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Franklin Open 7 (2024) 100112
Available online 27 May 2024
2773-1863/© 2024 The Author(s). Published by Elsevier Inc. on behalf of The Franklin Institute. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/).
Contents lists available at ScienceDirect
Franklin Open
journal homepage: www.elsevier.com/locate/fraope
Solution Strategy and Associated Results for Fuzzy Mellin Transformation
Payal Singh a,1,Kamal Hossain Gazi b,∗,1,Mostafijur Rahaman c,1,Tripti Basuri d,1,
Sankar Prasad Mondal b,1
aDepartment of Applied Sciences and Humanities, Faculty of Engineering and Technology, Parul University, Vadodara 391760, Gujarat, India
bDepartment of Applied Mathematics, Maulana Abul Kalam Azad University of Technology, West Bengal, Haringhata 741249, West Bengal, India
cDepartment of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, West Bengal, India
dUttar Dum Dum Vidyapith Girls (Pri section), Birati, West Bengal, India
ARTICLE INFO
Keywords:
Fuzzy valued function
Modified Hukuhara derivative
Fuzzy algebra
Fuzzy mellin transformation
ABSTRACT
The advancement of the Mellin transformation is that it exhibits a scale-invariant nature, and thus, it is
widely used in computer science. Fuzzy Mellin transformation can provide the purposes of such problems
in uncertain phenomena. The existing studies on fuzzy Mellin transformation consider the fuzzy level cut
approach, which makes the fuzzy valued functions into their crisp couplets regarding lower and uppercuts.
This paper reconstructs a distinct mathematical frame for the Mellin transformation in a completely fuzzy
environment in the sense of a modified Hukuhara derivative. In this paper, we mean the complete fuzzy
environment to describe fuzzy phenomena where the conversion of the fuzzy function to its crisp counterpart
is annulled. The proposed fuzzy Mellin transformation’s superiority over existing approaches is that it can be
used directly to fuzzy valued functions without converting them to their crisp counter version. The challenges
to dealing with fuzzy valued mappings and inputs without its level cut representation have been tackled in
this manuscript. Furthermore, numerical examples, possible applications, and further extensions of this work
are hinted at in this paper.
1. Introduction
The concept of fuzzy was given by Zadeh [1] in 1965. The con-
cept of fuzzy derivative was introduced by Chang and Zadeh [2] in
1972. After that, many fuzzy derivatives, Hukuhara derivative [3],
Seikkala derivative [4], strongly generalized derivative [5], General-
ized Hukuhara derivative [6], and Modified Hukuhara derivative [7],
etc. are proposed. Initially, people used Hukuhara derivative for ex-
plaining fuzzy differential equations but as time increases, the solution
becomes unbounded. To remove this disadvantage, strongly and gener-
alized Hukuhara derivatives are proposed, but under these derivatives,
one has to choose a solution from the obtained possible set of so-
lutions. Then, Purnima and Payal [7] proposed Modified Hukuhara
derivative, which gives a unique and bounded fuzzy solution although
time increases. The solution of fuzzy differential equation under the
above-mentioned derivatives is given using techniques like analytical,
numerical, transformation, and semi-analytical.
In the transformation technique, Fuzzy Laplace transform (FLT)
is proposed by Allahviranloo et al. [8] under generalized Hukuhara
derivative. Purnima and Payal [7] have redefined FLT under Modified
∗Corresponding author.
E-mail addresses: singhpayalmath@gmail.com (P. Singh), kamalhossain.gazi@makautwb.ac.in (K.H. Gazi), imostafijurrahaman@gmail.com (M. Rahaman),
triptibasuri92@gmail.com (T. Basuri), sankarprasad.mondal@makautwb.ac.in (S.P. Mondal).
1All authors equally contributed to this study.
Hukuhara derivative. Authors [9] worked on Fuzzy Mellin transform
with different approaches and give the relation between Mellin and the
two-sided Laplace Transform.
Data that is intrinsically ambiguous or fuzzy can be effectively
analysed and processed thanks to the novel mathematical framework
known as the fuzzy Mellin Transform. By adding fuzzy sets and fuzzy
numbers, it expands the possibilities of conventional signal processing
methods and offers a more accurate portrayal and interpretation of
real-world data that could include noise, imprecision, or ambiguity.
In this article, we propose a fuzzy Mellin transform in a complete
fuzzy environment. We deal with all the complexity involved in fuzzy
mapping and multiplication of fuzzy functions with fuzzy argument.
We prove all results in fuzzy setup, Using the Decomposition theorem,
in Klir [10].
1.1. Fuzzy set theory
The fuzzy set was first introduced by Lotfi A. Zadeh [1] in 1965. In
a fuzzy set, there is a membership value for every element in the set
https://doi.org/10.1016/j.fraope.2024.100112
Received 31 December 2023; Received in revised form 12 May 2024; Accepted 20 May 2024
Franklin Open 7 (2024) 100112
2
P. Singh et al.
Table 1
Comparative study of Fuzzy Mellin transform.
Authors Title Conversion in Limitation/ Advantage Remark
Crisp Fuzzy
[29]
Appadoo, S.S.
et al. (2014)
Mellin’s Transform and
Application to Some Time
Series Models
Yes No The contribution is to established the
means, skewness, variance and
kurtosis of fuzzy numbers in Mellin
transformation.
Using Fuzzy Mellin Transform to
understand statistical concept.
[9] Sun, X.
et al. (2016)
A Mellin transform method
for solving fuzzy
differential equations
Yes No The operator property is constructed
by fuzzy Mellin transforms and set
up solution strategies of differential
equations.
Using Laplace transform to
explore Mellin Transform.
[30] Ren, W.
et al. (2018)
Hyers–Ulam stability of
Hermite fuzzy differential
equations and fuzzy Mellin
transform
Yes No The contribution is to build the
solution strategy of the fuzzy Mellin
transformation.
Stability result for Hermite fuzzy
differential equation is proposed
and proved.
[31] Azhar,
N. et al.
(2022)
Solution of fuzzy fractional
order differential equations
by fractional Mellin
transform method
Yes No Solved the differential equations of
fractional order using the fuzzy
Mellin transformation.
Fuzzy Mellin Transform is
explored in fractional sense
This article
(2024)
Strategy for Fuzzy Mellin
Transformation
For proving
result crisp
form is used.
Yes The fuzzy Mellin transformation is
defined based on 𝛼−cut
representations of fuzzy numbers
incorporated with Hukuhara
derivative and Riemann integration.
Basic Concepts are proposed and
proved in complete fuzzy
environment.
which belongs to zero to one. After the fuzzy set was invented [11],
there are various properties, operations, theorems and extensions de-
veloped over time. A special type of fuzzy set extension is triangular
fuzzy set (TFS) introduced by Laarhoven et al. [12] in 1983. In TFS, the
membership function [13] is triangular in shape. Singh, P. et al. [13]
use triangular fuzzy number (TFN) in disease Malaria model. Ardil,
C. [14] applied TFN for evaluating and selecting in fighter aircraft.
Furthermore, TFN is used to solve the differential by Taylor series
expansion by Singh, P. et al. [15]. In addition, TFN also uses in
evaluating the mental health of the students by Wang, R. et al. [16] and
quality check of the hospital healthcare system by Liang, D. et al. [17].
Fuzzy set applied in multi criteria decision making (MCDM) analysis
by Mandal, S. et al. [18] in the education fields and Momena, A. F.
et al. [19] utilize in the technological field by edge computing model
set up. The fuzzy set is also used by Momena, A. F. [20] in sustainable
manufacturing industries. Additionally, the fuzzy set has been used by
Aryanezhad, M. [21] in power control and by Gazi, K. H. et al. [22] in
the healthcare sector.
1.2. Different Fuzzy mathematical transformation
There are several types of transformations to evaluate the differen-
tial equations by integral transformations. Laplace transformation [23],
Fourier transformation [23], Z transformation [23], Hilbert transforma-
tion [23], Hartley transformation [23], Sumudu transformation [24],
and Mellin transformation [24] are the major integral transformations
in the crisp environment.
Several transformations deal with the differential equations in the
fuzzy environment and determine their corresponding integral equa-
tions. Like, Fuzzy Laplace Transformation [25], Fuzzy Fourier Trans-
formation [26], Fuzzy Mellin Transformation [27], Fuzzy Transforma-
tion [28], etc. The problems are solved by Mellin transformations in a
completely fuzzy space in this study.
1.3. Fuzzy Mellin transformation
There are different studies conducted on fuzzy Mellin transform.
The recent articles on the Fuzzy Mellin transformation are given below
in Table 1. The below table describes their details with advantages and
limitations in fuzzy fields.
Comparison among the Laplace transform [32], Fourier transform
[33] and Mallin transform [31] under the fuzzy environment is de-
scribed in Table 2.
1.4. Motivation of this study
This study was conducted on Mellin transformation in a fuzzy
environment. The motivation of this research as follows:
•The Mellin transformation can be regarded as a multiplicative
replica of the two sided Laplace transformation. Mellin trans-
formation offers a better ability than integral transformations to
handle the complexity of the models in many cases of modelling
in computer science, theoretical and applied physics.
•The fuzzy Mellin transform is motivated by the necessity to deal
with uncertainty and imprecision in data. The classic (crisp)
Mellin transform is a useful mathematical tool for signal process-
ing, pattern detection, and image manipulation. However, it relies
on exact and deterministic data, which may not always correctly
reflect real-world scenarios.
•On the other hand, fuzzy set theory provides a very compre-
hensive mathematical tool to deal with uncertainty regarding
dynamical systems. A fruitful consequence would be introduced
by combining the ideas of Mellin transformation and fuzzy uncer-
tainty. On the other hand, we have found very few research on
the term ‘‘fuzzy Mellin transformation’’ in the literature.
•The incorporation of fuzziness into the Mellin transform allows
for the analysis and processing of data that is intrinsically ambigu-
ous or unclear. This is especially beneficial in applications like
image processing, where images may contain noise, ambiguity,
or changes that cannot be exactly described.
•Furthermore, fuzzy Mellin transformation was considered in a
parametric approach with crisp lower and upper cuts in almost
all of the existing texts. Our paper is the consequence of these
lacunae.
1.5. Objective & novelty of this research
Motivated by the mentioned possibilities and lacunae, we aimed
to redefine Mellin transformation in such fuzzy phenomena where the
parametric approach with crisp lower and upper cuts for the fuzzy
valued numbers and functions can be annulled. The fuzzy set and its
extensions in lower and upper cuts are discussed briefly. Furthermore,
the Hukuhara derivative and Riemann integration are discussed in
crisp and uncertain environment. With this objective, the following
contributions have been added to the literature by this paper:
Franklin Open 7 (2024) 100112
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P. Singh et al.
Table 2
Comparison among Laplace transform, Fourier transform and Mallin transform under fuzzy environment.
Fuzzy Laplace transform [32] Fuzzy Fourier transform [33] Fuzzy Mellin transform [31]
Fuzzy Laplace Transform is used basically
converting the time domain into the frequency
domain. So, it is used in signal processing.
The Fourier transform decomposes a function into
its constituent frequencies.
Fuzzy Mellin Transform is commonly used in
computer science for algorithm analysis due to its
scale in variance quality.
Fuzzy Laplace Transform can be obtained of
piecewise continuous and exponentially bounded
function.
It transforms a function from the time (or spatial)
domain into the frequency domain.
Fuzzy Mellin Transform is two sided Laplace
Transform.
The Fuzzy Laplace transform is particularly useful
for solving linear ordinary differential equations
with constant coefficients.
It is particularly useful in analysing periodic
signals and systems, as well as for solving partial
differential equations.
The scale in variance property is similar to the
Fourier Transform’s shift in variance property.
1. First, we redefine fuzzy Mellin transformation inspired by the
Modified Hukuhara derivative approach by which the conver-
sion of the fuzzy functions into their crisp couplets can be made
abandoned.
2. Second, the law of linearity, scaling and sifting properties, and
results of the modified Hukuhara derivative of fuzzy valued
functions for the defined fuzzy Mellin transformation have been
manifested explicitly.
3. Third, we try to give an expression of Mellin transform in fuzzy
set-up so that it is directly applicable to fuzzy problems without
converting it into it’s crisp part.
4. Fourth, defined fuzzy Mellin transformation based on 𝛼−cut rep-
resentations of fuzzy numbers integrated with Hukuhara deriva-
tive and Riemann integration.
5. Fifth, numerical examples, possible domains of applications and
future extensions are mentioned at the end of the discussion.
1.6. Key findings
In this section, we discuss the key findings of this study as follows:
•The Mellin transformation engages in mathematical analysis and
solving differential equations. On the other hand, the fuzzy set
analyses data that are inherently uncertain or imprecise. Then
the fuzzy Mellin transformation is particularly useful in real-world
scenarios where data may not be precisely defined.
•Demonstrated the Mellin Transform using Modified Hukuhara
derivative. This gives a unique and bounded fuzzy solution.
•The fuzzy setup provides derivative and shifting properties under
the fuzzy set up.
•The Fuzzy Mellin Transform is shown for purely fuzzy functions
in a fuzzy context.
Fig. 1 represents the flowchart of this study.
1.7. Structure of this paper
In this study, Section 1discusses on introduction and background
of this research. The basic concept of Mellin transformation is covered
in Section 2. The main results of Mellin transformations are described
in Section 3. Section 4contains some numerical description of Mellin
transformation. The application of the fuzzy Mellin transformation is
discussed in Section 5. Finally, Section 6covers the conclusion of this
study.
2. Basic concepts
Let (𝐸, 𝑑 )be the nonempty compact, convex and complete metric
space with the following properties [34], 𝐸= {𝑢 ∶R→[0,1] such that
𝑢 satisfies following properties}
•𝑢 is normal,
•𝑢 is fuzzy convex,
•𝑢 is upper semi-continuous,
•𝑠𝑢𝑝𝑝(𝑢) = {𝑥∈R∶𝑢(𝑥)0} is compact.
For 0< 𝛼 1, denote 𝛼𝑢 = {𝑥∈R∶𝑢(𝑥)𝛼}then from the above
properties follows, 𝛼𝑢 ∈𝑃(R)∀𝛼∈ (0,1] and the parametric form of 𝛼𝑢
is given by [𝑢, 𝑢].
Definition 1 (Fuzzy Set [22]).Assume 𝑋is a universal set of discourse.
A fuzzy set
𝐹define on 𝑋and defined as
𝐹= {(𝜉, 𝜇
𝐹(𝜉)) ∶ 𝜉∈𝑋}(1)
where 𝜇
𝐹(𝜉)is called membership function of fuzzy set
𝐹.
Definition 2 (𝛼−cut of Fuzzy set [35]).Consider
𝐹be a fuzzy set defined
in Definition 1. An 𝛼−cut of the fuzzy set
𝐹denoted as 𝛼
𝐹and defined
by a collection of all elements of
𝐹such that all its membership values
are greater than 𝛼, i.e.,
𝛼
𝐹= {𝜉∶𝜇
𝐹(𝜉)𝛼&𝜉∈𝑋}(2)
2.1. Fuzzy operations
The fuzzy addition and scalar multiplication [36] are defined as,
𝛼𝑢 ⊕ 𝛼𝑣 = [𝑢, 𝑢]+[𝑣, 𝑣]=[𝑢+𝑣, 𝑢 +𝑣],∀𝑢, 𝑣 ∈𝐸(3)
𝑘 ⊗ 𝛼𝑢 =𝑘× [𝑢, 𝑢]=[𝑘𝑢, 𝑘𝑢], 𝑘 > 0(4)
𝑘 ⊗ 𝛼𝑢 =𝑘× [𝑢, 𝑢]=[𝑘𝑢, 𝑘𝑢], 𝑘 < 0(5)
The fuzzy multiplication of two fuzzy variables,
𝛼𝑢 ⊗ 𝛼𝑣 =[𝑢, 𝑢]×[𝑣, 𝑣]
=min{𝑢𝑣, 𝑢𝑣, 𝑢𝑣, 𝑢𝑣},max{𝑢𝑣, 𝑢𝑣, 𝑢𝑣, 𝑢𝑣}(6)
where ∀𝑢, 𝑣 ∈𝐸.
The fuzzy difference,
𝛼𝑢 ⊖ 𝛼𝑣 = [𝑢, 𝑢]−[𝑣, 𝑣] = [min{𝑢−𝑣, 𝑢 −𝑣},max{𝑢−𝑣, 𝑢 −𝑣}] (7)
2.2. Fuzzy continuity
Let is fuzzy continuous [37] at a point (𝑡0, 𝑢0), for any fixed number
𝛼∈ (0,1] and any 𝜖 > 0,∃𝛿(𝜖, 𝛼 )such that 𝑑(
𝑓(𝑡, 𝑢),
𝑓(𝑡0, 𝑢0)) < 𝜖
whenever 𝑡−𝑡0< 𝛿(𝜖, 𝛼 )and 𝑑([𝑢],[𝑢0]) < 𝛿(𝜖, 𝛼) ∀𝑡∈ (𝑎, 𝑏)and 𝑢 ∈𝐸.
2.3. Generalized Hukuhara derivative
Definition 3 (Generalized Hukuhara Difference: [15]).Generalized
Hukuhara Differences [38] is defined as follows, considering 𝐸as
space of convex nonempty set of 𝑋, taking
𝐴,
𝐵∈𝐸then generalized
difference (gH) of
𝐴and
𝐵,
𝐴 ⊖𝑔
𝐵⇔
𝐴=
𝐵 ⊕
𝐶
𝐵=
𝐴 ⊕ (−1)
𝐶(8)
Franklin Open 7 (2024) 100112
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P. Singh et al.
Fig. 1. Flowchart of the Fuzzy Mellin Transformation.
On the basis of the above difference, the Generalized Hukuhara
derivative is given as follows.
Definition 4 (Generalized Hukuhara Derivative: [13]).Let fuzzy
mapping
𝑓∶𝐼→𝐸and 𝑡0∈𝐼then
𝑓is said to be Generalized
Hukuhara differentiable [13] at 𝑡0∈𝐼, ∃an element
𝑓(𝑡0) ∈ 𝐸and
given as,
lim
ℎ→0+
𝑓(𝑡0+ℎ)⊖𝑔
𝑓(𝑡0)
ℎ=
𝑓(𝑡0)(9)
2.4. Modified Hukuhara derivative
First, we define difference here,
𝛼𝑢 ⊖𝛼
𝑚ℎ 𝑣 =min{𝑢−𝑣, 𝑢 −𝑣, 𝑢 −𝑣, 𝑢 −𝑣},
max{𝑢−𝑣, 𝑢 −𝑣, 𝑢 −𝑣, 𝑢 −𝑣}(10)
In [7], a function
𝑓∶ (𝑎, 𝑏)→𝐸is said to be modified Hukuhara
differentiable [7] at 𝑡0∈ (𝑎, 𝑏) ∃ an element
𝑓(𝑡0) ∈ 𝐸such that for all
ℎ > 0sufficiently small ∃
𝑓(𝑡0+ℎ)⊖
𝑓(𝑡0),
𝑓(𝑡0)⊖
𝑓(𝑡0−ℎ)should exist
and the limits,
lim
ℎ→0+
𝑓(𝑡0+ℎ)⊖
𝑓(𝑡0)
ℎ= lim
ℎ→0−
𝑓(𝑡0)⊖
𝑓(𝑡0−ℎ)
ℎ=
𝑓(𝑡0)(11)
where, its equivalent parametric form is given as,
lim
ℎ→0+
𝛼
𝑓(𝑡0+ℎ)⊖𝛼
𝑓(𝑡0)
ℎ
=min lim
ℎ→0
𝑓(𝑡0+ℎ) − 𝑓(𝑡0)
ℎ,lim
ℎ→0
𝑓(𝑡0+ℎ) − 𝑓(𝑡0)
ℎ,
lim
ℎ→0
𝑓(𝑡0+ℎ) − 𝑓(𝑡0)
ℎ,lim
ℎ→0
𝑓(𝑡0+ℎ) − 𝑓(𝑡0)
ℎ
,
max lim
ℎ→0
𝑓(𝑡0) − 𝑓(𝑡0−ℎ)
ℎ,lim
ℎ→0
𝑓(𝑡0) − 𝑓(𝑡0−ℎ)
ℎ,
lim
ℎ→0
𝑓(𝑡0) − 𝑓(𝑡0−ℎ)
ℎ,lim
ℎ→0
𝑓(𝑡0) − 𝑓(𝑡0−ℎ)
ℎ
(12)
and
lim
ℎ→0−
𝛼
𝑓(𝑡0)⊖𝛼
𝑓(𝑡0−ℎ)
ℎ
=min lim
ℎ→0
𝑓(𝑡0) − 𝑓(𝑡0−ℎ)
ℎ,lim
ℎ→0
𝑓(𝑡0) − 𝑓(𝑡0−ℎ)
ℎ,
lim
ℎ→0
𝑓(𝑡0) − 𝑓(𝑡0−ℎ)
ℎ,lim
ℎ→0
𝑓(𝑡0) − 𝑓(𝑡0−ℎ)
ℎ
,
max lim
ℎ→0
𝑓(𝑡0+ℎ) − 𝑓(𝑡0)
ℎ,lim
ℎ→0
𝑓(𝑡0+ℎ) − 𝑓(𝑡0)
ℎ,
lim
ℎ→0
𝑓(𝑡0+ℎ) − 𝑓(𝑡0)
ℎ,lim
ℎ→0
𝑓(𝑡0+ℎ) − 𝑓(𝑡0)
ℎ
(13)
Franklin Open 7 (2024) 100112
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P. Singh et al.
Note 1. We need to select a minimum and maximum value from the limiting
case and these two limits
limℎ→0
(𝑓(𝑡0+ℎ) − 𝑓(𝑡0))
ℎ,limℎ→0
(𝑓(𝑡0)−𝑓(𝑡0−ℎ))
ℎdo not exist for all
fuzzy valued function.
Example 1. We consider the same example and it’s explanation as
in [7], The fuzzy initial value problem is as follows,
𝑑 𝑦
𝑑𝑡 =⊖ 𝑦;
𝑦(0) =(0.96,1,1.01),
𝛼𝑦0=[0.96 + 0.04𝛼, 1.01 − 0.01𝛼]
An attempt was made by authors in [11] to solve such an example,
and in [12], authors used the gH-derivative to fix it. Wherein the
optimal solution was chosen among the ones that were obtained. The
mH-derivative can be used to solve it as,
𝑦, 𝑦= − 𝑦, 𝑦,∀𝑡 > 0.(14)
Comparing component wise, we obtain,
𝑦 = min −𝑦, −𝑦
𝑦 = max −𝑦, −𝑦(15)
i.e.,
𝑦 = −𝑦
𝑦 = −𝑦(16)
After solving this, we have
𝑦(𝑡) = 𝑐1𝑒𝑡+𝑐2𝑒−𝑡
𝑦(𝑡)=−𝑐1𝑒𝑡+𝑐2𝑒−𝑡(17)
After putting the initial condition, they become,
𝑦(𝑡) = (0.025𝛼− 0.025)𝑒𝑡+ (0.985 + 0.015𝛼)𝑒−𝑡
𝑦(𝑡) = −(0.025𝛼− 0.025)𝑒𝑡+ (0.985 + 0.015𝛼)𝑒−𝑡(18)
which is exactly the same for 𝛼= 0 as in [12].
Example 2. We solve the fuzzy initial value problem using the Modi-
fied Hukuhara derivative is as follows,
𝑑 𝑧
𝑑𝑡 =⊖ 𝑧;
𝑧(0) =(1.0,1.2,1.4),
𝛼𝑧0=[1 + 0.02𝛼, 1.4−0.02𝛼]
Writing parametric form of equations using 𝛼−cut,
𝛼𝑑 𝑧
𝑑𝑡 =⊖𝛼𝑧
[
𝑍,
𝑧] = −[𝑧, 𝑧],∀𝑡 > 0
Comparing both intervals, the following equations are obtained.
𝑧 = min −𝑧, −𝑧
𝑧 = max −𝑧, −𝑧(19)
i.e.,
𝑧 = −𝑧
𝑧 = −𝑧(20)
After solving this, we have
𝑧(𝑡) = 𝑐1𝑒𝑡+𝑐2𝑒−𝑡
𝑧(𝑡)=−𝑐1𝑒𝑡+𝑐2𝑒−𝑡(21)
After putting the initial condition, they become,
𝑧(𝑡) = (0.2𝛼− 0.2)𝑒𝑡+ 1.2𝑒−𝑡
𝑧(𝑡) = −(0.2𝛼− 0.2)𝑒𝑡+ 1.2𝑒−𝑡(22)
2.5. Fuzzy Riemann integration
As in [39], let
𝑓(𝑥)be a fuzzy valued function on [𝛼, ∞) represented
by min 𝑓(𝑥, 𝑥), 𝑓 (𝑥, 𝑥),max 𝑓(𝑥, 𝑥), 𝑓 (𝑥, 𝑥)for any fixed value of
𝛼∈ (0,1]. Assume
min 𝑓(𝑥, 𝑥), 𝑓 (𝑥, 𝑥),max 𝑓(𝑥, 𝑥), 𝑓 (𝑥, 𝑥)both are Riemann inte-
grable on [𝑎, 𝑏]and assume there are two positive functions 𝑀and 𝑀
such that 𝑏
𝑎𝑓(𝑡)𝑀and 𝑏
𝑎𝑓(𝑡)𝑀then
𝑓(𝑥)is improper fuzzy
Riemann integrable [39] and it’s parametric form is defined as follows,
∞
𝛼
𝛼(
𝑓(𝑥))𝑑 𝑥 =∞
𝛼
min 𝑓(𝑥, 𝑥), 𝑓 (𝑥, 𝑥)𝑑𝑥,
∞
𝛼
max 𝑓(𝑥, 𝑥), 𝑓 (𝑥, 𝑥)𝑑𝑥(23)
3. Main result: Mellin transformation of Fuzzy valued function
In this section, we propose and establish fuzzy Mellin transforma-
tion. The Fuzzy Mellin transformation is used for solving the fuzzy
differential equations with integer [9] and polynomial coefficients [27].
To solve the multi order differential equations Butera, S. et al. [40] use
Fuzzy Mellin transformations. Butzer, P.L. et al. [41] represented an
application of fractional integration and differentiation by the Mellin
transformation.
Fuzzy Mellin transformation is also used in electric circuits by
Qayyum, M. et al. [42] and in biomedical fields by Singh, S. et al. [43].
From above studies motivated us to work on Fuzzy Mellin transforma-
tion on fuzzy valued functions.
3.1. Fuzzy riemann integration in sense of modified hukuhara derivative:
From the Modified Hukuhara derivative [7] in parametric form,
lim
ℎ→0+
𝛼
𝑓(𝑡0+ℎ)⊖𝛼
𝑓(𝑡0)
ℎ
=min lim
ℎ→0
𝑓(𝑡0+ℎ) − 𝑓(𝑡0)
ℎ,lim
ℎ→0
𝑓(𝑡0+ℎ) − 𝑓(𝑡0)
ℎ,
lim
ℎ→0
𝑓(𝑡0+ℎ) − 𝑓(𝑡0)
ℎ,lim
ℎ→0
𝑓(𝑡0+ℎ) − 𝑓(𝑡0)
ℎ
,
max lim
ℎ→0
𝑓(𝑡0) − 𝑓(𝑡0−ℎ)
ℎ,lim
ℎ→0
𝑓(𝑡0) − 𝑓(𝑡0−ℎ)
ℎ,
lim
ℎ→0
𝑓(𝑡0) − 𝑓(𝑡0−ℎ)
ℎ,lim
ℎ→0
𝑓(𝑡0) − 𝑓(𝑡0−ℎ)
ℎ
(24)
The right hand side expression of the above equation can be written
as,
min 1
ℎ𝑡0+ℎ
𝑎
𝐹(𝑡)𝑑𝑡−1
ℎ𝑡0
𝑎
𝐹(𝑡)𝑑𝑡,
1
ℎ𝑡0+ℎ
𝑎
𝐹(𝑡)𝑑𝑡 −1
ℎ𝑡0
𝑎
𝐹(𝑡)𝑑𝑡,
max 1
ℎ𝑡0+ℎ
𝑎
𝐹(𝑡)𝑑𝑡−1
ℎ𝑡0
𝑎
𝐹(𝑡)𝑑𝑡,
1
ℎ𝑡0+ℎ
𝑎
𝐹(𝑡)𝑑𝑡 −1
ℎ𝑡0
𝑎
𝐹(𝑡)𝑑𝑡
(25)
Franklin Open 7 (2024) 100112
6
P. Singh et al.
Now, using properties of the definite integral,
min 1
ℎ𝑡0+ℎ
𝑡0
𝐹(𝑡)𝑑𝑡−1
ℎ𝑡0
𝑎
𝐹(𝑡)𝑑𝑡 +1
ℎ𝑡0
𝑎
𝐹(𝑡)𝑑𝑡,
1
ℎ𝑡0+ℎ
𝑎
𝐹(𝑡)𝑑𝑡 −1
ℎ𝑡0
𝑎
𝐹(𝑡)𝑑𝑡 +1
ℎ𝑡0
𝑎
𝐹(𝑡)𝑑𝑡,
max 1
ℎ𝑡0+ℎ
𝑡0
𝐹(𝑡)𝑑𝑡−1
ℎ𝑡0
𝑎
𝐹(𝑡)𝑑𝑡 +1
ℎ𝑡0
𝑎
𝐹(𝑡)𝑑𝑡,
1
ℎ𝑡0+ℎ
𝑎
𝐹(𝑡)𝑑𝑡 −1
ℎ𝑡0
𝑎
𝐹(𝑡)𝑑𝑡 +1
ℎ𝑡0
𝑎
𝐹(𝑡)𝑑𝑡
(26)
This gives,
=min 1
ℎ𝑡0+ℎ
𝑡0
𝐹(𝑡)𝑑𝑡, 1
ℎ𝑡0+ℎ
𝑡0
𝐹(𝑡)𝑑𝑡,max 1
ℎ𝑡0+ℎ
𝑡0
𝐹(𝑡)𝑑𝑡, 1
ℎ𝑡0+ℎ
𝑡0
𝐹(𝑡)𝑑𝑡
(27)
Similarly,
lim
ℎ→0−
𝛼
𝑓(𝑡0)⊖𝛼
𝑓(𝑡0−ℎ)
ℎ
=min lim
ℎ→0
𝑓(𝑡0) − 𝑓(𝑡0−ℎ)
ℎ,lim
ℎ→0
𝑓(𝑡0) − 𝑓((𝑡0) − ℎ)
ℎ,
lim
ℎ→0
𝑓(𝑡0) − 𝑓(𝑡0−ℎ)
ℎ,lim
ℎ→0
𝑓(𝑡0) − 𝑓(𝑡0−ℎ)
ℎ
,
max lim
ℎ→0
𝑓(𝑡0) − 𝑓(𝑡0−ℎ)
ℎ,lim
ℎ→0
𝑓(𝑡0) − 𝑓(𝑡0−ℎ)
ℎ,
lim
ℎ→0
𝑓(𝑡0) − 𝑓(𝑡0−ℎ)
ℎ,lim
ℎ→0
𝑓(𝑡0) − 𝑓(𝑡0−ℎ)
ℎ
(28)
This gives,
min 1
ℎ𝑡0
𝑡0
𝐹(𝑡)𝑑𝑡, 1
ℎ𝑡0
𝑡0−ℎ
𝐹(𝑡)𝑑𝑡,max 1
ℎ𝑡0
𝑡0
𝐹(𝑡)𝑑𝑡, 1
ℎ𝑡0
𝑡0−ℎ
𝐹(𝑡)𝑑𝑡
(29)
Definition 5 (Crisp Mellin Transformation [44]).Let 𝑓∈L(R)be a
real-valued function. The Mellin transform of the real-valued function
𝑓(𝜏)is denoted by [𝑓(𝜏)](𝑠)and defined as,
[𝑓(𝜏)](𝑠)[𝑓(𝜏)](𝑠) = ∞
0
𝜏𝑠−1 ⋅𝑓(𝜏)𝑑𝜏 (30)
where 𝑠=𝑐+𝑖𝑡 ∈Cand 𝜏∈R.
Definition 6 (Fuzzy Mellin Transformation [9]).Let
𝑓∈L(R)be a
fuzzy-valued function. The fuzzy Mellin transform of the fuzzy-valued
function
𝑓(𝜏)is denoted by [
𝑓(𝜏)](𝑠)and defined as,
[
𝑓(𝜏)](𝑠) = ∞
0
𝜏𝑠−1 ⊙
𝑓(𝜏)𝑑𝜏 (31)
where 𝑠=𝑐+𝑖𝑡 ∈Cand 𝑡, 𝜏 ∈R.
Definition 7 (Fuzzy Mellin Transformation with 𝛼−cut representation
[30]).Consider
𝑓∈L(R)be a fuzzy-valued function with
𝛼−cut representation of
𝑓is
𝑓(𝜏)=[
𝑓𝑙(𝜏;𝛼),
𝑓𝑟(𝜏;𝛼)] where 𝛼∈ [0,1].
The fuzzy Mellin transform of the fuzzy-valued function
𝑓(𝜏)is denoted
by [
𝑓(𝜏)](𝑠)and defined as,
[
𝑓(𝜏)](𝑠)[
𝑓](𝑠;𝛼)
=∞
0
𝜏𝑠−1
𝑓𝑙(𝜏;𝛼)𝑑𝜏 , ∞
0
𝜏𝑠−1
𝑓𝑟(𝜏;𝛼)𝑑𝜏
=[
𝑓]𝑙(𝑠;𝛼),[
𝑓]𝑟(𝑠;𝛼)
(32)
Example 3. Let
𝑓(𝜏) = 𝑒−
𝛬𝜏 with [
𝛬]𝛼= [5 + 𝛼 , 7 − 𝛼]. We have
[𝑒−
𝛬𝜏 ](𝑠;𝛼) = (7 − 𝛼)−𝑠𝛤(𝑠),(5 + 𝛼)−𝑠𝛤(𝑠)
Definition 8. The inversion formula for the fuzzy Mellin transform can
be given as,
𝐹(𝑠) = −1[
𝐹(𝑠)](𝜏) = 1
2𝜋𝑖
⋅𝑝𝜈 𝑐+𝑖∞
𝑐−𝑖∞
𝜏−𝑠⋅
𝐹(𝑠)𝑑𝑠 (33)
Example 4. Let [
𝛬]𝛼= [5 + 𝛼 , 7 − 𝛼],𝛼∈ [0,1] be a triangular fuzzy
number.
−1
𝛬−𝑠⋅𝛤(𝑠)(𝑡) = 1
2𝜋𝑖
⋅𝑝𝜈 𝑐+𝑖∞
𝑐−𝑖∞
𝜏−𝑠⋅
𝛬−𝑠⋅𝛤(𝑠)𝑑𝑠
=𝑒−
𝛬⋅𝑡
Definition 9. Let
𝐹∈L[𝑎, 𝑏]and
𝐺∶R+→Cbe two functions. Then
fuzzy Mellin convolution product is denoted by
𝐹 ⋆
𝐺and defined as,
𝐹 ⋆
𝐺(𝜏) = ∞
0
𝐺𝜏
𝑠⋅
𝐹(𝑠)⋅
𝑑𝑠
𝑠(34)
Moreover, for 𝛼∈ [0,1],if
𝐹is a fuzzy-valued function, then we can get
𝐹 ⋆
𝐺(𝜏;𝛼) ∶= ∞
0
𝐺𝜏
𝑠
𝐹𝑙(𝑠;𝛼)1
𝑠𝑑𝑠,
∞
0
𝐺𝜏
𝑠
𝐹𝑟(𝑠;𝛼)1
𝑠𝑑𝑠
=
𝐹𝑙⋆
𝐺(𝜏;𝛼),
𝐹𝑟⋆
𝐺(𝜏;𝛼)
(35)
3.2. Theorem of Mellin transformation
This section discussed different theorems of Mellin transformation
and their proofs.
Theorem 1. If
𝑓∶L(R)→Cor
𝑓is a fuzzy valued function on (0,∞),
then the fuzzy Mellin transformation [
𝑓(𝜏)] is defined in complex plane
Cby this relation,
(
𝑓(𝜏)) = ∞
0
𝜏𝑠−1 ⊙
𝑓(𝜏)𝑑𝜏 =
𝐹(𝑠)(36)
or,
(
𝑓) = ∞
0
𝜏𝑝−1 ⊙
𝑓 𝑑𝜏 =
𝐹(𝑠)(37)
where 𝑠∈Cand 0<R(𝑠)<1.
Proof. The 𝛼−cut of the following fuzzy function and variable is
defined as,
𝛼
𝑓=[min{𝑓(𝜏,𝜏), 𝑓 (𝜏, 𝜏)},max{𝑓(𝜏 , 𝜏), 𝑓 (𝜏 , 𝜏)}] and
𝛼𝜏 = [min{𝜏, 𝜏},max{𝜏, 𝜏}].
Since 𝜏 > 0,𝛼𝜏 = [𝜏, 𝜏]. Using above notations, the parametric from
of (
𝑓(𝜏)).
(𝛼
𝑓(𝜏)) =min{𝑓(𝜏, 𝜏 ), 𝑓 (𝜏, 𝜏 )},
max{𝑓(𝜏,𝜏), 𝑓 (𝜏, 𝜏)}(38)
Franklin Open 7 (2024) 100112
7
P. Singh et al.
And, from the definition of Mellin transform [45] in crisp set up,
we get,
min 𝑓(𝜏,𝜏), 𝑓 (𝜏, 𝜏)=∞
0
[𝜏𝑠−1, 𝜏 𝑠−1]⊙min 𝑓(𝜏, 𝜏), 𝑓 (𝜏, 𝜏),
max 𝑓(𝜏,𝜏), 𝑓 (𝜏, 𝜏)𝑑 𝜏
(39)
For convenience, let 𝑓𝑙= min 𝑓(𝜏, 𝜏 ), 𝑓 (𝜏, 𝜏)and
𝑓𝑟= max 𝑓(𝜏,𝜏), 𝑓 (𝜏, 𝜏).
Thus, Eq. (38) becomes,
min 𝑓(𝜏,𝜏), 𝑓 (𝜏, 𝜏),max 𝑓(𝜏, 𝜏), 𝑓 (𝜏, 𝜏)
=∞
0
[𝜏𝑠−1, 𝜏 𝑠−1]⊙[𝑓𝑙, 𝑓𝑟]⊙[𝑑𝜏 , 𝑑𝜏 ]
(40)
Using fuzzy multiplication, we get,
min{𝑓(𝜏, 𝜏 ), 𝑓 (𝜏, 𝜏 )},max{𝑓(𝜏, 𝜏 ), 𝑓 (𝜏, 𝜏 )}
=∞
0𝜏𝑠−1𝑓𝑙𝑑 𝜏, 𝜏 𝑠−1𝑓𝑟𝑑𝜏 (41)
As we know the above equation is written in the sense of 𝛼−cut, so
using the Decomposition theorem as in Klir [10], we get,
[
𝑓(𝜏)] = ∞
0
𝜏𝑠−1 ⊙
𝑓(𝜏)𝑑𝜏 (42)
or,
[
𝑓] = ∞
0
𝜏𝑠−1 ⊙
𝑓 𝑑𝜏 =
𝐹(𝑠)(43)
The existence condition of fuzzy Mellin transform is fuzzy valued
function must be bounded and the value of 𝑠, must be in this interval
0<R(𝑠)<1.
3.3. Some important propositions
Proposition 1. If [
𝑓𝑗(𝜏)] =
𝐹𝑗(𝑠)for 0< 𝑅𝑒(𝑠)<1, then
𝑛
𝑗=1
𝑐𝑗⊙
𝑓𝑗(𝜏)(𝑠) =
𝑛
𝑗=1
𝑐𝑗⊙
𝑓𝑗(𝜏)(𝑠)(44)
Proof. In order to complete the proof of the proposition, we first
encounter the proof for addition. Instead of taking 𝑛elements for
addition without loss of generality, let 𝑛= 2.
Then we have
𝑓1(𝜏)⊕
𝑓2(𝜏)(𝑠, 𝛼)
=
𝑓1𝑙(𝜏, 𝛼 )⊕
𝑓2𝑙(𝜏, 𝛼 ),
𝑓1𝑟(𝜏, 𝛼 )⊕
𝑓2𝑟(𝜏, 𝛼 )(𝑠)
=∞
0
𝜏𝑠−1
𝑓1𝑙(𝜏, 𝛼 )𝑑𝜏 +∞
0
𝜏𝑠−1
𝑓2𝑙(𝜏, 𝛼 )𝑑𝜏,
∞
0
𝜏𝑠−1
𝑓1𝑟(𝜏, 𝛼 )𝑑𝜏 +∞
0
𝜏𝑠−1
𝑓2𝑟(𝜏, 𝛼 )𝑑𝜏
=∞
0
𝜏𝑠−1
𝑓1𝑙(𝜏, 𝛼 )𝑑𝜏, ∞
0
𝜏𝑠−1
𝑓1𝑟(𝜏, 𝛼 )𝑑𝜏
+∞
0
𝜏𝑠−1
𝑓2𝑙(𝜏, 𝛼 )𝑑𝜏, ∞
0
𝜏𝑠−1
𝑓2𝑟(𝜏, 𝛼 )𝑑𝜏
=
𝑓1(𝜏)(𝑠, 𝛼) +
𝑓2(𝜏)(𝑠, 𝛼)
So, we can write that
𝑓1(𝜏)⊕
𝑓2(𝜏)(𝑠) =
𝑓1(𝜏)(𝑠, 𝛼) +
𝑓2(𝜏)(𝑠)(45)
Proceeding similarly, we can establish that
𝑛
𝑗=1
𝑓𝑗(𝜏)(𝑠) =
𝑛
𝑗=1
𝑓𝑗(𝜏)(𝑠)(46)
The impact of scalar multiplication is not addressed yet. For doing
so, let 𝑐0. Then
𝑐
𝑓(𝜏, 𝛼 ) = 𝑐
𝑓𝑙(𝜏, 𝛼 ), 𝑐
𝑓𝑟(𝜏, 𝛼 )
and
𝑐
𝑓(𝜏, 𝛼 )(𝑠, 𝛼) = ∞
0
𝑐𝜏 𝑠−1
𝑓𝑙(𝜏, 𝛼 )𝑑𝜏,
∞
0
𝑐𝜏 𝑠−1
𝑓𝑟(𝜏, 𝛼 )𝑑𝜏
=𝑐∞
0
𝜏𝑠−1
𝑓𝑙(𝜏, 𝛼 )𝑑𝜏,
∞
0
𝜏𝑠−1
𝑓𝑟(𝜏, 𝛼 )𝑑𝜏
=𝑐
𝑓(𝜏, 𝛼 )(𝑠, 𝛼)
Again, consider 𝑐 < 0, then
𝑐
𝑓(𝜏, 𝛼 ) = 𝑐
𝑓𝑟(𝜏, 𝛼 ), 𝑐
𝑓𝑙(𝜏, 𝛼 )
and
𝑐
𝑓(𝜏, 𝛼 )(𝑠, 𝛼) = ∞
0
𝑐𝜏 𝑠−1
𝑓𝑟(𝜏, 𝛼 )𝑑𝜏,
∞
0
𝑐𝜏 𝑠−1
𝑓𝑙(𝜏, 𝛼 )𝑑𝜏
=𝑐∞
0
𝜏𝑠−1
𝑓𝑙(𝜏, 𝛼 )𝑑𝜏,
∞
0
𝜏𝑠−1
𝑓𝑟(𝜏, 𝛼 )𝑑𝜏
=𝑐
𝑓(𝜏, 𝛼 )(𝑠, 𝛼)
So, for any arbitrary 𝑐, we get
𝑐
𝑓(𝜏)(𝑠, 𝛼) = 𝑐
𝑓(𝜏, 𝛼 )(𝑠)(47)
Combining the result of the finite addition and scalar multiplication,
we will obtain the results of the proposition.
Note 2. Proposition 1describes the linearity property of the fuzzy Mellin’s
transformation in the proposed perspective of this article.
Proposition 2. If [
𝑓𝑗(𝜏)] =
𝐹𝑗(𝑠)for 0< 𝑅𝑒(𝑠)<1. We have
𝑓(𝜆𝜏)(𝑠) = 𝜆−𝑠
𝑓(𝜏)(𝑠)(48)
where 𝜆 > 0.
Proof. By Definition 6, we have
𝑓(𝜏)(𝑠, 𝛼) = ∞
0
𝜏𝑠−1
𝑓𝑙(𝜏, 𝛼 )𝑑𝜏, ∞
0
𝜏𝑠−1
𝑓𝑟(𝜏, 𝛼 )𝑑𝜏
Therefore,
𝑓(𝜆𝜏)(𝑠, 𝛼 ) = ∞
0
𝜏𝑠−1
𝑓𝑙(𝜆𝜏, 𝛼 )𝑑𝜏, ∞
0
𝜏𝑠−1
𝑓𝑟(𝜆𝜏, 𝛼 )𝑑𝜏
=∞
0𝜂
𝜆𝑠−1
𝑓𝑙(𝜂, 𝛼)𝑑 𝜂
𝜆,
∞
0𝜂
𝜆𝑠−1
𝑓𝑟(𝜂, 𝛼)𝑑 𝜂
𝜆
Let, 𝜆𝜏 =𝜂⇒𝑑𝜏 =𝑑 𝜂
𝜆
=𝜆−𝑠∞
0
𝜂𝑠−1
𝑓𝑙(𝜂, 𝛼)𝑑 𝜂, ∞
0
𝜂𝑠−1
𝑓𝑟(𝜂, 𝛼)𝑑 𝜂
=𝜆−𝑠
𝑓(𝜏)(𝑠, 𝛼)
Therefore, we can write
𝑓(𝜆𝜏)(𝑠) = 𝜆−𝑠
𝑓(𝜏)(𝑠)(49)
where 0< 𝑅𝑒(𝑠)<1and 𝜆 > 0.
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P. Singh et al.
Note 3. Proposition 2describes the scaling properties of the fuzzy Mellin
transformation in the proposed perspective of this article.
Proposition 3. If the first order derivative
𝑓of the fuzzy valued function
𝑓
in the sense of Modified Hukuhara derivative is integrable and 𝜏𝑠−1 ⊙
𝑓→0
as 𝜏→0and 𝜏→∞, then fuzzy Mellin transformation of
𝑓is
𝑓(𝜏)(𝑠) = −(𝑠− 1)
𝐹(𝑠− 1) (50)
where
𝐹(𝑠) =
𝑓(𝜏)(𝑠).
Proof. From the discussion of the earlier Section 3.2, we get
𝑓(𝜏)(𝑠) = ∞
0
𝜏𝑠−1 ⊙
𝑓(𝜏)𝑑𝜏 (51)
Performing integration by parts, in the Modified Hukuhara sense,
we get
𝑓(𝜏)(𝑠) = 𝜏𝑠−1 ⊙
𝑓(𝜏)∞
0− (𝑠− 1) ∞
0
𝜏𝑠−1 ⊙
𝑓(𝜏)𝑑𝜏
=−(𝑠− 1)
𝐹(𝑠− 1)
since, 𝜏𝑠−1 ⊙
𝑓→0as 𝜏→0and 𝜏→∞and
𝐹(𝑠) =
𝑓(𝜏)(𝑠) = ∞
0
𝜏𝑠−1 ⊙
𝑓(𝜏)𝑑𝜏
This completes the proof of Proposition 3.
Proposition 4. If
𝑓=
𝐹(𝑠)then 𝜏 ⊙
𝑓(𝜏)= −𝑠
𝐹(𝑠)where
𝑓
is the first order derivative of the fuzzy valued function
𝑓in the Modified
Hukuhara sense and 𝜏𝑠−1 ⊙
𝑓→0as 𝜏→0and 𝜏→∞.
Proof. The proof of Proposition 4 is the consequence of the result of
Proposition 3. So, we obtain the result as follows:
𝜏 ⊙
𝑓(𝜏)(𝑠) = ∞
0
𝜏𝑠−1 ⊙ 𝜏 ⊙
𝑓(𝜏)𝑑𝜏
=∞
0
𝜏𝑠⊙
𝑓(𝜏)𝑑𝜏
= − 𝑠
𝑓(𝑠)
(52)
Since, ∞
0𝜏𝑠⊙
𝑓(𝜏)𝑑𝜏 = −(𝑠− 1)
𝑓(𝑠− 1) by Proposition 3.
Proposition 5. If (
𝑓) =
𝐹(𝑠)then (𝜏 ⊙ 𝜏 ⊙
𝑓(𝜏)) = (−1)2𝑠(𝑠+ 1)
𝐹(𝑠)
provided 𝜏𝑠−1 ⊙
𝑓→0as 𝜏→0and 𝜏→∞.
In the mentioned statement,
𝑓denotes the second order fuzzy derivative
of
𝑓in the Modified Hukuhara derivative.
Proof. We omit the proof as it can be derived by successive integration
by parts methods and the pathway directed by Propositions 3 and 4.
Now, we solve some popular examples using the definition of fuzzy
Mellin transformation as in the next section (Section 4).
4. Numerical illustration
Now, using the above proposed results, we obtain fuzzy Mellin
transform of some fuzzy valued functions,
Example 5.
1
1 + 𝜏 =𝛤 𝑠𝛤 (1 − 𝑠) = 𝜋csc 𝑠𝜋 , 0<R(𝑠)<1(53)
Solution:
1
1 + 𝜏 =∞
0
𝜏𝑠−1 ⊙1
1 + 𝜏 𝑑 𝜏 (54)
Let, 𝜏 =𝜐
1 − 𝜐 ,𝜐 =𝜏
1 + 𝜏 then
=∞
0
(𝜐
1 − 𝜐 )𝑠−1 ⊙1
1 − 𝜐 ⊙ ⊖ 1
(1 − 𝜐)2𝑑 𝜐
=∞
0
𝜐𝑠−1 ⊙1
(1 − 𝜐)𝑠−2 𝑑 𝜐
=1
0
𝜐𝑠−1 ⊙(1 − 𝜐)1+(1−𝑠)𝑑 𝜐
(55)
This is well known expression of the 𝛽function.
𝛽(𝑠, 1 − 𝑠) = ⊖𝛤 𝑠𝛤 (1 − 𝑠) = 𝜋csc 𝑠𝜋, 0<R(𝑠)<1(56)
After taking the inverse Mellin Transform, we have the desired solution.
Example 6. If the function 𝑒−𝑛 𝜏 is integrable over (0,∞) then the fuzzy
Mellin transformation of 𝑒−𝑛 𝜏 is
(𝑒−𝑛 𝜏 ) = ∞
0
𝜏𝑠−1 ⊙ 𝑒−𝑛𝜏 𝑑 𝜏 (57)
Solution: The parametric form of the following fuzzy variables,
𝛼(𝑒−𝑛 𝜏 ) = [min{𝑒−𝑛𝜏 , 𝑒−𝑛𝜏 },max{𝑒−𝑛𝜏 , 𝑒−𝑛𝜏 }] (58)
and
𝛼𝜏𝑠−1 = [𝜏𝑠−1, 𝜏 𝑠−1],∀𝜏 > 0(59)
Since, 𝜏 > 0,min(𝑒−𝑛𝜏 , 𝑒−𝑛𝜏 ) = 𝑒−𝑛𝜏 ,max(𝑒−𝑛𝜏 , 𝑒−𝑛𝜏 ) = 𝑒−𝑛𝜏 .
Thus, using the Mellin transformation for crisp 𝑒−𝑛𝜏 [45], we get,
(𝑒−𝑛𝜏 ) = ∞
0
𝜏𝑠−1𝑒−𝑛𝜏 𝑑 𝜏 (60)
(𝑒−𝑛𝜏 ) = ∞
0
𝜏𝑠−1𝑒−𝑛𝜏 𝑑 𝜏 (61)
𝑒−𝑛𝜏 , 𝑒−𝑛𝜏 =∞
0𝜏𝑠−1𝑒−𝑛𝜏 𝑑 𝜏, 𝜏 𝑠−1𝑒−𝑛𝜏 𝑑𝜏 (62)
The equation is given in parametric form or 𝛼−cut form. So, using
the Decomposition theorem in Klir [10], Eq. (42) becomes,
𝑒−𝑛 𝜏 =∞
0
𝜏𝑠−1 ⊙ 𝑒−𝑛𝜏 𝑑 𝜏 (63)
After solving integration, we have 𝑒−𝑛 𝜏 =𝛤 𝑠
𝑛𝑠.
Example 7. If the function 𝑒−𝑖𝑛𝜏 = cos 𝑛 𝜏 ⊖ 𝑖 sin 𝑛 𝜏 is integrable over
(0,∞) then the fuzzy Mellin transformation of 𝑒−𝑖𝑛 𝜏 is,
(𝑒−𝑖𝑛 𝜏 ) = ∞
0
𝜏𝑠−1 ⊙ 𝑒−𝑖𝑛𝜏 𝑑 𝜏 (64)
(sin(𝑛 𝜏 ))=∞
0
𝜏𝑠−1 ⊙sin(𝑛 𝜏 )𝑑 𝜏 =𝑛−𝑠𝛤 𝑠 cos 𝜋𝑠
2(65)
and
(cos(𝑛 𝜏 ))=∞
0
𝜏𝑠−1 ⊙cos(𝑛 𝜏 )𝑑 𝜏 =𝑛−𝑠𝛤 𝑠 sin 𝜋𝑠
2(66)
Solution: To prove the above results, consider the complex fuzzy
valued function 𝑒−𝑖𝑛 𝜏 . As, we know that complex numbers are not com-
parable, for defining the parametric form of fuzzy complex function,
we focus on the real parts involved in it.
So, we use the following identity, 𝑒−𝑖𝑛 𝜏 = cos 𝑛 𝜏 ⊖ 𝑖 sin 𝑛 𝜏.
Let the parametric form of 𝑒−𝑖𝑛 𝜏 , as follows,
𝛼𝑒−𝑖𝑛 𝜏 =min(cos 𝑛(𝜏, 𝜏)),max(cos 𝑛(𝜏 , 𝜏))
⊖ 𝑖 min(sin 𝑛(𝜏, 𝜏)),max(sin 𝑛(𝜏 , 𝜏))
=min{min(cos 𝑛(𝜏, 𝜏 )) − 𝑖min(sin 𝑛(𝜏, 𝜏 ))},
max{max(cos 𝑛(𝜏, 𝜏 )) − 𝑖max(sin 𝑛(𝜏,𝜏))}
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P. Singh et al.
Now, we get,
𝛼𝑒−𝑖𝑛 𝜏 = [min (𝑒−𝑖𝑛(𝜏,𝜏),max(𝑒−𝑖𝑛(𝜏,𝜏))] (67)
Now, we apply crisp Mellin Transform [45] on above equation,
𝛼𝑒−𝑖𝑛 𝜏 =∞
0𝜏𝑠−1,𝜏𝑠−1
⊙min(𝑒−𝑖𝑛 𝜏 ),max(𝑒−𝑖𝑛 𝜏 )𝑑 𝜏, 𝑑 𝜏(68)
Using fuzzy multiplication,
𝛼𝑒−𝑖𝑛 𝜏 =∞
0𝜏𝑠−1 min(𝑒−𝑖𝑛(𝜏,𝜏))𝑑𝜏,
𝜏𝑠−1 max(𝑒−𝑖𝑛(𝜏,𝜏))𝑑𝜏(69)
For solving purposes, we consider,
𝛼𝑒−𝑖𝑛 𝜏 =∞
0
𝜏𝑠−1𝑒−𝑖𝑛𝜏 𝑑 𝜏, ∞
0
𝜏𝑠−1𝑒−𝑖𝑛𝜏 𝑑 𝜏(70)
Since 𝑠is a non fuzzy number so, we have only a single value
after solving the above interval. But we do calculations in a fuzzy
environment, so, keep all terms in interval form.
𝛤 𝑠
𝑛𝑠𝑖−𝑠,𝛤 𝑠
𝑛𝑠𝑖−𝑠=𝛤 𝑠
𝑛𝑠cos 𝑠𝜋
2+𝑖sin 𝑠𝜋
2,
𝛤 𝑠
𝑛𝑠cos 𝑠𝜋
2+𝑖sin 𝑠𝜋
2
Comparing real and imaginary parts,
∞
0𝜏𝑠−1 cos 𝑠𝜏𝑑 𝜏 =𝛤 𝑠
𝑛𝑠cos 𝑠𝜋
2
∞
0𝜏𝑠−1 sin 𝑠𝜏𝑑 𝜏 =𝛤 𝑠
𝑛𝑠sin 𝑠𝜋
2(71)
∞
0𝜏𝑠−1 sin 𝑠𝜏𝑑 𝜏 =𝛤 𝑠
𝑛𝑠sin 𝑠𝜋
2
∞
0𝜏𝑠−1 cos 𝑠𝜏𝑑 𝜏 =𝛤 𝑠
𝑛𝑠cos 𝑠𝜋
2(72)
Using Decomposition theorem in Klir [45], we get,
(sin 𝑛 𝜏 ) = ∞
0
𝜏𝑠−1 ⊙sin 𝑛 𝜏 𝑑 𝜏 =𝑛−𝑠𝛤 𝑠 cos 𝑛𝜋
2(73)
and,
(cos 𝑛 𝜏 ) = ∞
0
𝜏𝑠−1 ⊙cos 𝑛 𝜏 𝑑 𝜏 =𝑛−𝑠𝛤 𝑠 sin 𝑛𝜋
2(74)
Example 8.
(
2⊙ 𝑒−𝜏 2) = ∞
0
𝜏𝑠−1 ⊙
2⊙ 𝑒−𝜏 2⊗ 𝑑 𝜏 (75)
Solution: After solving the integration,
(
2⊗ 𝑒−𝜏 2) =
2⊗1
2𝛤(𝑠
2)
For verifying this value in a crisp setup, we can write the parametric
form of
2 = (1,2,3)
𝛼
2 = [1 + 𝛼 , 3 − 𝛼]⊗1
2𝛤(𝑠
2)(76)
At core, 𝛼= 0, the value is 𝛤(𝑠
2)which exactly matches with a crisp
one.
Example 9.
1
𝑒𝜏 −1 =∞
0
𝜏𝑠−1 ⊙1
𝑒𝜏 − 1𝑑 𝜏 (77)
Solution: We know that,
∞
𝑛=1
𝑒−𝑛 𝜏 =𝑒−𝜏 ⊕ 𝑒−2 𝜏 ⊕ 𝑒−3 𝜏 ⊕…
=𝑒−𝜏
1 − 𝑒−𝜏 =1
𝑒𝜏 − 1
(78)
or,
∞
𝑛=1
𝑒−𝑛 𝜏 =
∞
𝑛=1 ∞
0
𝜏𝑠−1 ⊙ 𝑒−𝑛𝜏 𝑑 𝜏
=
∞
𝑛=1
𝛤 𝑠
𝑛𝑠=𝛤 𝑠
𝑓(𝑠)
(79)
5. Application of Fuzzy Mellin transformation
The preceding section gives details of several mathematical exam-
ples solved by fuzzy Mellin transformation. In this point of view, all of
the mathematical encounter in Section 4belongs to the domain of ap-
plicability of the proposed theory. So, in this section, we mention some
specific possible applications of the proposed theory without detailing
the mathematical steps. Following maybe some notable applications of
the fuzzy Mellin transformation in the mentioned definition:
1. Fuzzy Mellin transformation exhibits a nice scale in variance
characteristic, which is analogous to that of Furrier transforma-
tion. This characteristic is very much desirable mathematical
tool for the logic involved with image recognition. An image
in an uncertain recognition phenomenon can be easily scaled
during the movement of two objects towards or away from
the Camera. Thus, the proposed theory may have impactful
relevance in the study of computer science.
2. It can be established using the proposed theory in this article that
the fuzzy Mellin transformations of the product of two fuzzy val-
ued independent random variables in probability theory is equal
to the product of their individual fuzzy Mellin transformation.
Thus, fuzzy Mellin transformation may be regarded as a very
impactful advance regarding the probability theory.
3. Suppose, we consider a fuzzy valued Riemann Zeta function
𝜁(𝑠).
Then
𝜁(𝑠)can be given by
𝜁(𝑠) = 1
𝛤(𝑠)∞
0
1
𝑒𝜏− 1 𝑑 𝜏 (80)
That, the fuzzy valued Riemann Zeta function can be expressed
by the fuzzy Mellin transformation of a fuzzy valued function
𝑓(𝜏) = 1
𝑒𝜏−1 .
4. Suppose, the potential in an infinite wedge is taken into a fuzzy
scenario. Then, the fuzzy potential function
𝜙(𝑟, 𝜃)will satisfy
the following fuzzy partial differential equation
𝑟2𝛿2
𝜙2
𝛿𝑟2+𝑟𝛿
𝜙
𝛿𝑟 +𝛿2
𝜙
𝛿𝜃2= 0 (81)
with infinite wedge 0< 𝑟 < inf and −∞ < 𝜃 < ∞having the
boundary conditions as follows:
𝜙(𝑟, 𝛼) =
𝑓(𝑟)
𝜙(𝑟, −𝛼) = 𝑔(𝑟)
where
𝑓and 𝑔 are two fuzzy valued functions and
𝜙(𝑟, 𝜃)→0as
𝑟→∞for all 𝜃∈ (−∞,∞).
The above problem has significant role in physics and engineer-
ing and it can be dealt with by the proposed theory of this article.
The problem can be a viewed as a fruitful future consequence of
the proposed theory.
5. Suppose,
𝑓be a fuzzy valued function. Then, Weyl functional
integration and derivative of order 𝛼can be defined as follows:
−𝛼
𝑓(𝜏)=1
𝛤 𝛼 ∞
𝜂
(𝜂−𝜏)𝛼−1
𝑓(𝜂)𝑑𝜂 (82)
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P. Singh et al.
and
𝛼
𝑓(𝜏)=(−1)𝑛
𝛤(𝑛−𝛼)
𝑑𝑛
𝑑𝜏 𝑛∞
𝜂
(𝜂−𝜏)(𝑛−𝛼−1)𝑑 𝜏 (83)
where 𝛼, (𝑛−𝛼)>0and the derivative and improper integrals
are given in Modified Hukuhara sense. If
𝑓(𝜏)(𝑠) =
𝐹(𝑠)be
the fuzzy Mellin transformation of
𝑓, it can be established that
−𝛼
𝑓(𝜏)(𝑠) = 𝛤(𝑠)
𝛤(𝑠+𝛼)
𝐹(𝑠+𝛼)(84)
and
𝛼
𝑓(𝜏)(𝑠) = 𝛤(𝑠)
𝛤(𝑠−𝛼)
𝐹(𝑠−𝛼)(85)
where 𝛤denotes the Gamma function.
6. Conclusion
Dealing with fuzzy numbers and fuzzy valued functions without
taking their level cuts was a significant challenge. In this context,
this present research work has contributed an uncertain representation
of the Mellin transformation in a newly designed fuzzy frame where
all the fuzzy operations have been done without taking parametric
representations consisting of the crisp components. This paper describes
the properties of the newly defined fuzzy Mellin transformation, such
as the law of linearity, scaling and sifting properties, and results of the
modified Hukuhara derivative of fuzzy valued functions. The adequate
examples and notes also justify the proposed theory. The proposed
approach has provided hints for the future in the following directions:
First, the idea of fuzzy Laplace transformation is widely used to solve
fuzzy differential equations and concerned mathematical models. Fuzzy
Mellin transformation can replace that of Laplace due to the proposed
theory. Second, the results of the fractional derivatives and differen-
tial equations can be reviewed in the light of the introduced Fuzzy
Mellin transformation. Third, the Mellin transformation is widespread
in computer science and quantum field theory. Again, modelling in
such domains incurred uncertainty. The newly proposed fuzzy Mellin
transformation may be a fruitful mathematical tool in this context.
CRediT authorship contribution statement
Payal Singh: Conceptualization, Data curation, Formal analysis,
Methodology, Resources, Visualization, Writing – original draft. Kamal
Hossain Gazi: Data curation, Formal analysis, Investigation, Method-
ology, Software, Supervision, Writing – original draft, Writing – review
& editing. Mostafijur Rahaman: Conceptualization, Formal analysis,
Investigation, Supervision, Validation, Visualization, Writing – origi-
nal draft, Writing – review & editing. Tripti Basuri: Data curation,
Methodology, Resources, Software, Writing – original draft. Sankar
Prasad Mondal: Conceptualization, Funding acquisition, Investigation,
Project administration, Resources, Software, Supervision, Validation,
Visualization, Writing – review & editing.
Declaration of competing interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Data availability
All the necessary data are cited in the article.
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Dr. Payal Singh working as an Assistant Professor in
Department of Applied Sciences and Humanities, Parul Uni-
versity. She has 13 years of teaching experience. She has
cleared the GSET exam in Mathematics. She teaches real
analysis, complex analysis, functional analysis, and applied
PDE in post graduation classes. Her research area is fuzzy
dynamical systems, mathematical modelling and fractional
calculus.
Kamal Hossain Gazi is a Research Scholar at Maulana Abul
Kalam Azad University of Technology, West Bengal, India.
He was qualified CSIR NET in June 2020 & June 2021.
He received UGC JRF under the Maulana Azad National
Fellowship (MANF) scheme in August 2022. He completed
his B.Sc. (Honours) and M.Sc. (Pure) degrees from Aliah
University, Kolkata in 2016 and 2018, respectively. He
already published more than 10 research articles in various
reputed journals. His research interests are Fuzzy sets, Op-
timization, Fuzzy Algebra, Linear Programming, Differential
Equations, Operations Research, Data Analysis, MCDM and
Mathematical Modelling.
Mostafijur Rahaman has submitted his Ph.D. thesis re-
cently after completing research in the Department of
Mathematics, Indian Institute of Engineering Science and
Technology, Shibpur, India. He was an awardee of UGC-
JRF and UGC-SRF for his Ph.D. tenure by University Grant
Commission (UGC). Earlier, he did M.Sc. in Mathematics
(Pure) and B.Sc. (Honours) in Mathematics from Univer-
sity of Kalyani, India in 2017 and 2015, respectively. He
received UGC Merit Scholarship for Rank Holder during
his M. Sc. for securing first position among first classes
in B. Sc. (Honours) in Mathematics. His research interests
include supply chain and inventory planning in uncertain
environments, differential equations under uncertainty and
fractional ordered calculus, etc. Till date, He published
Twenty four research articles in journals of repute and seven
chapters in edited books. He is an active reviewer of many
journals of repute.
Tripti Basuri is an Assistant Teacher in Uttar Dumdum
Vidyapith (Girls) Primary Section, West Bengal, India. She
completed her B.Sc in Mathematics Hons from St. Paul’s
Cathedral Mission College (under the University of Calcutta)
in 2013, and M.Sc in Mathematics from Bethune College
affiliated to the University of Calcutta in 2015. She also
cleared the exams such as NET (rank-65) in 2020, SET in
2020 and GATE in 2021 in Mathematical Science. Now, she
is working as a researcher and her areas of interest are
fuzzy sets and it’s applications, MCDM, Optimization theory,
SDGs, and Educational policies.
Dr. Sankar Prasad Mondal is an Assistant Professor in
the Department of Applied Mathematics, Maulana Abul
Kalam Azad University of Technology, West Bengal, India.
Previously he was working as an Assistant Professor in
the Department of Mathematics in Midnapore College (Au-
tonomous) and National Institute of Technology, Agartala.
He completed his B.Sc. in Mathematics Hons. From Krishna-
gar Government College (under Kalyani University) in 2008,
M.Sc. in Applied Mathematics from Bengal Engineering and
Science University, Shibpur in 2010 and Ph.D from Indian
Institute of Engineering Science and Technology, Shibpur in
2014. He has 10 years of teaching and 14 years of research
experience in the field of operations research, differential
equations, fuzzy sets, mathematical biology, fuzzy differen-
tial equations, Soft Computing, Artificial Intelligence, and
Optimization theory. He already published more than 150
research papers in reputed journals, book chapters and
conferences. The Start-Up Research Grant FRPS (UGC) was
received in 2019 August.
