Open access peer-reviewed chapter

# Study of a Dynamical Problem under Fuzzy Conformable Differential Equation

Written By

Reviewed: 15 June 2022 Published: 26 August 2022

DOI: 10.5772/intechopen.105904

From the Edited Volume

## Qualitative and Computational Aspects of Dynamical Systems

Edited by Kamal Shah, Bruno Carpentieri and Arshad Ali

Chapter metrics overview

View Full Metrics

## Abstract

The notion of inclusion by generalized conformable differentiability is used to analyze fuzzy conformable differential equations (FCDE). This idea is based on expanding the class of conformable differentiable fuzzy mappings, and we use generalized lateral conformable derivatives to do so. We’ll see that both conformable derivatives are distinct and that they lead to different FCDE solutions. The approach’s utility and efficiency are demonstrated with an example.

### Keywords

• fuzzy fractional differential equation
• conformable fractional derivative
• fuzzy number

## 1. Introduction

Aubin and Cellina [1] established the notion of differential inclusions systemically. They looked at the existence and qualities of differential inclusion solutions of the form [2].

u'tΦutoru'tΦtut.E1

In this paper we will consider the conformable fractional differential equation.

uqt=Φtutuκ0u0κ,κ01E2

where t0a and u0 is a fuzzy number. uq is the conformable fractional derivative of u of order γ01 [3, 4, 5]. There are numerous options for defining a fuzzy fractional derivatives and, as a result, see [6, 7, 8, 9], studying Eq. (2). [10, 11, 12, 13, 14, 15, 16] constructed the generalized derivative of a set value function and investigated it, while [17, 18, 19, 20] explored the generalized conformable fractional derivative.

The objective of this research is to see if fuzzy solutions exist using conformable differential inclusion, using the generalized conformable differentiability concept This idea is based on expanding the class of differentiable fuzzy mappings, and we use lateral conformable derivatives to do so. We will see that both derivatives are different and they lead us to different solutions from an FCDE.

## 2. Preliminaries

We’ll go through a few definitions now that will come in handy later in the paper. Let us start with a definition. RF the class of fuzzy subsets of the real axis η:R01 satisfying the following properties:

1. η is normal,

2. η is convex fuzzy set,

3. η is upper semicontinuous,

4. η0=clxRηx>0 is compact.

Then RF is called the space of fuzzy numbers [21].

If η is a fuzzy set, we define ηκ=xRηxκ the κ-level sets of η, with 0<κ1. Also, if ηRF then κ-cut of η denoted by ηκ=η1κη2κ.

For η,νRF and λR the sum η+ν and the product λη are defined by κ01,

η+νκ=η1κ+ν1κη2κ+ν2κ,E3
ληκ=ληκ=λη1κλη2κ,λ0;λη2κλη1κ,λ<0,E4

Let d:RF×RFR+0 by the following equation [22].

dην=supκ01dHηκνκ,forallη,νRF,E5
=supκ01max|η1κν1κ|,|η2κν2κ|E6

where dH is the Hausdorff metric.

The following properties are well-known see [22, 23]: η,ν,ω,ρRF and λR.

dη+ων+ω=dηνanddην=dνη,dληλν=λdην,dη+νω+ρdηω+dνρ.E7

And RFd is a complete metric space.

Theorem 1. [1, 22] Let η:0aRF and ηtκ=η1κtη2κt be Seikkala differentiable. Then, η1κt and η2κt are differentiable and

ηtκ=η1κtη2κt,κ01.E8

Definition 1. [24] Let η:0aRF. bcηtdt,b,c0a is the fuzzy integral, defined by

bcηtdtκ=bcη1κtdtbcη2κtdt,E9

for all 0κ1. In [24], if η:0aRF is continuous, it is fuzzy integrable.

Theorem 2. [22, 25] If ηRF, then:

1. ηκ2ηκ1,if0κ1κ21;E10

2. κk01 is a increasing sequence which converges to κ,

ηκ=k1ηκk.E11

Alternatively, if ϒκ={η1κη2κ;κ(0,1]} is a closed real intervals i and ii, then ϒκ defined a fuzzy number ηRF such that ηκ=ϒκ..

## 3. Fuzzy conformable differentiability and integral

The funcion Φ:abRF is called fuzzy function. The κ-level representation of fuzzy function Φ given by Φtκ=ϕ1κtϕ2κt,tab,κ01.

Definition 2. [17] Let Φ:0aRF be a fuzzy function. γth orderfuzzy conformable derivative” of Φ is defined by

TγΦt=limε0+Φt+εt1γΦtε=limε0+ΦtΦtεt1γε.E12

for all t>0,γ01. Let Φγt stands for TγΦt. Hence

Φγt=limε0+Φt+εt1γΦtε=limε0+ΦtΦtεt1γε.E13

If Φ is γ-differentiable in some 0a, and limt0+Φγt exists, then

Φγ0=limt0+ΦγtE14

and the limits (in the metric d).

Remark 1. [26]

1. If Φ is γ-differentiable then the multivalued mapping Φκ is γ-differentiable for all κ01 and

TγΦκ=Φγtκ,E15

where TγΦκ is denoted from the conformable fractional derivative of Φκ of order γ.

1. ηκνκ,κ01 does not imply the existence of Hukuhara difference (H-difference) ην.

Theorem 3. [26]

Let Φ:0aRF, Φtκ=ϕ1κtϕ2κt,κ01.

1. If Φ is γ1-differentiable, then ϕ1κt and ϕ2κt are γ-differentiable and

Φγ1tκ=ϕ1κγtϕ2κγtE16

2. If Φ is γ2-differentiable, then ϕ1κt and ϕ2κt are γ-differentiable and

Φγ2tκ=ϕ2κγtϕ1κγt.E17

Theorem 4. [17] Let γ01:.

1. If Φ is 1-differentiable and Φ is γ1-differentiable, then

Tγ1Φt=t1γD11ΦtE18

2. If Φ is 2-differentiable and Φ is γ2-differentiable, then

Tγ2Φt=t1γD21ΦtE19

Theorem 5. If Φ:0aRF is γ-differentiable then it is continuous.

Proof. Denote Φκt=ϕ1κtϕ2κt,κ01. Then ϕ1κt and ϕ2κt are continuous at t0, so Φ is continuous at t0.

If ε>0 and κ01, we have:

Φt0+εt01γΦt0κ=ϕ1κt0+εt01γϕ1κt0ϕ2κt0+εt01γϕ2κt0

Dividing and multiplying by ε, we have:

Φt0+εt01γΦt0κ=ϕ1κt0+εt01γϕ1κt0εεϕ2κt0+εt01γϕ2κt0εε

Similarly, we obtain:

Φt0Φt0εt01γκ=ϕ1κt0ϕ1κt0εt01γεεϕ2κt0ϕ2κt0εt01γεε

Then

limε0+Φt0+εt01γΦt0κ=limε0+ϕ1κt0+εt01γϕ1κt0εlimε0+ε,limε0+ϕ2κt0+εt01γϕ2κt0εlimε0+ε

Similarly, we obtain:

limε0+Φt0Φt0εt01γκ=limε0+ϕ1κt0ϕ1κt0εt01γεlimε0+ε,limε0+ϕ2κt0ϕ2κt0εt01γεlimε0+ε

Let h=εt01γ. Then

limh0+Φt0+hΦt0κ=ϕ1κγt00ϕ2κγt00

Similarly, we obtain:

limh0+Φt0hκ=Φt0κ

which implies that

limh0+Φt0+hκ=Φt0κ

Similary, we obtain:

limh0+Φt0hκ=Φt0κ

Hence, Φ is continuous at t0. □

Remark 2. If Φ:0aRF is γ-differentiable and Φγ for all γ01 is continuous, then we denote ΦC10aRF.

Theorem 6. Let γ01 and if Φ,Ψ:0aRF are γ-differentiable and λR then

1. TγΦ+Ψt=TγΦt+TγΨtE20

2. TγλΦt=λTγΦt.E21

proof. We present the details only for case i, since the other case is anlogous. Since Φ is γ1-differentiable it follows that Φt+εt1γΦt exists i.e. there exists u1tεt1γ such that

Φt+εt1γ=Φt+u1tεt1γE22

Analogously since Ψ is γ1-differentiable there exists v1tεt1γ such that

Ψt+εt1γ=Ψt+v1tεt1γ

and we get

Φt+εt1γ+Ψt+εt1γ=Φt+Ψt+u1tεt1γ+v1tεt1γE23

that is the H-difference

Φt+εt1γ+Ψt+εt1γΦt+Ψt=u1tεt1γ+v1tεt1γE24

By similar reasoning we get that there exist u2tεt1γ and v2tεt1γ such that

Φt=Φtεt1γ+u2tεt1γΨt=Ψtεt1γ+v2tεt1γ

and so

Φt+Ψt=Φtεt1γ+Ψtεt1γ+u2tεt1γ+v2tεt1γ

that is the H-difference

Φt+ΨtΦtεt1γ+Ψtεt1γ=u2tεt1γ+v2tεt1γE25

We observe that

limε0+u1tεt1γε=limε0+u2tεt1γε=Φγtandlimε0+v1tεt1γε=limε0+v2tεt1γε=Ψγt.

Finally, by multiplying (24) and (25) with 1ε and passing to limit with limε0+ we get that Φ+Ψ is γ1-differentiable and TγΦ+Ψt=TγΦt+TγΨt The case when Φ and Ψ are γ2-differentiable is similar to the previous one. □

Definition 3. Let ΦC0aRFL10aRF, Define the fuzzy fractional integral for γ01.

IγΦt=I1tγ1Φt=0tΦs1γsds,E26

where the integral is the usual Riemann improper integral.

Theorem 7. TγIγΦt, for t0, where Φ is any continuous function in the domain of Iγ.

Proof. Since Φ is continuous, then IγΦt is clearly conformable differentiable. Hence,

TγIγΦtκ=t1γddtIγΦtκ=t1γddt0tϕ1κxx1γdxt1γddt0tϕ2κxx1γdx=t1γϕ1κtt1γt1γϕ2κtt1γ=Φtκ

Theorem 8. Let γ01 and Φ be γ-differentiable in 0a and assume that the conformable derivative Φγ is integrable over 0a. Then s0a we have.

Φs=Φa+IγΦγtE27

Proof. Let γ01 and κ01 be fixed. We will demonstrate this.

Φκs=Φκa+IγΦκγE28

where Φκγ is conformable derivative of Φκ, the equation is then obtained by applying Theorems 3 and 4.

Φκs=Φκa+IγΦκγ=Φκa+Iγt1γΦκ'

by (26) we have

Φκs=Φκa+Iγt1γΦκ'=Φκa+0stγ1t1γΦκ'

So

Φκs=Φκa+0sΦκ'E29

where Φκ' is the derivative of Φκ, For a fuzzy mapping, the (29) is likewise true Φ:0aRF. In [1], the equality (28) now follows Theorem (8).

## 4. Solutions via conformable differential inclusions

We consider the fuzzy conformable differential equation.

uγt=Φtut,u0=u0,γ01,E30

where Φ:0a×RFRF is generated from a continuous function using Zadeh’s extension principle. ψ:0a×RR.

Let Φtu can be calculated at the level, i.e., κ01.

Φtuκ=ψtuκ

for all t0a,uRF and κ01. We interpret the fuzzy initial value problem (30) as a set of differential inclusions, following Diamonds [7, 10].

vγκt=ψtvκt,vκ0u0κE31

The reachable sets, under reasonable assumptions.

ϒκt=vκtvκisasolutionof31,

are κ-cuts of a fuzzy set ut, which we call a solution of (30). If we assume that the solutions to the initial value problem are unique, vγκt=ψtvκt,vκ0=v0, it follows that ϒκt=w1tw2t, where.

wγ1t=ψtw1t,w10=u01κandwγ2t=ψtw2t,w20=u02κ

Theorem 9. The fuzzy solution and solution by differential inclusions solution using the first form are equivalent if ψ is nondecreasing with respect to the second argument.

proof. For each κ01 and γ01, we think ut]κ=u1κtu2κt and u0]κ=u01κu02κ. Since g is continuous and

Φtutκ=ψtut)κ=ψtu1κtu2κt,

then ψ(t,u1κtu2κt) is compact and connected i.e. a closed bounded interval.

As ψ is nondecreasing, we have that

ψtu1κtu2κt=ψtu1κtψtu2κt

As a result, the conformable differential system for boundary functions of fuzzy solution is uncoupled into two initial value problems:

uγ1κt=ψtu1κt,u1κ0=u01κ,uγ2κt=ψtu2κt,u2κ0=u02κ,

This results in u1κ=w1 and u2κ=w2.

Now, we offer the following result as an extension of the preceding theorem to the class of differentiable functions with regard to the second form (17).

Theorem 10. If ψ is nonincreasing with respect to the second argument then, using the derivative in the second form (17), the fuzzy solution of (30) and the solution via differential inclusions are identical.

proof. Let κ01 and γ01, we consider.

ut]κ=u1κtu2κtandu0)κ=u01κu02κ.

So

Φtutκ=ψtut)κ=ψtu1κtu2κt

and ψ is continuous, and ψtu1κtu2κt is a closed bounded interval. Since ψ is nonincreasing, it follows that

ψtu1κtu2κt=ψtu2κtψtu1κt

Consequently, from (31), we have the conformable differential system.

uγ2κt=ψtu2κt,u2κ0=u02κ,uγ1κt=ψtu1κt,u1κ0=u01κ,E32

If uγt is consider in the form (2), we have that

uγtκ=uγ2κtuγ1κt=ψtu2κtψtu1κt

The proof is complete after obtaining the differential system (32).

Example 1. Consider the fuzzy initial value problem.

uγt=u2t,u0=u0E33

Where u0 is a traingular fuzzy number u0κ=1+κ3κ. Since u2 is continuous and we are operating on RF, we can solve the equation levelwise.

Since u2 is increasing when x>0, we have to solve a conformable differential system for γ01.

u1κγt=u1κ2t,u1κ0=1+κ,E34
u2κγt=u2κ2t,u2κ0=3κ,E35

where utκ=u1κu2κ.

The solutions are

u1κt=1κtγγ+tγγκ1andu2κt=3+κ3tγγtγγκ1

We can see this u2κt< for t<13 and

0u1κtu2κt

for these values of t.

As a result, there is a fuzzy solution to the fuzzy initial value problem. ut for 0t<13.

## 5. Conclusion

The fuzzy conformable differential inclusions (FCDI) are introduced, which have been used by various authors to solve FDE for γ=1 [2, 6]. It also has the advantage that the solutions derived using FCDI appear to be more intuitive than other conformable derivative solutions first form (9), [19]. It’s also worth noting that this interpretation has a drawback in that we cannot discuss the fuzzy conformable derivative. Instead, we address this obstacle by utilizing the fuzzy conformable derivative in the second form (17), and the fuzzy solution and the solution via conformable differential inclusions are identical.

## Conflicts of interest

The authors declare that they have no conflicts of interest.

## Data availability

The data used to support the findings of this study are available from the corresponding author upon request.

## References

1. 1. Kaleva O. A note on fuzzy differential equations. Nonlinear Analysis. 2006;64:895-900
2. 2. Diamond P. Time-dependent differential inclusions, cocycle attractors and fuzzy differential equations. IEEE Transactions on Fuzzy Systems. 1999;7:734-740
3. 3. Abdeljawad T. On conformable fractional calculus. Journal of Computational and Applied Mathematics. 2015;279:57-66
4. 4. Khalil R, Al Horani M, Yousef A, Sababheh M. A new definition of fractional derivative. Journal of Computational and Applied Mathematics. 2014;264:65-70
5. 5. Unal E, Gokdogan A. Solution of conformable fractional ordinary differential equations via differential transform method. International Journal for Light and Electron Optics. 2017;128:264-273
6. 6. Abbasbandy S, Nieto JJ, Alavi M. Tuning of reachable set in one dimensional fuzzy differential inclusions. Chaos, Solitons-Fractals. 2005;26:1337-1341
7. 7. Arshad S, Lupulescu V. On the fractional differential equations with uncertainty. Nonliniear Analysis. 2011;74:3685-3693
8. 8. Harir A, Melliani S, Chadli LS, Minchev E. Solutions of fuzzy fractional heatlike and wave-like equations by variational iteration method. International Journal of Contemporary Mathematical Sciences. 2020;15(1):11-35
9. 9. Harir A, Melliani S, Chadli LS. Fuzzy fractional evolution equations and fuzzy solution operators. Advanced Fuzzy Systems. 2019;2019:10. DOI: 10.1155/2019/5734190
10. 10. Bede B, Gal SG. Almost periodic fuzzy number valued functions. Fuzzy Sets and Systems. 2004;147:385-403
11. 11. Bede B, Gal SG. Generalizations of the differentiability of fuzzy number value functions with applications to fuzzy differential equations. Fuzzy Sets and Systems. 2005;151:581-599
12. 12. Goo HY, Park JS. On the continuity of the Zadeh extensions. Journal of the Chungcheong Mathematical Society. 2007;20(4):525-533
13. 13. Shah K, Arfan M, Ullah A, Mdallal Q, Ansari KJ, Abdeljawad T. Computational study on the dynamics of fractional order differential equations with applications. Chaos, Solitons and Fractals. 2022;157:111955
14. 14. Shah K, Naz H, Sarwar M, Abdeljawad T. On spectral numerical method for variable-order partial differential equations. AIMS Mathematics. 2022;7(6):10422-10438
15. 15. Shah K, Ali A, Zeb S, Khan A, Alqudah MA, Abdeljawad T. Study of fractional order dynamics of nonlinear mathematical model. Alexandria Engineering Journal. 2022;61(12):11211-11224
16. 16. Shahid A, Khan A, Shah K, Alqudah MA, Abdeljawad T, Islam Su. On computational analysis of highly nonlinear model addressing real world applications. Results in Physics. 2022;36:105431
17. 17. Harir A, Melliani S, Chadli LS. Fuzzy generalized conformable fractional derivative. Advanced Fuzzy Systems. 2019;2020:7. DOI: 10.1155/2020/1954975
18. 18. Seikkala S. On the fuzzy initialvalue problem. Fuzzy Sets and Systems. 1987;24:319-330
19. 19. Harir A, Melliani S, Chadli LS. The fractional differential equations with uncertainty by conformable derivative. European Journal of Pure and Applied Mathematics. 2022;15(2):557-571. DOI: 10.29020/nybg.ejpam.v15i2.4299
20. 20. Harir A, Melliani S, Chadli LS. Fuzzy conformable fractional semigroups of operators. International Journal of Differential Equations. 2020, 2020:6. DOI: 10.1155/2020/8836011
21. 21. Diamond P, Kloeden PE. Metric Spaces of Fuzzy Sets: Theory and Applications. Singapore: World Scienific; 1994
22. 22. Harir A, Melliani S, Chadli LS. Existence, uniqueness and approximate solutions of fuzzy fractional differential equations. In: Fuzzy Systems—Theory and Applications. London, UK: IntechOpen; 2020. DOI: 10.5772/intechopen.94000
23. 23. Puri ML, Ralescu DA. Differentials of fuzzy functions. Journal of Mathematical Analysis and Applications. 1983;91:552-558
24. 24. Song S, Guo L, Feng C. Global existence of solutions to fuzzy differential equations. Fuzzy Sets and Systems. 2000;115:371-376
25. 25. Negoita CV, Ralescu DA. Applications of Fuzzy Sets to System Analysis. Basel: Birkhauser; 1975
26. 26. Harir A, Melliani S, Chadli LS. Analytic solution method for fractional fuzzy conformable Laplace transforms. SeMA. 2021;78:401-414. DOI: 10.1007/s40324-021-00240-7

Written By