1. Introduction
Many researchers have studied the use of fuzzy partial differential and integral equations. These equations are an excellent tool for modeling vagueness and misinterpretation of knowledge-based systems [
1], control systems [
2], image processing [
3], industrial automation, power engineering, robotics [
4], artificial intelligence [
5], and consumer electronics. Buckley and Feuring developed the first definition of fuzzy partial differential equations [
6]. The difference method for solving fuzzy partial differential equations is proposed in [
7]. Nemati and Matinfar [
8] constructed an implicit finite difference approach to solving complex fuzzy parabolic differential equations. An explicit numerical solution to the fuzzy hyperbolic and parabolic equation has been introduced in [
9]. Recently, Arqub et al. [
10] applied the reproducing kernel algorithm to find a solution to fuzzy boundary value problems. In [
11], the fuzzy Fredholm–Volterra integro-differential equations are solved by using the adaptation of the reproducing kernel algorithm. Various numerical methods have been used to address fuzzy problems [
12,
13,
14,
15].
Integral transforms have been used in solving many types of equations, increasing their importance and need for research in their application. The fuzzy version of classical General transform is introduced by Rashid et al. [
16]. Ullah et al. [
17] proposed the fuzzy Yang transformation as a method to find a solution to second-order fuzzy differential equations of integer and fractional order.
Recently, some researchers have introduced double fuzzy transforms for solving fuzzy partial differential equations. Stabestari and Ezzati constructed a double fuzzy Laplace transform that is applicable to solve the fuzzy wave equation [
18]. The double fuzzy Elzaki transform is used to find the solution of the fuzzy Poission’s equation and the fuzzy Telegraph Equation [
19]. A fuzzy solution to the Telegraph equation is obtained in [
20] by applying the fuzzy Sawi transform. In work [
21], by use of a double fuzzy Aboodh transform, we derive the solution of a fuzzy partial differential equation of first order.
In our study, the aim was to introduce the use of a new double fuzzy integral transform, called the double fuzzy Yang–General transform (DFY-GT). To achieve this, we prove some of the basic properties of DFY-GT and compute values of DFY-GT for some functions. New theorems related to partial gH-derivatives are established. The originality of this paper comes from using a combination between the single fuzzy transforms of Yang and General, where the new double fuzzy Yang–General transformation has the advantages of two transforms.
The advection–diffusion equation is the model that can be used to simulate natural processes. The advection is due to the movement of materials from one region to another and diffusion is due to the movement of materials from higher concentration to low concentration. Different papers have appeared to solve and use this equation [
22,
23,
24,
25].
In this research, we are implemented DFY-GT to solve the following linear fuzzy advection–diffusion equation
where
is a constant diffusion coefficient and
is constant advection velocity. A simple formula for the solution of the above equation is obtained and applied to solve a numerical example in order to display the efficiency of this new approach.
The rest of this paper is organized as follows: In
Section 2, we present the basic concepts that will be used in the main part of the paper. In
Section 3, we discuss the fundamental facts and properties of single fuzzy Yang and General transforms. In
Section 4, we introduce a new integral transformation, the DFY-GT, which combines the fuzzy Yang transform and the General fuzzy transform, and we define some properties of this transform. In
Section 5, we apply a DFY-GT to the fuzzy linear advection–diffusion equation, and we find a formula for the exact solution.
Section 6 provides a numerical experiment that demonstrates the effectiveness of the proposed method. Finally, the concluding remarks and comments are contained in
Section 7.
2. Basic Concepts
In this section, we introduce some definitions and theorems that will be used in most of the paper.
We will denote the set of fuzzy numbers. A fuzzy number is a mapping and satisfies the conditions
- (i)
is normal, i.e., there exists for which ;
- (ii)
is upper semi continuous in the higher half;
- (iii)
for all and ;
- (iv)
is compact.
Definition 1 ([
26])
. The r-level set of fuzzy number is defined byThe core of the fuzzy number w is the set of elements of that have membership grade 1
, i.e., A fuzzy set w is a fuzzy number if and only if the r-levels are nonempty compact intervals of the form .
Definition 2 ([
26])
. An ordered pair is called a parametric form of fuzzy number w, if the functions satisfy the conditions: - (i)
, are bounded monotonic left continuous for all and right continuous for ;
- (ii)
, are non-decreasing and non-increasing, respectively;
- (iii)
for all .
Definition 3 ([
26])
. An ordered foursome is called a trapezoidal fuzzy number w, if and . The r-levels of this fuzzy number are . A triangular fuzzy number is obtained if . Let
, where
and
and
. The addition
and the scalar multiplication
are defined
The subtraction of fuzzy numbers
v and
w is defined as addition, that is,
The Hukuhara difference (
H-difference) of two fuzzy numbers
v and
w has been introduced as a fuzzy number
if and only if
. The
H-difference is unique, but it does not always exist. If
exists, then its parametric form is
Definition 4 ([
26])
. The generalized Hukuhara difference (gH-difference) of two fuzzy numbers v and w is the fuzzy number u if it exists and Sufficient conditions for the existence of the
-difference are obtained in [
26].
For an interval
, we define the norm
Then, the Hausdorff distance between two fuzzy numbers
v and
w is given as
The metric D is well defined, since the -difference of the intervals always exists. Hence is a complete metric space.
The following properties for the
-difference are given in [
27].
Proposition 1. Suppose that v and w are fuzzy numbers, then
- (i)
if exists, then or ;
- (ii)
if exists, then it is unique;
- (iii)
if exists in the sense , then exists in the sense and vice versa;
- (iv)
;
- (v)
;
- (vi)
if if and only if .
2.1. Fuzzy Function of One-Variable
In the following part of the article, we introduce some definitions and some properties of the fuzzy-valued function
. The endpoint functions
are called the upper and lower functions of the fuzzy-valued function
for each
. Then, the parametric form of the function
is
A fuzzy-valued function is a function that produces an image of a crisp domain in a fuzzy set.
Example 1. Consider two crisp sets, and .
A fuzzy-valued function g maps the elements in A to the power set in the following manner.where , , , . The function g maps element to element with degree , to element with , and to element with . Now, we apply the r-cut operation to the fuzzy-valued function. Lemma 1 ([
18])
. Let be a fuzzy-valued function and real numbers and such that or , then Let
be a fuzzy-valued function with parametric form
. Suppose that, for all fixed
, the crisp functions
and
are integrable on
, for every
, and that there exist two positive constants
and
such that
for every
. Then
is fuzzy Riemann integrable on
, its improper fuzzy integral
and
Lemma 2 ([
18])
. Let or . If fuzzy-valued functions are improper fuzzy Rieman integrable on , then - (i)
;
- (ii)
.
Based on the gH-difference, we obtain the following definition for the gH-derivative.
Definition 5 ([
26])
. Let and k be such that . Then, the generalized Hukuhara derivative (-derivative) of a fuzzy-valued function at the point is fuzzy number defined asif limit exists. Definition 6 ([
26])
. The fuzzy-valued function is called - (i)
gH-differentiable
at the point if - (ii)
gH-differentiable at the point ifprovided that the functions and are differentiable at the point .
Now, we will give some basic properties for the gH-differentiable function.
Theorem 1 ([
28])
. Let the fuzzy-valued functions be gH-differentiable, then is gH-differentiable, and Theorem 2 ([
28])
. Let the fuzzy-valued function be -differentiable and the functions be differentiable. Then Theorem 3 ([
26])
. Let be -differentiable in the interval . Then Theorem 4 ([
29])
. Let be -differentiable in the interval and be differentiable functions in the interval . Then 2.2. Fuzzy Function of Two-Variable
Let be the parametric form of the fuzzy function for all .
Definition 7 ([
29])
. We say that the fuzzy-valued function is continuous at the point if for each there is such that whenever Definition 8 ([
28])
. Let the point and real constants h, k be such that , . Then, the first generalized Hukuhara partial derivative (-p-derivative) of fuzzy function at with respect to x and t are fuzzy numbers and defined byprovided that limits exist in Equations and . Definition 9 ([
28])
. Let be a fuzzy-valued function, and , both be partial differentiable at the point with respect to t. Then, we say that is
- (i)
-p-differentiable at the point with respect to t if - (ii)
gH-p-differentiable at the point with respect to t if
Theorem 5 ([
28])
. Let the fuzzy-valued function be -p-differentiable with respect to t and . Then, exists and Theorem 6 ([
29])
. Let be a fuzzy-valued function. Suppose that is convergent for each and is convergent for each . Then 3. Basic Definitions and Theorems for Yang and General Fuzzy Transforms
In this section, we give the definitions and some fundamental properties of the fuzzy Yang and fuzzy General transforms.
3.1. Fuzzy Yang Transform
The definition of fuzzy Yang transform (FYT) is introduced in [
17].
Definition 10. Let and the function be fuzzy improper integrable on for . Then, the FYT of is defined aswhere x and α are transform variables. Definition 11. The fuzzy inverse Yang transform is given bywhere the function must be analytic for all α such that . Definition 12. We say that the fuzzy-valued function is of exponential order if there exists a constant , such that for all Theorem 7. Let the fuzzy function be continuous in every interval of exponential order . Then, the FYT of exists for all α provided that .
Proof. Using Definition 10 and the properties of fuzzy improper integral, we obtain
where
. □
In [
30], the classical Yang transform is applied on some special functions.
- (i)
;
- (ii)
where m is positive integer;
- (iii)
where ;
- (iv)
where ;
- (v)
where .
Using Definition 10, we obtain the following useful properties of the FYT.
Theorem 8. Let for fuzzy-valued functions exist a FYT. Then, the FYT of functions and exist and
- (i)
;
- (ii)
where the real numbers and such that or .
Proof. We prove case (ii). Using Lemma 2, we obtain
Analogously, we prove case (i). □
Theorem 9. Let the fuzzy-valued function be continuous of exponential order and be continuous in every interval . Then
- (i)
;
- (ii)
,
where and .
Proof. Using the definition of the improper fuzzy Riemann integral and Theorem 4, we obtain
The fuzzy-valued function
is of exponential order
. Then there exist
and
, such that
for all
. Hence
for
.
By the above equation and Proposition 1, we obtain
We will prove Equation (ii). Using Definition 10 and (
10), we have
□
Corollary 1. Let be a fuzzy-valued function. Then, the FYT for partial derivatives of is as follows
- (i)
;
- (ii)
,
where .
3.2. Fuzzy General Transform
The fuzzy General transform is introduced in [
16].
Definition 13. Let and the function be improper fuzzy Riemann integrable on for some . Then, the fuzzy General transform (FGT) of a function is defined aswhere and are positive real functions. If
and
, then FGT (
11) gives the FYT (
8).
Definition 14. The fuzzy inverse General transform is given bywhere must be analytic for all β such that . Theorem 10. Let the fuzzy-valued function be a continuous function in every finite interval of exponential order . Then, the FGT of exists for all β, such that .
Proof. From Definition 12, it follows that there exist
and
, such that
Hence
where
and
□
The classical General transform for some special functions is given in [
31].
- (i)
;
- (ii)
where n is a positive integer;
- (iii)
where ;
- (iv)
where ;
- (v)
where .
Using Definition 13, we obtain the basic properties of the FGT.
Theorem 11. Let for fuzzy-valued functions exist an FGT. Then, the FGT of functions and exists and
- (i)
;
- (ii)
,
where or .
Proof. Analogously in Theorem 8. □
Theorem 12. Let the fuzzy-valued function be continuous of exponential order and be continuous in every interval . Then
- (i)
;
- (ii)
,
where and we have.
Proof. Analogously in Theorem 9. □
Corollary 2. Let be a fuzzy-valued function. Then, the FYT for partial derivatives of is as follows:
- (i)
;
- (ii)
,
where .
4. Double Fuzzy Yang-General Transform
In this part, we introduce DFY-GT, which combines the Yang and General first-order fuzzy transforms. The definition and some of the fundamental properties of this integral transform are presented.
Definition 15. Let , the function be improper fuzzy integrable on for some and . Then, the fuzzy Yang–General transform of a function is defined aswhere is a real positive function. Definition 16. Double fuzzy inverse Yang–General transform is denoted by and Definition 17. A fuzzy function is called exponentially ordered , , if there exist positive real constants such that for all , Theorem 13. Let the fuzzy function be a continuous in of exponential order , . Then, the DFY-GT of exists for all α and with , .
Proof. Let
. Then
The improper double fuzzy integral converges for all , . Thus, exists. □
Lemma 3. Let and the fuzzy function . Then, Proof. Using Definition 15, we find
□
Lemma 4. Let . Then, we havefor and . Proof. Using Definition 15, we have
□
By using Yang and General transform on some special functions, we obtain
- (i)
- (ii)
where are positive integers;
- (iii)
where ;
- (iv)
where
Now, we present some properties of DFY-GT.
Using Theorems 8 and 11, we obtain that double fuzzy Yang–General transform is a linear transformation.
Theorem 14. Let be fuzzy-valued functions. Then,
- (i)
;
- (ii)
,
where or .
From Theorem 14, it follows that the double fuzzy inverse Yang–General transform is also a linear transformation.
Theorem 15. Let be a periodic function of periods ξ and η such thatand exists. Then, Proof. Using the Definition 15 and properties of improper fuzzy integral, we find
Putting
and
on second integral and using the periodicity of the function
, we obtain
By substituting into the above equation, we obtain
This equation can be simplified into
□
Theorem 16. Let exists. Thenwhereis the Heaviside function. Proof. By Definition 15, we obtain
Substituting
and
, we obtain
□
Theorem 17. Let . Then
- (i)
;
- (ii)
;
- (iii)
;
- (iv)
;
- (v)
;
- (vi)
Proof. Using Definition 15 and Corollary 1, we obtain
Similarly, we prove the case
. Indeed,
Now, we will prove case (iii). Applying Definition 15 and case (ii) of Corollary 1, we have
By Definition 15 and case (ii) of Corollary 2, we obtain case (iv). Really,
We will prove the case (v).
The proof of case (vi) is analogous to case (v). □
5. Applications of Double Fuzzy Yang–General Transform
In this section, we illustrate the application of DFY-GT and obtain the solution framework for the linear fuzzy advection–diffusion equation given by
with the initial condition
and the boundary conditions
The constant is the diffusion coefficient and the constant is the advection velocity.
Applying DFY-GT on both sides of Equation (
16), we obtain
Let
. Using the differentiation property of the DFY-GT (Theorem 17), we obtain
We apply the fuzzy Yang transform to Equation (
17) and the fuzzy General transform to Equation (
18).
Operation of fuzzy Yang transform to the initial condition yields
Fuzzy General transform of the boundary conditions is given by
Substituting into the above equation, we obtain
By using Proposition 1, we obtain
Applying the inverse DFY-GT on both sides of Equation (
19), we obtain
.
6. Examples
Example 2. Here, we consider a numerical example solved using the presented method to demonstrate the application of DFY-GT. The initial and boundary conditions are given as In this case, and fuzzy functions Then, applying FYT to Equation (21) and FGT to Equation (22), we find By using Equation (19), we obtain We can simplify the above equation as Finally, taking the inverse DFY-GT of the above equation, we find Thus, the solution for fuzzy advection–diffusion Equations (20)–(22) is 7. Conclusions
In this paper, a new double fuzzy transform called DFY-GT was introduced. Some fundamental properties of this transform are presented. New theorems related to the existence, linearity, periodicity, and gH-partial derivatives were proven. These results were used to obtain a new simple formula for solving the linear fuzzy advection–diffusion equation. Finally, we constructed a numerical example and obtained the exact solution of the equation considered applying the new double fuzzy integral transform.
This work can be extended to the case where the fuzzy linear advection–diffusion equation is two-dimensional. Furthermore, we can consider analytical solutions of fuzzy nonlinear advection–diffusion problems. In addition, future applications of the DFY-GT can be developed to solve fuzzy parabolic integro-differential equations.
Author Contributions
Conceptualization, A.G. and S.I.C.; methodology, A.G. and S.I.C.; validation, A.G. and S.I.C.; formal analysis, A.G., S.I.C. and M.S.; writing—original draft preparation, A.G., S.I.C. and M.S.; writing—review and editing A.G., S.I.C. and M.S.; funding acquisition A.G., S.I.C. and M.S. All authors have read and agreed to the published version of the manuscript.
Funding
This study is financed by the European Union-Next Generation EU through the National Recovery and Resilience Plan of the Republic of Bulgaria, project DUECOS BG-RRP-2.004-0001-C01.
Data Availability Statement
No new data were created or analyzed in this study. Data sharing is not applicable to this article.
Conflicts of Interest
The authors declare no conflicts of interest.
References
- Di Martino, F.; Perfilieva, I.; Sessa, S. A Summary of F-Transform Techniques in Data Analysis. Electronics 2021, 10, 1771. [Google Scholar] [CrossRef]
- Babuška, R. Fuzzy Modeling for Control; International Series in Intelligent Technologies; Springer: Berlin/Heidelberg, Germany, 1998; p. 260. [Google Scholar]
- Lu, J.; Ma, G.; Zhang, G. Fuzzy Machine Learning: A Comprehensive Framework and Systematic Review. IEEE Trans. Fuzzy Syst. 2024, 32, 3861–3878. [Google Scholar] [CrossRef]
- Napole, C.; Derbeli, M.; Barambones, O. Experimental Analysis of a Fuzzy Scheme against a Robust Controller for a Proton Exchange Membrane Fuel Cell System. Symmetry 2022, 14, 139. [Google Scholar] [CrossRef]
- Kadham, S.; Alkiffai, A. Model Tumor Response to Cancer Treatment Using Fuzzy Partial SH-Transform: An Analytic Study. Int. J. Math. Comput. Sci. 2022, 18, 23–28. [Google Scholar]
- Buckley, J.; Feuring, T. Introduction to fuzzy partial differential equations. Fuzzy Sets Syst. 1999, 105, 241–248. [Google Scholar] [CrossRef]
- Allahveranloo, T. Difference methods for fuzzy partial differential equations. Comput. Methods Appl. Math. 2006, 2, 233–242. [Google Scholar] [CrossRef]
- Nemati, K.; Matinfar, M. An implicit method for fuzzy parabolic partial differential equations. J. Nonlinear Sci. Appl. 2008, 1, 61–71. [Google Scholar] [CrossRef]
- Allahviranloo, T.; Kermani, M.A. Numerical methods for fuzzy linear partial differential equations under new definition for derivative. Iran. J. Fuzzy Syst. 2010, 7, 33–50. [Google Scholar]
- Arqub, O.A.; Al-Smadi, M.; Momani, S.; Hayat, T. Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems. Soft Comput. 2017, 21, 7191–7206. [Google Scholar] [CrossRef]
- Arqub, O.A. Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm-Volterra integrodifferential equations. Neural Comput. Appl. 2017, 28, 1591–1610. [Google Scholar] [CrossRef]
- Gumah, G.; Al-Omari, S.; Baleanu, D. Soft computing technique for a system of fuzzy Volterra integro-differential equations in a Hilbert space. Appl. Numer. Math. 2020, 152, 310–322. [Google Scholar] [CrossRef]
- Gumah, G.; Naser, M.F.M.; Al-Smadi, M.; Al-Omari, S.K.Q.; Baleanu, D. Numerical solutions of hybrid fuzzy differential equations in a Hilbert space. Appl. Numer. Math. 2020, 151, 402–412. [Google Scholar] [CrossRef]
- Gumah, G. Numerical solutions of special fuzzy partial differential equations in a reproducing kernel Hilbert space. Comput. Appl. Math. 2022, 41, 80. [Google Scholar] [CrossRef]
- Gumah, G. Reproducing kernel Hilbert space method to solve fuzzy partial Volterra integro-differential equations. Filomat 2024, 38, 8553–8564. [Google Scholar]
- Rashid, S.; Rehana Ashraf, R.; Hammouch, Z. New generalized fuzzy transform computations for solving fractional partial differential equations arising in oceanography. J. Ocean. Eng. Sci. 2023, 8, 55–78. [Google Scholar] [CrossRef]
- Ullah, A.; Ullah, A.; Ahmad, S.; Van Hoa, N. Fuzzy Yang transform for second order fuzzy differential equations of integer and fractional order. Phys. Scr. 2023, 98, 044003. [Google Scholar] [CrossRef]
- Stabestari, R.M.; Ezzati, R. The Fuzzy Double Laplace Transforms and their Properties with Applications to Fuzzy Wave Equation. New Math. Natural Comp. 2021, 17, 319–338. [Google Scholar] [CrossRef]
- Kshirsagar, K.A.; Nikam, V.R.; Gaikwad, S.B.; Tarate, S.A. The double fuzzy Elzaki transform for solving fuzzy partial differential equations. J. Chungcheong Math. Soc. 2022, 35, 2. [Google Scholar]
- Georgieva, A.T.; Pavlova, A. Application of the Double Fuzzy Sawi Transform for Solving a Telegraph Equation. Symmetry 2023, 15, 854. [Google Scholar] [CrossRef]
- Abaas Alshibley, S.T.; Hashem Nouri, A.; Ali, A.J. Double Fuzzy Aboodh Transform for Solving Fuzzy Partial Differential Equations. J. Comput. Anal. Appl. 2024, 33, 78–84. [Google Scholar]
- Boztosun, I.; Charafi, A. An analysis of the linear advection-diffusion equation using mesh-free and mesh-dependent methods. Eng. Anal. Bound. Elem. 2002, 26, 889–895. [Google Scholar]
- Mojtabi, A.; Deville, M.O. One-dimensional linear advection-diffusion equation: Analytical and finite element solutions. Comput. Fluids 2015, 107, 189–195. [Google Scholar]
- Andallah, L.; Khatun, M. Numerical solution of advection-diffusion equation using finite difference schemes. Bangladesh J. Sci. Ind. Res. 2020, 55, 15–22. [Google Scholar]
- Ara, K.N.I.; Rahaman, M.M.; Alam, M.S. Numerical Solution of Advection Diffusion Equation Using Semi-Discretization Scheme. Appl. Math. 2021, 12, 1236–1247. [Google Scholar]
- Bede, B.; Stefanini, L. Generalized differentiability of fuzzy-valued functions. Fuzzy Sets Syst. 2013, 230, 119–141. [Google Scholar]
- Stefanini, L. A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets Syst. 2010, 161, 1564–1584. [Google Scholar]
- Allahviranloo, T.; Gouyndeh, Z.; Ahmand, A.; Hasanoglu, A. On fuzzy solutions of heat equation based on generalized Hukuhara differentiability. Fuzzy Sets Syst. 2015, 265, 1–23. [Google Scholar]
- Gouyandeh, Z.; Allahviranloo, T.; Abbasbandy, S.; Armand, A. A fuzzy solution of heat equation under generalized Hukuhara differentiability by fuzzy Fourier transform. Fuzzy Sets Syst. 2017, 309, 81–97. [Google Scholar]
- Yang, X.J. A new integral transform method for solving steady heat-transfer problem. Therm. Sci. 2016, 20, 639–642. [Google Scholar]
- Jafari, H. A new general integral transform for solving integral equation. J. Adv. Res. 2021, 32, 133–138. [Google Scholar]
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