International Journal of Fuzzy Logic and Intelligent Systems 2020; 20(3): 247-254
Published online September 25, 2020
https://doi.org/10.5391/IJFIS.2020.20.3.247
© The Korean Institute of Intelligent Systems
V. Padmapriya1,2, M. Kaliyappan3, and A. Manivannan3
1Research Scholar, Vellore Institute of Technology, Chennai Campus, India
2New Prince Shri Bhavani Arts and Sciences College, Chennai, India
3Division of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Chennai Campus, India
Correspondence to :
V. Padmapriya (v.padmapriya2015@vit.ac.in)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper, a novel technique is proposed to solve fuzzy fractional delay differential equations (FFDDEs) with initial condition and source function, which are fuzzy triangular functions. The obtained solution is a fuzzy set of real functions. Each real function satisfies an FFDDE with a specific membership degree. A detailed algorithm is provided to solve the FFDDE. The proposed method has been elucidated in detail through numerical illustrations. Graphs are plotted using MATLAB.
Keywords: Fractional differential equation (FDE), Fuzzy delay differential equation (FDDE), Fuzzy fractional differential equation (FFDE), Fuzzy fractional delay differential equation (FFDDE), Fuzzy set
Delay differential equation (DDE) is a type of differential equation in which the derivative of an unknown function at any time depends on the solution at prior times. DDE is mainly used in a large number of system models in physics, biology, and engineering. In addition, DDE is applied in many practical systems such as automatic control, traffic models, neuroscience, and lasers. References [1] and [2] presented a complete study of DDEs and their applications.
On the other hand, when a dynamical phenomenon with a delay is modeled using ordinary DDEs, the model is not always accurate. In general, the initial conditions or parameters of the equations are incomplete or vague. This limitation leads to the study of fuzzy DDEs (FDDEs). In recent years, the theory and application of FDDEs have been developed by many researchers. Lupulescu introduced a DDE with a fuzzy case in [3]. Consequently, Khastan et al. [4] and Hoa et al. [5] presented the results of the existence and uniqueness of solutions of generalized FDDEs. Moreover, Hoa et al. [6] introduced fuzzy delay integro-differential equations with generalized Hukuhara differentiability. Random fuzzy delay integro-differential equations were investigated in [7] and [8].
In many real-world tasks, the memory and hereditary properties of differential operators are broadly perceived to be well anticipated through fractional operators. Fractional order is an extension of the classical order of differentiation and integration. In the last few decades, numerous phenomena in different fields of mathematics, economics, engineering, and science have been more precisely described via fractional derivatives, and fractional differential equations (FDEs) have come up as another powerful device for modeling many difficult types of complex systems. Podlubny [9] provided a major contribution to this field. The existence, uniqueness result, and some integral transforms of FDEs are presented in [9]. Later, Kilbas et al. [10] presented the idea of fractional calculus. A large amount of literature has been dedicated to the theory and applications of FDEs [11–13]. Currently, fractional chaotic systems are gaining considerable interest among many researchers in the different fields of science, mathematics, and engineering. Huang et al. [14] analyzed the behavior of a chaotic system of a fractional order love model with an external environment. Moreover, Huang et al. [15] studied the relationship of Romeo and Juliet love fractional model using a fuzzy function.
The idea of Riemann–Liouville FDE with an uncertainty was introduced by Agarwal et al. [16]. Consequently, many authors presented results on the existence and uniqueness of solutions for fuzzy FDEs (FFDEs) using numerous methods [17–20]. Mazandarani et al. [21] used the modified Euler method to solve FFDEs with a Caputo derivative. Salahshour et al. [22] obtained the solution of FFDEs using the fuzzy Laplace transform method. Ahmed et al. [23] provided analytical and numerical solutions of FFDEs using the Zadeh’s extension principle.
In the present work, our aim is to obtain a solution for FDDEs under a fractional derivative. Many authors solved fuzzy fractional DDEs (FFDDEs) using some numerical methods. For instance, Hoa [24] proposed a solution of FFDDEs that involve the Caputo gH-differentiability using the Adams–Bashforth–Moulton method and also proposed the modified Euler method to solve FFDDEs with an initial condition in [25].
In all the above-cited studies, the source functions and initial conditions are considered as a fuzzy number-valued function. For this function, the derivative is considered as a gH-derivative. This gH-derivative has two types, namely, [(i)-gH] and [(ii)-gH] differentiability. The disadvantages of this derivative are the difficulty in determining which type of derivative to choose and whether the solution is unique. Another difficulty of the gH-derivative is that it may convert the fuzzy problem into two crisp corresponding problems, and we need to choose one solution from a set of solutions that more appropriately fits the problem.
Therefore, our main motivation is to avoid the difficulties in choosing the fuzzy derivative. Then, we enhance the methodology proposed by Fatullayev et al. [26] for FDDEs to FFDDEs. The authors in [26] solved fuzzy problems with an initial condition and a forcing function as a set of fuzzy real functions using the method of Gasilov et al. [27]. Consequently, in [28] the authors provided solutions for fuzzy problems with boundary conditions in which the forcing functions were fuzzy functions.
To the best of our knowledge, this is the first study where this technique is utilized for solving FFDDEs. In the present work, our main objective is to obtain a solution for FDDEs with a fractional derivative. Recently, fuzzy triangular membership function is a beneficial device in many designing applications [29]. Therefore, in this approach, we consider that the initial condition and source functions are in fuzzy triangular forms. Moreover, we prove that if the initial conditions and source functions are fuzzy triangular functions, then the value of the solution is also a fuzzy triangular function. According to this methodology, we present the existence and uniqueness results for of FFDDEs.
This paper is structured as follows: The basic definition and notation of the fractional derivatives, fuzzy sets and fuzzy functions are presented in Section 2. The proposed method for solving FFDDEs is described in Section 3. Numerical applications are performed to test the validity and reliability of the present algorithm in Section 4. Finally, the conclusion is drawn in Section 5.
In this section, we provide certain essential definitions and properties of the fuzzy fractional concept.
The Riemann–Liouville fractional integral of
The Caputo derivative of order
We provide fixed set
We also consider
According to the geometric explanation of a fuzzy number,
In this study,
A fuzzy set can also be represented through the definition of
The
In our work, we express a fuzzy function as a set of fuzzy real functions in which every real function possesses a specific membership degree. This function is defined by Gasilov et al. [27].
Let
Fuzzy set
The interpretation of triangular fuzzy function
The fuzzy triangular function is a fuzzy triangular number for each time
We consider the following FFDDE:
where
Fuzzy set
where
We can explain
Let us consider that
As stated in Definition 3, solution
for
Let us express
Because
Then, we split FFDDE (
Fractional non-homogeneous equation
Fractional homogeneous equation involving an initial function, which is a triangular fuzzy function
Fractional non-homogeneous equation involving a fuzzy source function and whose initial value is zero
Next, we develop an algorithm to solve these three equations.
We consider fractional non-homogeneous equation, i.e.,
By performing fractional integration on both sides of
Next, we solve
Let us consider that
Let us also consider that
Let us consider
According to Definition 3, functions
However, the definition of triangular fuzzy function implies that bunch
Moreover, we apply the linear property for the fractional order (see [30]). Then, if
This result implies that set
Fuzzy triangular function
Next, we solve
We let
We also let
Let us consider
The proof of this theorem can be derived from the same procedure used in Theorem 1.
Next, we provide the algorithm for solving FFDDE (
Let us consider the following FFDDE:
where
We represent the source function and initial value as
First, we find a solution to the following fractional associated non-homogeneous equation:
where
By solving
Solution
Second, we derive the solution to the fractional homogeneous equation involving fuzzy initial function
Then, we solve the following fractional equation:
For
The obtained solution from
Next, we derive the solution for the fractional non-homogeneous equation involving a fuzzy source function.
Then, we solve the crisp problems.
For
The obtained solution from
Finally, we derive the solution of
Solution
In this study, we obtained the solutions of an FFDDE according to the method proposed by Fatullayev et al. [26]. In the present work, we considered that the initial condition and source function are triangular fuzzy functions. Meanwhile, the FFDDE was converted into three equations based on the initial and source functions. Finally, we combined all the solutions of the abovementioned three equations to obtain a solution of the FFDDE. Graphs were plotted for the fuzzy solution. The obtained solutions for the FFDDE were expressed as a fuzzy set of real functions. We recommended an efficient method to compute the fuzzy solution set. The numerical example demonstrated that the proposed method is efficient and accurate for application to FFDDEs.
No potential conflict of interest relevant to this article was reported.
Solution function and
Fuzzy solution in terms of the initial values and
Fuzzy solution relative to the source function and its
E-mail: v.padmapriya2015@vit.ac.in
E-mail: kaliyappan.m@vit.ac.in
E-mail: manivannan.a@vit.ac.in
International Journal of Fuzzy Logic and Intelligent Systems 2020; 20(3): 247-254
Published online September 25, 2020 https://doi.org/10.5391/IJFIS.2020.20.3.247
Copyright © The Korean Institute of Intelligent Systems.
V. Padmapriya1,2, M. Kaliyappan3, and A. Manivannan3
1Research Scholar, Vellore Institute of Technology, Chennai Campus, India
2New Prince Shri Bhavani Arts and Sciences College, Chennai, India
3Division of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Chennai Campus, India
Correspondence to:V. Padmapriya (v.padmapriya2015@vit.ac.in)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper, a novel technique is proposed to solve fuzzy fractional delay differential equations (FFDDEs) with initial condition and source function, which are fuzzy triangular functions. The obtained solution is a fuzzy set of real functions. Each real function satisfies an FFDDE with a specific membership degree. A detailed algorithm is provided to solve the FFDDE. The proposed method has been elucidated in detail through numerical illustrations. Graphs are plotted using MATLAB.
Keywords: Fractional differential equation (FDE), Fuzzy delay differential equation (FDDE), Fuzzy fractional differential equation (FFDE), Fuzzy fractional delay differential equation (FFDDE), Fuzzy set
Delay differential equation (DDE) is a type of differential equation in which the derivative of an unknown function at any time depends on the solution at prior times. DDE is mainly used in a large number of system models in physics, biology, and engineering. In addition, DDE is applied in many practical systems such as automatic control, traffic models, neuroscience, and lasers. References [1] and [2] presented a complete study of DDEs and their applications.
On the other hand, when a dynamical phenomenon with a delay is modeled using ordinary DDEs, the model is not always accurate. In general, the initial conditions or parameters of the equations are incomplete or vague. This limitation leads to the study of fuzzy DDEs (FDDEs). In recent years, the theory and application of FDDEs have been developed by many researchers. Lupulescu introduced a DDE with a fuzzy case in [3]. Consequently, Khastan et al. [4] and Hoa et al. [5] presented the results of the existence and uniqueness of solutions of generalized FDDEs. Moreover, Hoa et al. [6] introduced fuzzy delay integro-differential equations with generalized Hukuhara differentiability. Random fuzzy delay integro-differential equations were investigated in [7] and [8].
In many real-world tasks, the memory and hereditary properties of differential operators are broadly perceived to be well anticipated through fractional operators. Fractional order is an extension of the classical order of differentiation and integration. In the last few decades, numerous phenomena in different fields of mathematics, economics, engineering, and science have been more precisely described via fractional derivatives, and fractional differential equations (FDEs) have come up as another powerful device for modeling many difficult types of complex systems. Podlubny [9] provided a major contribution to this field. The existence, uniqueness result, and some integral transforms of FDEs are presented in [9]. Later, Kilbas et al. [10] presented the idea of fractional calculus. A large amount of literature has been dedicated to the theory and applications of FDEs [11–13]. Currently, fractional chaotic systems are gaining considerable interest among many researchers in the different fields of science, mathematics, and engineering. Huang et al. [14] analyzed the behavior of a chaotic system of a fractional order love model with an external environment. Moreover, Huang et al. [15] studied the relationship of Romeo and Juliet love fractional model using a fuzzy function.
The idea of Riemann–Liouville FDE with an uncertainty was introduced by Agarwal et al. [16]. Consequently, many authors presented results on the existence and uniqueness of solutions for fuzzy FDEs (FFDEs) using numerous methods [17–20]. Mazandarani et al. [21] used the modified Euler method to solve FFDEs with a Caputo derivative. Salahshour et al. [22] obtained the solution of FFDEs using the fuzzy Laplace transform method. Ahmed et al. [23] provided analytical and numerical solutions of FFDEs using the Zadeh’s extension principle.
In the present work, our aim is to obtain a solution for FDDEs under a fractional derivative. Many authors solved fuzzy fractional DDEs (FFDDEs) using some numerical methods. For instance, Hoa [24] proposed a solution of FFDDEs that involve the Caputo gH-differentiability using the Adams–Bashforth–Moulton method and also proposed the modified Euler method to solve FFDDEs with an initial condition in [25].
In all the above-cited studies, the source functions and initial conditions are considered as a fuzzy number-valued function. For this function, the derivative is considered as a gH-derivative. This gH-derivative has two types, namely, [(i)-gH] and [(ii)-gH] differentiability. The disadvantages of this derivative are the difficulty in determining which type of derivative to choose and whether the solution is unique. Another difficulty of the gH-derivative is that it may convert the fuzzy problem into two crisp corresponding problems, and we need to choose one solution from a set of solutions that more appropriately fits the problem.
Therefore, our main motivation is to avoid the difficulties in choosing the fuzzy derivative. Then, we enhance the methodology proposed by Fatullayev et al. [26] for FDDEs to FFDDEs. The authors in [26] solved fuzzy problems with an initial condition and a forcing function as a set of fuzzy real functions using the method of Gasilov et al. [27]. Consequently, in [28] the authors provided solutions for fuzzy problems with boundary conditions in which the forcing functions were fuzzy functions.
To the best of our knowledge, this is the first study where this technique is utilized for solving FFDDEs. In the present work, our main objective is to obtain a solution for FDDEs with a fractional derivative. Recently, fuzzy triangular membership function is a beneficial device in many designing applications [29]. Therefore, in this approach, we consider that the initial condition and source functions are in fuzzy triangular forms. Moreover, we prove that if the initial conditions and source functions are fuzzy triangular functions, then the value of the solution is also a fuzzy triangular function. According to this methodology, we present the existence and uniqueness results for of FFDDEs.
This paper is structured as follows: The basic definition and notation of the fractional derivatives, fuzzy sets and fuzzy functions are presented in Section 2. The proposed method for solving FFDDEs is described in Section 3. Numerical applications are performed to test the validity and reliability of the present algorithm in Section 4. Finally, the conclusion is drawn in Section 5.
In this section, we provide certain essential definitions and properties of the fuzzy fractional concept.
The Riemann–Liouville fractional integral of
The Caputo derivative of order
We provide fixed set
We also consider
According to the geometric explanation of a fuzzy number,
In this study,
A fuzzy set can also be represented through the definition of
The
In our work, we express a fuzzy function as a set of fuzzy real functions in which every real function possesses a specific membership degree. This function is defined by Gasilov et al. [27].
Let
Fuzzy set
The interpretation of triangular fuzzy function
The fuzzy triangular function is a fuzzy triangular number for each time
We consider the following FFDDE:
where
Fuzzy set
where
We can explain
Let us consider that
As stated in Definition 3, solution
for
Let us express
Because
Then, we split FFDDE (
Fractional non-homogeneous equation
Fractional homogeneous equation involving an initial function, which is a triangular fuzzy function
Fractional non-homogeneous equation involving a fuzzy source function and whose initial value is zero
Next, we develop an algorithm to solve these three equations.
We consider fractional non-homogeneous equation, i.e.,
By performing fractional integration on both sides of
Next, we solve
Let us consider that
Let us also consider that
Let us consider
According to Definition 3, functions
However, the definition of triangular fuzzy function implies that bunch
Moreover, we apply the linear property for the fractional order (see [30]). Then, if
This result implies that set
Fuzzy triangular function
Next, we solve
We let
We also let
Let us consider
The proof of this theorem can be derived from the same procedure used in Theorem 1.
Next, we provide the algorithm for solving FFDDE (
Let us consider the following FFDDE:
where
We represent the source function and initial value as
First, we find a solution to the following fractional associated non-homogeneous equation:
where
By solving
Solution
Second, we derive the solution to the fractional homogeneous equation involving fuzzy initial function
Then, we solve the following fractional equation:
For
The obtained solution from
Next, we derive the solution for the fractional non-homogeneous equation involving a fuzzy source function.
Then, we solve the crisp problems.
For
The obtained solution from
Finally, we derive the solution of
Solution
In this study, we obtained the solutions of an FFDDE according to the method proposed by Fatullayev et al. [26]. In the present work, we considered that the initial condition and source function are triangular fuzzy functions. Meanwhile, the FFDDE was converted into three equations based on the initial and source functions. Finally, we combined all the solutions of the abovementioned three equations to obtain a solution of the FFDDE. Graphs were plotted for the fuzzy solution. The obtained solutions for the FFDDE were expressed as a fuzzy set of real functions. We recommended an efficient method to compute the fuzzy solution set. The numerical example demonstrated that the proposed method is efficient and accurate for application to FFDDEs.
Solution function and
Fuzzy solution in terms of the initial values and
Fuzzy solution relative to the source function and its
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