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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

Numerical Solutions of Fuzzy Fractional Delay Differential Equations

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)

Received: February 28, 2020; Revised: June 10, 2020; Accepted: August 27, 2020

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 [1113]. 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 [1720]. 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.

Definition 1 [9, 10]

The Riemann–Liouville fractional integral of y(χ) with any order α > 0 is expressed as

Iχαy(χ)=1Γ(α)0χ(χ-s)α-1y(s)ds.

Definition 2 [9, 10]

The Caputo derivative of order α (0 < α < 1) of real-valued function y(χ) is defined as

Dαy(χ)=Iχ1-α[dy(χ)dχ]=1Γ(1-α)0χ(χ-s)-αdy(s)dsds.

2.1 Fuzzy-Set Theory

We provide fixed set U, and a fuzzy set à on U with membership function μà can be defined as à = {(x, μÃ(x)) | xU}.

We also consider a, b, and c as real numbers with acb. The membership function of triangular fuzzy number ũ = (a, c, b) is defined as

μA˜(x)={x-ac-a,a<x<c,1,x=c,b-xb-c,c<x<b,0,otherwise.

According to the geometric explanation of a fuzzy number, c is the vertex, u = a is the left end point, and ū = b is the right end point.

In this study, ũ can be represented by ũ = ucr + ũun, where ucr is the real part (ucr = c) and ũun is the uncertain part (ũun = (ac, 0, bc)).

A fuzzy set can also be represented through the definition of r-cut. For each 0 < r ≤ 1, the r-cut of à is expressed as Ar = {xU | μÃ(x) ≥ r}. For r = 0, A0 = closure(support(Ã)), where support(Ã) = {xU | μÃ(x) > 0}.

The r-cuts of fuzzy number ũ = (a, c, b) are intervals ur = [ur, ūr], where ur = a + r(ca) and ūr = b + r(cb). This ur and ūr can be written as ur = c + (1 – r)(ac) and ūr = c + (1 – r)(bc). Then, [ur, ūr] = c + (1 – r)[ac, bc].

2.2 Fuzzy Triangular Function

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 Fa(•), Fb(•), and Fc(•) be functions that are continuous on interval I.

Fuzzy set = 〈Fa, Fc, Fb〉 is called a fuzzy triangular function if it is determined by the following membership function:

μF˜(y(.))={r,y=Fa+r(Fc-Fa)and 0<r<1,1,y=Fc,r,y=Fb+r(Fc-Fb)and 0<r<1,0,otherwise.

The interpretation of triangular fuzzy function = 〈Fa, Fc, Fb〉 is presented by Gasilov et al in [27].

The fuzzy triangular function is a fuzzy triangular number for each time χI. Then, we can represent it using the following equation:

F˜(χ)=(min{Fa(χ),Fc(χ),Fb(χ)},Fc(χ),max{Fa(χ),Fc(χ),Fb(χ)}).

We consider the following FFDDE:

{DCαy(χ)=m(χ)y(χ-τ)+F˜(χ),χ>0,0<α<1,y(χ)=Φ˜(χ),-τχ0,

where m(χ) is a continuous crisp function and τ > 0 is the delay value. Φ̃(χ) = 〈Φa, Φc, Φb〉 and (χ) = 〈Fa, Fc, Fb〉 are triangular fuzzy functions defined on [τ, 0] and (0, ∞), respectively, and CDα denotes the Caputo derivative operator.

Definition 3 [26]

Fuzzy set is the solution of FFDDE (6) on U, and its membership function is defined by

μY˜(y())=min{μΦ˜(ϕ()),μF˜(DCαy(χ)-m(χ)y(χ-τ))},

where ϕ(χ) = y(χ).

We can explain Eq. (7) as follows:

Let us consider that y(•) is a function on U. Then, we calculate ϕ(χ) = y(χ) for τχ ≤ 0 and find μφ̃(ϕ(•)) = μ1. Consequently, for given function y(χ), we calculate f(χ) = CDαy(χ) – m(χ)y(χτ ) for χ > 0 and find μ (f(•)) = μ2. Next, we determine membership degree μ for function y(•), which is calculated as μ = min{μ1, μ2}. Then, we define function set y(•) with membership degree μ as fuzzy solution .

As stated in Definition 3, solution is a collection of fuzzy real functions y(χ). A function y(χ) has a positive membership degree if it satisfies the following equation:

{DCαy(χ)=m(χ)y(χ-τ)+f(χ),χ>0,0<α<1,y(χ)=ϕ(χ),-τχ0,

for ϕsupp(Φ̃) and fsupport( ).

Let us express as = fcr + , where fcr = Fc, = 〈fa, 0, fb〉 = 〈FaFc, 0, FbFc〉. Let us also express Φ̃ as Φ̃(χ) = ϕcr + ϕ, where ϕcr = Φc, ϕ = 〈ϕa, 0, ϕb〉 = 〈Φa – Φc, 0, Φb – Φc〉.

Because Eq. (6) is linear, we apply the principle of superposition into it.

Then, we split FFDDE (6) into three equations as follows:

  • Fractional non-homogeneous equation

    {DCαy(χ)=m(χ)y(χ-τ)+fcr(χ),χ>0,0<α<1,y(χ)=ϕcr(χ),-τχ0,

  • Fractional homogeneous equation involving an initial function, which is a triangular fuzzy function

    {DCαy(χ)=m(χ)y(χ-τ),χ>0,0<α<1,y(χ)=ϕ˜(χ),-τχ0.

  • Fractional non-homogeneous equation involving a fuzzy source function and whose initial value is zero

    {DCαy(χ)=m(χ)y(χ-τ)+f˜(χ),χ>0,0<α<1,y(χ)=0,-τχ0.

Next, we develop an algorithm to solve these three equations.

3.1 Fractional Non-Homogeneous Equation

We consider fractional non-homogeneous equation, i.e., Eq. (9). We use the method of steps to solve Eq. (9). First, we obtain the solution in the interval [0, τ ]. For 0 ≤ χτ, we obtain

{DCαy(χ)=m(χ)ϕcr(χ-τ)+fcr(χ),y(0)=ϕcr(0)..

By performing fractional integration on both sides of Eq. (12), we obtain unique solution y(χ) = ϕcr(0) + Jα[m(χ) ϕcr(χτ )] + Jα[fcr(χ)] for 0 ≤ χτ, where Jα denotes the fractional integral operators.

3.2 Fractional Homogeneous Equation Involving Initial Fuzzy Function

Next, we solve Eq. (10) in which the initial condition is fuzzy function φ̃ = 〈ϕa, 0, ϕb〉.

Let us consider that ya(χ) is the solution of the following equation:

{DCαy(χ)=m(χ)y(χ-τ),χ>0,0<α<1,y(χ)=ϕa(χ),-τχ0.

Let us also consider that yb(χ) is the solution of the following equation:

{DCαy(χ)=m(χ)y(χ-τ),χ>0,0<α<1,y(χ)=ϕb(χ),-τχ0.
Theorem 1

Let us consider Eq. (10), where φ̃ = 〈ϕa, 0, ϕb〉 is a triangular fuzzy function and m(χ), ϕa(χ), and ϕb(χ) are continuous functions. If ya(χ) and yb(χ) are solutions of Eqs. (13) and (14), respectively, then Eq. (10) has unique solution ϕ = 〈ya, 0, yb〉.

Proof

According to Definition 3, functions ya(χ) and yb(χ) with non-zero membership degree from bunch ϕ are solutions to Eqs. (13) and (14) for ϕa and ϕb from bunch φ̃, respectively.

However, the definition of triangular fuzzy function implies that bunch φ̃ = 〈ϕa, 0, ϕb〉 consists of functions a and b (k = 1– r and 0 ≤ r ≤ 1) because Φc = 0.

Moreover, we apply the linear property for the fractional order (see [30]). Then, if y(χ) is a solution function of CDαy(χ) = m(χ)y(χτ ) with initial condition y(χ) = ϕ(χ), then ky(χ) is a solution function of CDαy(χ) = m(χ)y(χτ ) with initial condition y(χ) = (χ), where k is a constant number.

This result implies that set ϕ contains functions kya and kyb (0 ≤ k ≤ 1). Hence, ϕ is a fuzzy triangular function derived by ϕ = 〈ya, 0, yb〉. Thus, the theorem is proven.

Remark 1

Fuzzy triangular function ϕ = 〈ya, 0, yb〉 can be expressed as a triangular fuzzy number at time χ as follows:

Y˜ϕ(χ)=(min{ya(χ),0,yb(χ)},0,max{ya(χ),0,yb(χ)}}.

Next, we solve Eq. (11) in which fuzzy source function = 〈fa, 0, fb〉 and the initial condition is zero.

We let yu(χ) be the solution to the following problem:

{DCαy(χ)=m(χ)y(χ-τ)+fa(χ),χ>0,0<α<1,y(χ)=0,-τχ0.

We also let yv(χ) be the solution to the following problem:

{DCαy(χ)=m(χ)y(χ-τ)+fb(χ),χ>0,0<α<1,y(χ)=0,-τχ0.

Theorem 2

Let us consider Eq. (11). Here, = 〈fa, 0, fb〉 is a triangular fuzzy function, and m(χ), fa(χ), and fb(χ) are continuous functions. If yu(χ) and yv(χ) are solutions of Eqs. (15) and (16), respectively, then Eq. (11) has unique solution f = 〈yu, 0, yv〉.

Proof

The proof of this theorem can be derived from the same procedure used in Theorem 1.

Next, we provide the algorithm for solving FFDDE (6).

  • Step 1: We express the source function and initial value as

    F˜(χ)=fcr+fa,0,fband Φ˜(χ)=ϕcr+ϕa,0,ϕb.

  • Step 2: We find solution ycr(χ) of Eq. (9) using the step method or any other numerical methods.

  • Step 3: We determine solutions ya(χ) and yb(χ) to Eqs. (13) and (14), respectively. Then, the solution of Eq. (10) can be defined as

    Y˜ϕ(χ)=(min{ya(χ),0,yb(χ)},0,max{ya(χ),0,yb(χ)}}.

  • Step 4: We find solutions yu(χ) and yv(χ) to Eqs. (15) and (16), respectively. Then, the solution of Eq. (11) can be expressed as

    Y˜f(χ)=(min{yu(χ),0,yv(χ)},0,max{yu(χ),0,yv(χ)}}.

  • Step 5: The solution of FFDDE (6) is expressed as follows:

    Y˜(χ)=ycr(χ)+Y˜ϕ(χ)+Y˜f(χ).

Let us consider the following FFDDE:

{DCαy(χ)=λχy(χ-1)+(χ2,2χ2,3χ2),χ[0,1],0<α<1,y(χ)=Φ˜(χ)=χ+1,2χ+2,3χ+3,χ[-1,0],

where λ∈[−1, 1] \ {0}.

We represent the source function and initial value as (χ) = fcr +〈fa, 0, fb〉 = 2χ2+ 〈–χ2, 0, χ2〉 and Φ̃(x) = ϕcr + 〈ϕa, 0, ϕb〉 = 2χ+2+ 〈–χ – 1, 0, χ + 1〉.

First, we find a solution to the following fractional associated non-homogeneous equation:

{DCαy(χ)=λχy(χ-1)+fcr,χ[0,1],0<α<1,y(χ)=ϕcr=2χ+2,χ[-1,0],

where fcr = 2χ2.

By solving Eq. (19), we obtain the following solution:

ycr(χ)=2+4(λ+1)Γ(3+α)χ2+α.

Solution ycr(χ) is shown in Figure 1 (dashed black color line).

Second, we derive the solution to the fractional homogeneous equation involving fuzzy initial function

{DCαy(χ)=λχy(χ-1),χ[0,1],0<α<1,y(χ)=ϕa,0,ϕb=-χ-1,0,χ+1,χ[-1,0].

Then, we solve the following fractional equation:

{DCαy(χ)=λχy(χ-1),χ[0,1],0<α<1,y(χ)=ϕ(χ),χ[-1,0].

For ϕ(χ) = ϕa(χ), we obtain solution ya(χ)=-1-2λΓ(3+α)χ2+α, and for ϕ(χ) = ϕb(χ), we obtain solution yb(χ)=1+2λΓ(3+α)χ2+α.

The obtained solution from Eq. (21) represents triangular fuzzy function ϕ = 〈ya, 0, yb〉. Solution ϕ and its r-cuts are shown in Figure 2, and for time χ, we obtain

Y˜ϕ(χ)=(min{ya(χ),0,yb(χ)},0,max{ya(χ),0,yb(χ)}).

Next, we derive the solution for the fractional non-homogeneous equation involving a fuzzy source function.

{DCαy(χ)=λχy(χ-1)+fa,0,fb,χ[0,1],0<α<1,y(χ)=0,χ[-1,0].

Then, we solve the crisp problems.

{DCαy(χ)=λχy(χ-1)+f(χ),χ[0,1],0<α<1,y(χ)=0,χ[-1,0].

For f(χ) = fa(χ), we obtain solution yu(χ)=-2Γ(3+α)χ2+α, and for f(χ) = fb(χ), we obtain solution yv(χ)=2Γ(3+α)χ2+α.

The obtained solution from Eq. (24) is fuzzy triangular function f = 〈yu, 0, yv〉. Solution f and its r-cuts are shown in Figure 3. For time t, we obtain

Y˜f(χ)=(min{yu(χ),0,yv(χ)},0,max{yu(χ),0,yv(χ)}).

Finally, we derive the solution of Equation (18) in the following form:

Y˜(χ)=ycr(χ)+Y˜ϕ(χ)+Y˜f(χ).

Solution (χ) and its r-cuts are shown in Figure 1.

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.

Fig. 1.

Solution function and r-cuts: black color line—0-cut; dotted blue color line—0.3-cut; dashed-dotted red color line—0.7-cut; and dashed black color line—1.0-cut. λ = 1 and α = 0.5.


Fig. 2.

Fuzzy solution in terms of the initial values and r-cuts: black color line—0-cut; dotted blue color line—0.3-cut; dashed—dotted red color line—0.7-cut; and dashed black color line—1.0-cut. λ = 1 and α = 0.5.


Fig. 3.

Fuzzy solution relative to the source function and its r-cuts: black color line—0-cut; dotted blue color line—0.3- cut; dashed–dotted red color line—0.7-cut; and dashed black color line—1.0-cut. λ = 1 and α = 0.5.


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V. Padmapriya is a Ph.D Research scholar in Division of Mathematics, Vellore Institute of Technology-Chennai Campus, India. She is working as Assistant professor in Department of Statistics, New prince Shri Bhavani Arts and Science College, Chennai, India. Her research interest includes Fuzzy differential equations and Fuzzy fractional differential equations.

E-mail: v.padmapriya2015@vit.ac.in


M. Kaliyappan is working as a Professor of Mathematics, Division of mathematics, School of Advanced Sciences, VIT Chennai, India. He has over 23 years of experience in teaching and research. His research interest includes Approximation theory, Numerical computing, Differential equations, Fuzzy differential equations, Fractional differential equations, Fuzzy fractional differential equations and Optimization theory.

E-mail: kaliyappan.m@vit.ac.in


A. Manivannan received the Ph.D. degree in Mathematics from the Madurai Kamaraj University, Madurai, Tamilnadu, India in 2015. He is currently an Assistant Professor in the Division of Mathematics, Vellore Institute of Technology-Chennai Campus, India. His research interests include Stability theory, Neural networks, Complex systems, Fuzzy functional differential equations, Fractional differential equations.

E-mail: manivannan.a@vit.ac.in


Article

Original Article

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.

Numerical Solutions of Fuzzy Fractional Delay Differential Equations

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)

Received: February 28, 2020; Revised: June 10, 2020; Accepted: August 27, 2020

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.

Abstract

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

1. Introduction

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 [1113]. 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 [1720]. 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.

2. Basics of Fuzzy and Fractional Calculus

In this section, we provide certain essential definitions and properties of the fuzzy fractional concept.

Definition 1 [9, 10]

The Riemann–Liouville fractional integral of y(χ) with any order α > 0 is expressed as

Iχαy(χ)=1Γ(α)0χ(χ-s)α-1y(s)ds.

Definition 2 [9, 10]

The Caputo derivative of order α (0 < α < 1) of real-valued function y(χ) is defined as

Dαy(χ)=Iχ1-α[dy(χ)dχ]=1Γ(1-α)0χ(χ-s)-αdy(s)dsds.

2.1 Fuzzy-Set Theory

We provide fixed set U, and a fuzzy set à on U with membership function μà can be defined as à = {(x, μÃ(x)) | xU}.

We also consider a, b, and c as real numbers with acb. The membership function of triangular fuzzy number ũ = (a, c, b) is defined as

μA˜(x)={x-ac-a,a<x<c,1,x=c,b-xb-c,c<x<b,0,otherwise.

According to the geometric explanation of a fuzzy number, c is the vertex, u = a is the left end point, and ū = b is the right end point.

In this study, ũ can be represented by ũ = ucr + ũun, where ucr is the real part (ucr = c) and ũun is the uncertain part (ũun = (ac, 0, bc)).

A fuzzy set can also be represented through the definition of r-cut. For each 0 < r ≤ 1, the r-cut of à is expressed as Ar = {xU | μÃ(x) ≥ r}. For r = 0, A0 = closure(support(Ã)), where support(Ã) = {xU | μÃ(x) > 0}.

The r-cuts of fuzzy number ũ = (a, c, b) are intervals ur = [ur, ūr], where ur = a + r(ca) and ūr = b + r(cb). This ur and ūr can be written as ur = c + (1 – r)(ac) and ūr = c + (1 – r)(bc). Then, [ur, ūr] = c + (1 – r)[ac, bc].

2.2 Fuzzy Triangular Function

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 Fa(•), Fb(•), and Fc(•) be functions that are continuous on interval I.

Fuzzy set = 〈Fa, Fc, Fb〉 is called a fuzzy triangular function if it is determined by the following membership function:

μF˜(y(.))={r,y=Fa+r(Fc-Fa)and 0<r<1,1,y=Fc,r,y=Fb+r(Fc-Fb)and 0<r<1,0,otherwise.

The interpretation of triangular fuzzy function = 〈Fa, Fc, Fb〉 is presented by Gasilov et al in [27].

The fuzzy triangular function is a fuzzy triangular number for each time χI. Then, we can represent it using the following equation:

F˜(χ)=(min{Fa(χ),Fc(χ),Fb(χ)},Fc(χ),max{Fa(χ),Fc(χ),Fb(χ)}).

3. Solution of Fuzzy Fractional Delay Differential Equations

We consider the following FFDDE:

{DCαy(χ)=m(χ)y(χ-τ)+F˜(χ),χ>0,0<α<1,y(χ)=Φ˜(χ),-τχ0,

where m(χ) is a continuous crisp function and τ > 0 is the delay value. Φ̃(χ) = 〈Φa, Φc, Φb〉 and (χ) = 〈Fa, Fc, Fb〉 are triangular fuzzy functions defined on [τ, 0] and (0, ∞), respectively, and CDα denotes the Caputo derivative operator.

Definition 3 [26]

Fuzzy set is the solution of FFDDE (6) on U, and its membership function is defined by

μY˜(y())=min{μΦ˜(ϕ()),μF˜(DCαy(χ)-m(χ)y(χ-τ))},

where ϕ(χ) = y(χ).

We can explain Eq. (7) as follows:

Let us consider that y(•) is a function on U. Then, we calculate ϕ(χ) = y(χ) for τχ ≤ 0 and find μφ̃(ϕ(•)) = μ1. Consequently, for given function y(χ), we calculate f(χ) = CDαy(χ) – m(χ)y(χτ ) for χ > 0 and find μ (f(•)) = μ2. Next, we determine membership degree μ for function y(•), which is calculated as μ = min{μ1, μ2}. Then, we define function set y(•) with membership degree μ as fuzzy solution .

As stated in Definition 3, solution is a collection of fuzzy real functions y(χ). A function y(χ) has a positive membership degree if it satisfies the following equation:

{DCαy(χ)=m(χ)y(χ-τ)+f(χ),χ>0,0<α<1,y(χ)=ϕ(χ),-τχ0,

for ϕsupp(Φ̃) and fsupport( ).

Let us express as = fcr + , where fcr = Fc, = 〈fa, 0, fb〉 = 〈FaFc, 0, FbFc〉. Let us also express Φ̃ as Φ̃(χ) = ϕcr + ϕ, where ϕcr = Φc, ϕ = 〈ϕa, 0, ϕb〉 = 〈Φa – Φc, 0, Φb – Φc〉.

Because Eq. (6) is linear, we apply the principle of superposition into it.

Then, we split FFDDE (6) into three equations as follows:

  • Fractional non-homogeneous equation

    {DCαy(χ)=m(χ)y(χ-τ)+fcr(χ),χ>0,0<α<1,y(χ)=ϕcr(χ),-τχ0,

  • Fractional homogeneous equation involving an initial function, which is a triangular fuzzy function

    {DCαy(χ)=m(χ)y(χ-τ),χ>0,0<α<1,y(χ)=ϕ˜(χ),-τχ0.

  • Fractional non-homogeneous equation involving a fuzzy source function and whose initial value is zero

    {DCαy(χ)=m(χ)y(χ-τ)+f˜(χ),χ>0,0<α<1,y(χ)=0,-τχ0.

Next, we develop an algorithm to solve these three equations.

3.1 Fractional Non-Homogeneous Equation

We consider fractional non-homogeneous equation, i.e., Eq. (9). We use the method of steps to solve Eq. (9). First, we obtain the solution in the interval [0, τ ]. For 0 ≤ χτ, we obtain

{DCαy(χ)=m(χ)ϕcr(χ-τ)+fcr(χ),y(0)=ϕcr(0)..

By performing fractional integration on both sides of Eq. (12), we obtain unique solution y(χ) = ϕcr(0) + Jα[m(χ) ϕcr(χτ )] + Jα[fcr(χ)] for 0 ≤ χτ, where Jα denotes the fractional integral operators.

3.2 Fractional Homogeneous Equation Involving Initial Fuzzy Function

Next, we solve Eq. (10) in which the initial condition is fuzzy function φ̃ = 〈ϕa, 0, ϕb〉.

Let us consider that ya(χ) is the solution of the following equation:

{DCαy(χ)=m(χ)y(χ-τ),χ>0,0<α<1,y(χ)=ϕa(χ),-τχ0.

Let us also consider that yb(χ) is the solution of the following equation:

{DCαy(χ)=m(χ)y(χ-τ),χ>0,0<α<1,y(χ)=ϕb(χ),-τχ0.
Theorem 1

Let us consider Eq. (10), where φ̃ = 〈ϕa, 0, ϕb〉 is a triangular fuzzy function and m(χ), ϕa(χ), and ϕb(χ) are continuous functions. If ya(χ) and yb(χ) are solutions of Eqs. (13) and (14), respectively, then Eq. (10) has unique solution ϕ = 〈ya, 0, yb〉.

Proof

According to Definition 3, functions ya(χ) and yb(χ) with non-zero membership degree from bunch ϕ are solutions to Eqs. (13) and (14) for ϕa and ϕb from bunch φ̃, respectively.

However, the definition of triangular fuzzy function implies that bunch φ̃ = 〈ϕa, 0, ϕb〉 consists of functions a and b (k = 1– r and 0 ≤ r ≤ 1) because Φc = 0.

Moreover, we apply the linear property for the fractional order (see [30]). Then, if y(χ) is a solution function of CDαy(χ) = m(χ)y(χτ ) with initial condition y(χ) = ϕ(χ), then ky(χ) is a solution function of CDαy(χ) = m(χ)y(χτ ) with initial condition y(χ) = (χ), where k is a constant number.

This result implies that set ϕ contains functions kya and kyb (0 ≤ k ≤ 1). Hence, ϕ is a fuzzy triangular function derived by ϕ = 〈ya, 0, yb〉. Thus, the theorem is proven.

Remark 1

Fuzzy triangular function ϕ = 〈ya, 0, yb〉 can be expressed as a triangular fuzzy number at time χ as follows:

Y˜ϕ(χ)=(min{ya(χ),0,yb(χ)},0,max{ya(χ),0,yb(χ)}}.

3.3 Fractional Non-Homogeneous Equation Involving Fuzzy Source Function and Zero Initial Value

Next, we solve Eq. (11) in which fuzzy source function = 〈fa, 0, fb〉 and the initial condition is zero.

We let yu(χ) be the solution to the following problem:

{DCαy(χ)=m(χ)y(χ-τ)+fa(χ),χ>0,0<α<1,y(χ)=0,-τχ0.

We also let yv(χ) be the solution to the following problem:

{DCαy(χ)=m(χ)y(χ-τ)+fb(χ),χ>0,0<α<1,y(χ)=0,-τχ0.

Theorem 2

Let us consider Eq. (11). Here, = 〈fa, 0, fb〉 is a triangular fuzzy function, and m(χ), fa(χ), and fb(χ) are continuous functions. If yu(χ) and yv(χ) are solutions of Eqs. (15) and (16), respectively, then Eq. (11) has unique solution f = 〈yu, 0, yv〉.

Proof

The proof of this theorem can be derived from the same procedure used in Theorem 1.

3.4 Algorithm for the Solution

Next, we provide the algorithm for solving FFDDE (6).

  • Step 1: We express the source function and initial value as

    F˜(χ)=fcr+fa,0,fband Φ˜(χ)=ϕcr+ϕa,0,ϕb.

  • Step 2: We find solution ycr(χ) of Eq. (9) using the step method or any other numerical methods.

  • Step 3: We determine solutions ya(χ) and yb(χ) to Eqs. (13) and (14), respectively. Then, the solution of Eq. (10) can be defined as

    Y˜ϕ(χ)=(min{ya(χ),0,yb(χ)},0,max{ya(χ),0,yb(χ)}}.

  • Step 4: We find solutions yu(χ) and yv(χ) to Eqs. (15) and (16), respectively. Then, the solution of Eq. (11) can be expressed as

    Y˜f(χ)=(min{yu(χ),0,yv(χ)},0,max{yu(χ),0,yv(χ)}}.

  • Step 5: The solution of FFDDE (6) is expressed as follows:

    Y˜(χ)=ycr(χ)+Y˜ϕ(χ)+Y˜f(χ).

4. Example

Let us consider the following FFDDE:

{DCαy(χ)=λχy(χ-1)+(χ2,2χ2,3χ2),χ[0,1],0<α<1,y(χ)=Φ˜(χ)=χ+1,2χ+2,3χ+3,χ[-1,0],

where λ∈[−1, 1] \ {0}.

We represent the source function and initial value as (χ) = fcr +〈fa, 0, fb〉 = 2χ2+ 〈–χ2, 0, χ2〉 and Φ̃(x) = ϕcr + 〈ϕa, 0, ϕb〉 = 2χ+2+ 〈–χ – 1, 0, χ + 1〉.

First, we find a solution to the following fractional associated non-homogeneous equation:

{DCαy(χ)=λχy(χ-1)+fcr,χ[0,1],0<α<1,y(χ)=ϕcr=2χ+2,χ[-1,0],

where fcr = 2χ2.

By solving Eq. (19), we obtain the following solution:

ycr(χ)=2+4(λ+1)Γ(3+α)χ2+α.

Solution ycr(χ) is shown in Figure 1 (dashed black color line).

Second, we derive the solution to the fractional homogeneous equation involving fuzzy initial function

{DCαy(χ)=λχy(χ-1),χ[0,1],0<α<1,y(χ)=ϕa,0,ϕb=-χ-1,0,χ+1,χ[-1,0].

Then, we solve the following fractional equation:

{DCαy(χ)=λχy(χ-1),χ[0,1],0<α<1,y(χ)=ϕ(χ),χ[-1,0].

For ϕ(χ) = ϕa(χ), we obtain solution ya(χ)=-1-2λΓ(3+α)χ2+α, and for ϕ(χ) = ϕb(χ), we obtain solution yb(χ)=1+2λΓ(3+α)χ2+α.

The obtained solution from Eq. (21) represents triangular fuzzy function ϕ = 〈ya, 0, yb〉. Solution ϕ and its r-cuts are shown in Figure 2, and for time χ, we obtain

Y˜ϕ(χ)=(min{ya(χ),0,yb(χ)},0,max{ya(χ),0,yb(χ)}).

Next, we derive the solution for the fractional non-homogeneous equation involving a fuzzy source function.

{DCαy(χ)=λχy(χ-1)+fa,0,fb,χ[0,1],0<α<1,y(χ)=0,χ[-1,0].

Then, we solve the crisp problems.

{DCαy(χ)=λχy(χ-1)+f(χ),χ[0,1],0<α<1,y(χ)=0,χ[-1,0].

For f(χ) = fa(χ), we obtain solution yu(χ)=-2Γ(3+α)χ2+α, and for f(χ) = fb(χ), we obtain solution yv(χ)=2Γ(3+α)χ2+α.

The obtained solution from Eq. (24) is fuzzy triangular function f = 〈yu, 0, yv〉. Solution f and its r-cuts are shown in Figure 3. For time t, we obtain

Y˜f(χ)=(min{yu(χ),0,yv(χ)},0,max{yu(χ),0,yv(χ)}).

Finally, we derive the solution of Equation (18) in the following form:

Y˜(χ)=ycr(χ)+Y˜ϕ(χ)+Y˜f(χ).

Solution (χ) and its r-cuts are shown in Figure 1.

5. Conclusion

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.

Fig 1.

Figure 1.

Solution function and r-cuts: black color line—0-cut; dotted blue color line—0.3-cut; dashed-dotted red color line—0.7-cut; and dashed black color line—1.0-cut. λ = 1 and α = 0.5.

The International Journal of Fuzzy Logic and Intelligent Systems 2020; 20: 247-254https://doi.org/10.5391/IJFIS.2020.20.3.247

Fig 2.

Figure 2.

Fuzzy solution in terms of the initial values and r-cuts: black color line—0-cut; dotted blue color line—0.3-cut; dashed—dotted red color line—0.7-cut; and dashed black color line—1.0-cut. λ = 1 and α = 0.5.

The International Journal of Fuzzy Logic and Intelligent Systems 2020; 20: 247-254https://doi.org/10.5391/IJFIS.2020.20.3.247

Fig 3.

Figure 3.

Fuzzy solution relative to the source function and its r-cuts: black color line—0-cut; dotted blue color line—0.3- cut; dashed–dotted red color line—0.7-cut; and dashed black color line—1.0-cut. λ = 1 and α = 0.5.

The International Journal of Fuzzy Logic and Intelligent Systems 2020; 20: 247-254https://doi.org/10.5391/IJFIS.2020.20.3.247

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