then
́
for all
. Dividing both sides by
gives
́
and
. As we let
, we get
́
for all
.
suppose ́
When
for all
by the mean value
version of Taylor's theorem, we have
́
́
for some
[
]
́
Now to establish
in general dimension, we recall that convexity
is equivalent to convexity along all lines i.e.,
31
is convex if
is convex for all
and for all
, we
just proved this happens if and only if
́́
́
for all
and for all
Hence,
such that
.
is convex if and only if
for all
3.2 Properties of Log Convex - Bessel Function
In the recent years many geometric and monotonic properties of
some special functions. Especially, some inequalities and monotonic
properties functions are useful in engineering, physics, probability and
statistics, and economics. It is known that log-concavity and log-convexity
properties have a crucial role in economics. Comprehensive information
about the log-concavity and the log-convexity properties can be found in
[37].
In this section, motivated by some earlier results which are given in
[39, 57] our main aim is to present some monotonic and log-concavity
properties of generalized k-Bessel functions.
Now, we would like to give the definition of logarithmic concavity of
a function.
Definition 3.3 [37]
A function
function
is said to be log-concave on interval
is a concave function on
To show log-concavity of a function
on the interval
sufficient to show one of the following two conditions:
i.
if the
monotone decreasing on
32
.
, it is
ii.
.
Also, the following lemma due to Biernacki and Krzyż [11] will be
used in order to prove some monotonic properties of the mentioned
functions.
Lemma 3.3 [11]
∑
Consider the power series
where
and
for all
. Further suppose that both series
. If the sequence {
converge on
}
is increasing (or
decreasing), then the function
decreasing) on
∑
and
is also increasing (or
.
Proof:
Suppose
∑
∑
then
∑
∑
∑
(
)
∑
the numerator of
|
which is
|
|
|
since
( |
|
is increasing.
33
|
|)
since
and at the denominator of
So
.
That conclude
Then
∑
is increasing in
.
is increasing in
Theorem 3.4
The following results hold true for generalization -Bessel function
1) For
the function
increasing on
is
for some fixed
2) For
the function
increasing on
is
for some fixed
3) The function
.
is log- convex on
fixed
and
is decreasing for
{
and
}.
b) The function
is decreasing on
and
.
c) The function
for
)
Then
a) The function
for
for some
(
. Here
4) Suppose that
[
.
is decreasing on
{
and
34
}.
Proof: (1)
Suppose that
∑
∑
( )
(
)
( )
(
)
(
)( )
(
)( )
(
)( )
(
)( )
By taking logarithmic for both sides
(
)( )
(
)( )
(
)( )
(
)( )
Then the logarithmic differentiation
( )
(
( )
)
(
)
( )
( )
́ (
)
(
)
That yield,
(
Here
(
)
(
is the k-digamma function and defined by
35
))
∑
where
is the Euler-Mascheronis constant
∑
∑
(
)
(
)
∑
∑
is increasing on
if
since
, hence
for all
.
This, in particular implies that the sequence {
}
{
}
increasing and hence the conclusion follows for lemma (3.3).
(2) The result follow from lemma (3.3), if the sequence
{
}
{
} is increasing for
Here,
( )
(
)
( )
(
{
)
(
}
(
)
36
)
is
{
}
(
)
(
)
Which together with the identity
{
{
}
gives
(
)
(
)
}
where
(
)
(
)
Take the logarithmic for above relation, we get
(
)
(
)
Take the differentiation for above relation we get
(
)
(
́
́
)
(
)
(
(
)
)
(
)
37
́
́
(
(
)
(
))
(
)
(
)
If
(
(
)
(
))
(
)
(
)
Then
is increasing and consequently the sequence. {
}
is increasing.
Finally, the result follows form lemma (3.3).
(3) We know that sum of log-convex function is log-convex function by
Theorem (3.1).
Thus, to prove the result it is enough to show that
(
)
( )
(
)
is log-convex.
A logarithmic represent
(
)
(
( )
)
A logarithm differentiation of
with respect to
38
yields
(
(
)
( )
(
)
)
(
)
(
( )
(
)
∑
(
)
(
(
)
(
(
)
)
)
(
(
(
Since
(
)
(
) (
∑
and
)
) (
(
for all
)
́ (
)
∑
∑
( )
)
(
∑
( )
) (
)
)
)
.
by theorem (3.2)
by definition (3.2) and remark we conclude
is log-convex.
39
is convex.
is log-convex hence
)
(4) Denote
∑
and
∑
where
( )
(
with
and
the sequence {
}
To apple lemma (3.3) consider
defined by
( )
(
)
( )
Where
(
)
Now let the logarithm for two sides
(
)
)
40
Algorithmic differentiation of
with respect to
(
)
(
yields
́
́
)
(
)
and its follows that, view of increasing properties of
{
and
and
} the function
and also, the sequence {
decreasing on
on
}
is
is decreasing.
By lemma (3.3) the function
is decreasing.
In the case (b) and (c) the sequence {
}
reduce to
( )
(
)
from case (a)
(
)
Now let the logarithm for two sides
(
41
)
Algorithmic differentiation of
(
with respect to
yields
)
(
)
( )
(
)
( ) Now for
If
( )
inequality holds if
required that
{
while for
}
42
this
it's
Chapter Four
Certain Fractional Kinetic Equation
Involving the -Bessel Function
43
4.1 Introduction
In recent years the solution of the fractional kinetic equations attracts
the attention many workers due to their importance in the field of applied
science such as astrophysics, dynamical systems, control systems and
mathematical physics. The kinetic equations of fractional order have been
used to determine certain physical phenomenon. Especially, the kinetic
equations describe the continuity of motion of substance. Therefore, a
large number of articles in the solution of these equations have been
published in the literature.
In view of great importance of the kinetic equation certain astrophysical
problems have largely motivated our present study. The fractional
differential equation between rate of change of the reaction, the destruction
rate and the production rate were established by Haubold and Mathai [23]
given as follows:
4.2 Fractional Kinetic Equations Involving Generalized -Bessel
Function via Laplace Transform
In this section, we have established further generalization of fractional
kinetic equations involving
-Bessel function. Solutions of these
generalized fractional kinetic equations were obtained in terms of Mittag–
Leffler function using Laplace transform. Some special cases also contain
the generalized Bessel function and Struve function.
Generalized Bessel, Lommel, Struve and Lommel-Wright function
have originated from concrete problems in Mechanics, Physics,
Engineering and Astronomy.
44
Kachhia and Prajapati [27] generalized fractional kinetic equation
involving Lommel-Wright function and obtained solution explicitly in the
following theorems:
Theorem 4.1 [27]
and | |
If
then the
solution of equation
is given by
∑
(
where
( )
)
is the generalized Mittag-Leffler function.
Theorem 4.2 [27]
and | |
If
then the
solution of equation
is given by
∑
where
(
)
(
)
is the generalized Mittag-Leffler function.
Theorem 4.3 [27]
If
| |
and
then the solution of equation
45
is given by
∑
(
(
)
where
)
is the generalized Mittag-Leffler function.
Now, we formulate our theorems and proofs for the solutions of
fractional kinetic equations involving -Bessel type function under study as
follows
Theorem 4.4
( )
If
(
)
( )
| |
(
)
then the solution of
equation
is given by
∑
( )
where
is the generalized Mittag-Leffler function.
Proof:
Applying Laplace transform we have
{
{
}
}
{
∫
}
}
( )
∑
(
{
{
}
46
)
{
}
{
∫
∑
(
}
)
{
}
∑
(
{
)
}
∑
(
)
Obtain that,
∑
{
}
∑
∑
(
)
Taking inverse Laplace transform
∑
{∑
(
)
∑
∑
(
)
47
}
∑
∑
(
)
∑
(
)
Theorem 4.5
( )
If
(
)
| |
( )
then the solution of
equation
is given by
∑
(
(
)
)
where
is the generalized Mittag-Leffler function.
Proof:
Applying Laplace transform we have
{
}
{
}
{
}
{
}
(
∫
∑
(
{
{
)
)
}
}
∫
∑
(
)
48
{
}
∑
{
}
(
)
(
)
∑
We obtained know,
∑
{
}
(
)
∑
∑
(
{
)
}
(
)
∑
∑
(
)
Taking inverse Laplace transform
(
)
∑
{∑
(
)
(
)
∑
∑
(
)
49
}
(
)
∑
∑
(
)
(
)
∑
(
)
Theorem 4.6
If
| |
and
then the solution of equation
is given by
∑
(
(
where
)
)
is the generalized Mittag-Leffler function.
Proof:
Applying Laplace transform we have
{
}
{
}
{
}
{
}
(
∫
)
∑
(
{
)
}
50
{
}
∫
∑
(
{
)
}
∑
{
}
{
(
)
(
)
∑
}
(
)
∑
∑
(
)
Taking inverse Laplace transform
(
)
∑
{∑
(
}
)
(
)
∑
∑
(
)
(
)
∑
∑
(
)
51
∑
(
(
)
)
Corollary 4.7
If
and
then the
solution of equation
is given by
∑
( )
𝑘
𝑘
𝑘
𝑘
Figure 4.1: (Graph of equation 4.14)
Corollary 4.8
If
and
solution of equation
is given by
52
then the
( )
∑
𝑘
𝑘
𝑘
𝑘
Figure 4.2: (Graph of equation 4.16)
Corollary 4.9
If
and
the solution of equation
is given by
∑
( )
53
then
𝑘
𝑘
𝑘
𝑘
Figure 4.3: (Graph of equation 4.18)
4.3 Fractional Kinetic Equations Involving Generalized -Bessel
Function via Sumudu Transform
Principle aim of this section is to develop fractional kinetic equations
involving generalized -Bessel function via Sumudu transform.
Bessel functions and their extensions are very important in
application point of view. Recently many authors study extensions of
Bessel functions and study various properties using different approaches
[12,14-16]
Romero et al. [48] introduced and study the -Bessel function of the
first kind as follows:
∑
where
( )
and
54
Here, we intorduce more generalized
using the -Pochhammer symbol
-Bessel function
in the following form: [17]
( )
∑
(
where
)
and
Recently, Watugala [62,63] introduced Sumudu integral transform is
defined as follows [4,5,9,10]
[
where set
]
∫
is the class of exponentially bounded functions
:
{
and
and
are some positive real constants.
The Sumudu transform is the theoretical dual of the Laplace
transform defined by
[
]
∫
We will investigate the solution of the generalized fractional kinetic
equations by considering generalized -Bessel function.
Agarwal et al. [2] generalized fractional kinetic equation is stablished as:
Theorem 4.10
If
and
then the solution of equation
is given by the following formula
55
∑
(
)
(
where
)
is the generalized Mittag-Leffler function.
Theorem 4.11
If
and
then the solution of equation
is given by the following formula
∑
(
)
(
)
where
is the generalized Mittag-Leffler function.
Theorem 4.12
If
and
then the solution of equation
is given by the following formula
∑
(
)
( )
where
is the generalized Mittag-Leffler function.
56
Now, we find the solution to the generalization of fractional kinetic
equations.
Theorem 4.13
( )
If
(
)
( )
| |
then the solution of equation
is given by
( )
∑
(
where
)
is the generalized Mittag-Leffler function.
Proof:
Applying the Sumudu transform we have
{
}
{
}
{
}
{
( )
{∑
(
{
∫
}
)
}
(
∑
(
)
)
57
}
∫
∑
(
)
∑
(
)
∑
∑
(
)
∑
∑
(
)
Taking inverse
{
}
∑
{∑
(
)
∑
∑
(
)
∑
∑
(
)
(
)
∑
58
}
Theorem 4.14
( )
If
( )
| |
(
)
then the solution of equation
is given by
(
)
∑
(
where
)
is the generalized Mittag-Leffler function.
Proof:
Applying the sumudu transform, we have
{
}
(
)
∑
(
{
{
)
}
(
∫
}
)
∑
(
)
(
)
∫
∑
(
)
59
(
)
∑
(
(
)
)
∑
∑
(
)
Taking inverse
{
}
(
)
∑
{∑
(
)
(
)
∑
∑
(
)
(
)
∑
∑
(
)
(
)
∑
(
)
Theorem 4.15
If
| |
and
then the solution of equation
60
}
is given by
∑
(
(
where
)
)
is the generalized Mittag-Leffler function.
Proof:
Applying the Sumudu transform
{
}
(
)
∑
{
(
{
)
}
(
∫
}
)
∑
(
)
∫
∑
(
)
∑
∑
(
)
Taking inverse Sumudu transform
{
}
∑
{∑
(
)
61
}
(
)
∑
∑
(
)
(
)
∑
∑
(
)
(
)
∑
(
)
Corollary 4.16
If
and
solution of equation
is given by
∑
( )
62
then the
𝑘
𝑘
𝑘
𝑘
Figure 4.4: (Graph of equation 4.37)
Corollary 4.17
If
and
solution of equation
is given by
∑
( )
63
then the
𝑘
𝑘
𝑘
𝑘
Figure 4.5: (Graph of equation 4.39)
Corollary 4.18
If
and
the solution of equation
is given by
∑
( )
64
then
𝑘
𝑘
𝑘
𝑘
Figure 4.6: (Graph of equation 3.41)
65
References
[1] Agarwal P., Chand M. and Singh G., (2016): "Certain fractional
kinetic equations involving the product of generalized k-Bessel
function", Alexandria Eng. J., vol. 55 (4), pp. 3053–3059.
[2] Agarwal P., Ntouyas S. K., S. Jain, Chand M. and Singh G., (2018):
"Fractional kinetic equations involving generalized Bessel function
via Sumudu transform", Alexandria Engineering Journal, vol. 57, pp.
1937–1942.
[3] Ali M., Khan W. A. and Khan I. A., (2020): "On certain integral
transforminvolving generalized
Bessel-Maitland function with
applications”, J. Fract. Calc. Appl. 11(1), 82-90.
[4] Asiru M.A., (2001): "Sumudu transform and the solution of integral
equations of convolution type", Int. J. Math. Edu. Sci. Technol., vol.
32, pp. 906–910.
[5] Asiru M.A., (2002): "Further properties of the Sumudu transform and
its applications", Int. J. Math. Edu. Sci. Technol., vol. 33, pp. 441–
449.
[6] Atangana A. and Baleanu D., (2016): "New fractional derivatives
with non-local and non-singular kernel: theory and application to
heat transfer model", Therm. Sci., On Line-First (00),
[7] Atangana A., (2016): "On the new fractional derivative and
application to nonlinear Fisher’s reaction-diffusion equation",
66
Applied Math. and Comput. Journal (Elsevier), vol. 273, pp. 948–
956.
[8] Atangana A., (2016): "Chaos in a simple nonlinear system with
Atangana–Baleanu derivatives with fractional order", Chaos, Solitons
Fract., vol. 89, pp. 447–454.
[9] Belgacem F.B.M., Karaballi A.A. and Kalla S.L., (2003): "Analytical
investigations of the Sumudu transform and applications to integral
production equations", Math. Probl. Eng., vol. 3, pp. 103– 118.
[10] Belgacem F.B.M., (2006): "A.A. Karaballi, Sumudu transform
fundamental properties investigations and applications". Int. J. Appl.
Math. Stoch. Anal., article ID 91083, pp. 1–23.
[11] Biernacki M. and Krzyż J., (1955): "On the monotonity of certain
functionals in the theory of analytic functions", Ann. Univ. Mariae
Curie-Skłodowska Sect. A, vol. 9, pp. 135–147.
[12] Boudjelkha M.T., (2005): "Extended Rieman Bessel functions",
Discr. Contin. Dyn. Syst. Ser., vol., pp. 121–130.
[13] Boyd S. and Vandenberghe L., (2004): "Convex Optimization".
Cambridge University Press, pp.99–113.
67
[14] Cerutti R.A., (2012) "On the k-Bessel functions", Int. Math. Forum,
vol. 7(38), pp. 1851–1857.
[15] Choi J., Agarwal P., (2013): "Certain unified integrals associated
with Bessel functions", Bound. Value Probl., vol. 1, pp. 1–9.
[16] Choi J., Agarwal P., (2013): "Certain unified integrals involving a
product of Bessel functions of the first kind", Honam Math. J., vol.
35(4), pp. 667–677.
[17] Diaz R. and Pariguan E., (2007): "On hypergeometric functions and
-Pochhammer symbol", Divulg. Mat., vol. 15(2), pp. 179–192.
[18] Fox
C.,
(1928)
The
asymptotic
expansion
of
generalized
hypergeometric functions. Proc.London Math. Soc., 27(4), 389-400.
[19] Gehlot K. S. (2013) and Prajapati J. C., Fractional calculus of
generalized k-Wright function. J. Fract. Calc. Appl., 4, 283-289.
[20] Ghayasuddin, M, Khan, W. A, (2018) A new extension of Bessel
Maitland function and its properties, Matematicki Vesnik,1-11.
[21] Hansen P.A. (1843) “Ermittelung der absoluten Strungen in Ellipsen
von beliebiger Excentricitt und Neigung, I.” Schriften der Sternwarte
Seeberg. Gotha.
68
[22] Hardy, G. H., (1988): "Littlewood", Cambridge, England: Cambridge
University Press, pp. 198-200, 2nd Edition.
[23] Haubold H.J. and Mathai A.M.,(2000): "The fractional kinetic
equation and thermonuclear functions", Astrophys. Space Sci., vol.
327, pp. 53–63.
[24] Haubold H.J., Mathai A.M. and Saxena R.K.,(2002): "On fractional
kinetic equations", Astrophys. Space Sci., vol. 282, pp. 281–287.
[25] Haubold H.J., Mathai A.M.
and Saxena R.K.,(2004): "On
generalized fractional kinetic equations", Physica A., vol. 344, pp.
657–664.
[26] Haubold H.J., Mathai A.M. and Saxena R.K.,(2006): "Solution of
generalized fractional reaction-diffusion equations", Astrophys. Space
Sci., vol. 30, pp. 305–313.
[27] Kachhia K.B. and Prajapati J.C.,(2016): "On generalized fractional
kinetic equations involving generalized Lommel-Wright functions",
Alexandria Engineering Journal (elsevier), vol. 55, pp. 29532957.
[28] Kataetbeh Q.D., Belgacem F.B.M., (2011): "Applications of the
Sumudu transform to differential equations". Nonlinear Studies, vol.
18(1), pp.99-112.
69
[29] Khan, W. A, Nisar, K S, (2018): Beta type integral formula
associated with Wright generalized Bessel function, Acta Math. Univ.
Comenianae, 1, 117-125.
[30] Khan, W. A, Nisar, K S, Choi, J, (2017): An integral formula of the
Mellin transform type involving the extended Wright Bessel function,
Far East J. Math., 102, 2903-2912.
[31] Khan, W. A, Nisar, K S, (2018): Unified integral operator involving
generalized Bessel-Maitland function, Proc. Jangjeon Mathe. Soc. 21,
339 - 346.
[32] Kokologiannaki C. G. and V. Krasniqi, (2013): "Some properties of
the K -Gamma function", Matematiche (Catania), vol. 68(1), pp. 13–
22.
[33] Koning D.E. (2015):" Fractional Calculus", Faculty of mathematics
and natural sciences, University of Groningen.
[34] Kourganoff V., (1973): "Introduction to the Physics of Stellar
Interiors", D. Reidel Publishing Company, Dordrecht.
[35] Kumar D., Purohit S.D., Secer A. and Atangana A.,(2015) "On
Generalized Fractional Kinetic Equations Involving Generalized
Bessel Function of the First Kind", Math. problem Eng., vol.(7)
289389, .
70
[36] Mathai A. M., Saxena R. K. and Haubold H. J., (2010): "The Hfunction, Theory and Applications", Springer, New York.
[37] Mathews G. B. and Gray A., (1895): "A Treatise on Bessel Functions
and Their Applications to Physics", London: Macmillan and co..
[38] Mohtasami Borzadaran G.R. and Mohtasami Borzadaran H.A., (2011
) Log-concavity property for some well-known distributions, Surv.
Math. Appl. 6, 203–219,.
[39] Mondal S.R. and Akel M.S., Differential equation and inequalities of
the generalized k-Bessel functions, J. Inequal. Appl., 2018:175. doi:
10.1186/s13660-018-1772-1.
[40] Nisar K. S. and Mondal S. R., (2017): "Certain unified integral
formulas involving the generalized modified k-Bessel function of first
kind". Commun. Korean Math. Soc., vol. 32, pp. 47-53.
[41] Nisar K.S., Khan W.S. and Ghayasuddin M., (2018): Certain integral
transforms of generalized k-Bessel function. Anal. Theory Appl.,34(2)
,165-174.
[42] Oteiza M.B.M., Kalla S. and Conde S., (1986): "Un estudio sobre la
function Lommel-Maitland", Revista Te´ c. Facul. Ingenieria Univ.
Zulia, vol. 9(2), pp.33–40.
71
[43] Owa, S., (1978): "On the distorsion theorems", I Kyungpook Math.
J., vol. 18 (1), pp.53-59.
[44] Panchat K. S. and Bodkhe S. D., (2016): "On Sumudu transform of
fractional derivative and its applications to fractional differential
equations", Asian Journal of Mathematics and Computer, vol. 11(1),
pp.69-77.
[45] Paneva-Konovska J.,(2007): "Theorems on the convergence of series
in generalized Lommel-Wrihgt functions", Fract. Calculus Appl.
Anal., vol. 10(1), pp. 60-74.
[46] Pathak R.S., (1966): "Certain convergence theorems and asymptotic
properties
of
a
generalization
of
Lommel
and
Matiland
transformations", Proc. Nat. Acad. Sci., vol. A-36(1), pp. 81–86.
[47] Prieto A.I., Romero S.S. and Srivastava H.M., (2007): "Some
fractional calculus result involving the generalized Lommel-Wright
and related functions", Appl. Math. Lett., vol. 20, pp. 17–22.
[48] Rainville E. D., (1960): "Special Functions. Macmillan", New York.
[49] Romero L.G., Dorrego G.A. and Cerutti R. A., (2012): "The k-Bessel
function of first kind", Inter. Math. forum, vol. 38(7), pp. 1854-1859.
72
[50] Romero L.G., Dorrego G.A. and Cerutti R.A.,(2014): "The kBessel
function of first kind", Int. Math. Forum, vol. 38(7), pp. 1859– 1854.
[51] Saichev A. and Zaslavsky M.,(1997): "Fractional kinetic equations:
solutions and applications", Chaos, vol. 7, pp. 753–764.
[52] Samko S.G., Kilbas A. and Marichev O., (1990): "Fractional Integral
and Derivatives: Theory and Applications", Gordon and Breach Sci.
Publ., New York.
[53] Saxena R.K. and Kalla S.L.,(2008): "On the solutions of certain
fractional kinetic equations", Appl. Math. Comput., vol. 199, pp. 504–
511.
[54] Schilling, R. L., (2019): "Measures, Integrals & Martingales",
Cambridge University Press, Cambridge, 2nd Edition, Dresden, May.
[55] Schlmilch, O.X., (1857): " Ueber die Bessel’schen Function" , Z. fr
Math. u. Phys., vol. 2, pp. 137-165.
[56] Simon, C.P., and Blume, L., (1994): "Mathematics for Economists",
W.W. Norton & company. Inc., United States, First Edition.
[57] Sneddon I.N., (1979): "The Use of Integral Transform", Tata Mc
Graw Hill, New Delhi, India.
73
[58] Srivastava
H.
M.,
(2007):
"Some
Fox-Wright
generalized
hypergeometric functions and associated families of convolution
operators". Applic. Anal. Disc. Math., vol. , pp. 56-71.
[59] Srivastava H.M. amd Saxena R.K., (2001): "Operators of fractional
integration and their applications", Appl. Math. Comput., vol. 118, pp.
1–52.
[60] Toklu E., (2019): "Radii of starlikeness and convexity of generalized
-Bessel functions", arXiv:1902.09979.
[61] Watson G. N., (1922): "A Treatise on the Theory of Bessel
Functions". Cambridge University Press.
[62] Watugala G. K., (1993): " Sumudu transform: a new integral
transform to solve differential equations and control engineering
problems". Int. J. Math. Educ. Sci. Technol, vol. 43, pp. 35-43.
[63] Watugala G. K., (2001): " The Sumudu transform for function of two
variable". Math. Eng. Ind., vol. 8, pp. 293-302.
[64] Whittakar E.T. and Watson G.N., (1962): "A Course of Modern
Analysis", Cambridge Univ.Press.
[65] Wright E. M., (1940): "The asymptotic expansion of integral
functions defined by Taylor series". Philos.Trans Roy. Soc. London.
Ser.A, vol. 238, pp. 423-451.
74
[66] Zaslavsky G.M.,(1994): "Fractional kinetic equation for Hamiltonian
chaos", Physica D., vol. 76, pp.110–122.
[67] Zhanga X. and Jiangb W., (2012): " Some properties of log-convex
function and applications for the exponential function". Computers &
Mathematics with Applications, vol. 63(6), pp.11
75