# Fuzzy Conformable Fractional Differential Equations

Authors: Atimad Harir; Said Melliani; Lalla Saadia Chadli
Journal: International Journal of Differential Equations
 (2021)

---

## Abstract

In this study, fuzzy conformable fractional differential equations are investigated. We study conformable fractional differentiability, and we define fractional integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy fractional differential equation by using the concept of conformable differentiability. This concept is based on the enlargement of the class of differentiable fuzzy mappings; for this, we consider the lateral Hukuhara derivatives of orderq∈0,1.


---

## Body

## 1. Introduction

Fractional calculus is generalization of differentiation and integration to an arbitrary order. The derivative for fuzzy-valued mappings was developed by [1] that generalized and extended the concept of Hukuhara differentiability (H-derivative) for set-valued mappings to the class of fuzzy mappings. Subsequently, using the H-derivative [2, 3] started to develop a theory for FDE. The concept of the fuzzy fractional derivative was introduced by [4] and developed by [5–11], but these researchers tried to put a definition of a fuzzy fractional derivative. Most of them used an integral from the fuzzy fractional derivative, two of which are the most popular ones, Riemann-Liouville definition and Caputo definition [12–14]. All definitions above satisfy the property that the fuzzy fractional derivative is linear. This is the only property inherited from the first fuzzy derivative by all of the definitions. However, the following are some of the setbacks of the other definitions [15]. The fuzzy conformable derivative may facilitate some computations:(i)
It satisfies all concepts and rules of an ordinary derivative such as quotient, product, and chain rules while the other fractional definitions fail to meet these rules(ii)
It can be extended to solve exactly and numerically fractional differential equations and systems easily and efficientlyAnd it was introduced and developed in [16, 17]. The objective of this study is to present some results for fuzzy conformable differentiability and fuzzy fractional integrability of such functions; we study the fuzzy fractional differential equations (FFDEs) by using this derivative and give an existence and uniqueness theorem for a solution of FFDEs.


## 2. Preliminaries

Let us denote byℝℱ=u:ℝ⟶0,1 the class of fuzzy subsets of the real axis satisfying the following properties:(i)
u is normal, i.e, there exists x0∈ℝ such that ux0=1(ii)
u is fuzzy convex, i.e, for x,y∈ℝ and 0<λ≤1,(1)uλx+1−λy≥minux,uy.(iii)
u is upper semicontinuous(iv)
u0=clx∈ℝ|ux>0 is compactThen,ℝℱ is called the space of fuzzy numbers. Obviously, ℝ⊂ℝℱ. For 0<α≤1, denote uα=x∈ℝ|ux≥α; then, from (i) to (iv), it follows that the α-level set uα∈PKℝ for all 0≤α≤1 is a closed bounded interval which is denoted by uα=u1α,u2α. By PKℝ, we denote the family of all nonempty compact convex subsets of ℝ and define the addition and scalar multiplication in PKℝ as usual.Theorem 1 (see [7]).
Ifu∈ℝℱ, then(i)
uα∈PKℝ for all 0≤α≤1(ii)
uα2⊂uα1 for all 0≤α1≤α2≤1(iii)
αk⊂0,1 is a nondecreasing sequence which converges to α, and then,(2)uα=∩k≥1uαk.
Conversely, ifAα=u1α,u2α;α∈0,1 is a family of closed real intervals verifying (i) and (ii), then Aα defined a fuzzy number u∈ℝℱ such that uα=Aα for 0<α≤1 and u0=∪0<α≤1Aα¯⊂A0.Lemma 1 (see [18]).
Letu,v:ℝℱ⟶0,1 be the fuzzy sets. Then, u=v if and only if uα=vα for all α∈0,1.The following arithmetic operations on fuzzy numbers are well known and frequently used below. Ifu,v∈ℝℱ, then(3)u+vα=u1α+v1α,u2α+v2α,λuα=λuα=λu1α,λu2α,ifλ≥0,λu2α,λu1α,ifλ<0.Definition 1 (see [19, 20]).
Letu,v∈ℝℱ. If there exists w∈ℝℱ such as u=v+w, then w is called the H-difference of u,v, and it is denoted as u⊖v.Definition 2 (see [21]).
Let we denote(4)0¯=1,t=0,0,t≠0.
Defined:ℝℱ×ℝℱ⟶ℝ+∪0 by the equation(5)du,v=supα∈0,1dHuα,vα,for allu,v∈ℝℱ,where dH is the Hausdorff metric.(6)dHuα,vα=maxu1α−v1α,u2α−v2α.
It is well known thatℝℱ,d is a complete metric space. We list the following properties of du,v:(7)du+w,v+w=du,v,du,v=dv,u,dku,kv=kdu,v,du,v≤du,w+dw,v,for all u,v,w∈ℝℱ and λ∈ℝ.
LetAk be a sequence in PKℝ converging to A. Then, theorem in [2] gives us an expression for the limit.Theorem 2 (see[2]).
IfdAk,A⟶0 as k⟶∞, then(8)A=∩k≥1∪m≥kAm¯.


## 3. Fuzzy Conformable Fractional Differentiability and Fuzzy Fractional Integral

### 3.1. Fuzzy Conformable Fractional Differentiability

Now, we present our new definition, which is the simplest and most natural and efficient definition of fractional derivative of orderq∈0,1.Definition 3 (see[17]).
LetF:0,a⟶ℝℱ be a fuzzy function, and qth order fuzzy conformable fractional derivative of F is defined by(9)TqFt=limε⟶0+Ft+εt1−q⊖Ftε=limε⟶0+Ft⊖Ft−εt1−qε,for all t>0,q∈0,1. Let Fqt stands for TqFt. Hence,(10)Fqt=limε⟶0+Ft+εt1−q⊖Ftε=limε⟶0+Ft⊖Ft−εt1−qε.
IfF is q-differentiable in some 0,a and limt⟶0+Fqt exists, then(11)Fq0=limt⟶0+Fqt,and the limits (in the metric d).Remark 1.
From the definition, it directly follows that ifF is q-differentiable, then the multivalued mapping Fα is q-differentiable for all α∈0,1 and(12)TqFα=Fqtα.where TqFα is denoted from the conformable fractional derivative of Fα of order q.Theorem 3 (see[17]).
LetF:0,a⟶ℝℱ be q-differentiable. Denote Fαt=f1αt,f2αt,α∈0,1. Then, f1αt and f2αt are q-differentiable and(13)Fqtα=f1αqt,f2αqt.Theorem 4.
LetF:0,a⟶ℝℱ is q-differentiable on 0,a. If t1,t2∈0,a with t1≤t2, then there exists λ∈ℝℱ such that Ft2=Ft1+λ.Proof.
For eachs∈t1,t2, there exists δs>0 such that the H-differences Fs+εs1−q⊖Fs and Fs⊖Fs−εs1−q exist for all 0≤ε<δs. Then, we can find a finite sequence t1=s1<s2<⋯<sn=t2 such that the family Isi=si−δsi,si+δsi|i=1,2,…,n covers t1,t2 and Isi∩Isi+1≠∅. Pick xi∈Isi∩Isi+1,i=1,2,…,n−1, such that si<xi<si+1. Then,(14)Fsi+1=Fxi+A1=Fsi+A2+A1=Fsi+λi,i=1,2,…,n−1,for some A1,A2,λi∈ℝℱ. Hence,(15)Ft2=Ft1+∑i=1n−1λi=Ft1+λ.Theorem 5.
IfF:0,a⟶ℝℱ is q-differentiable, then it is continuous.Proof.
Lett,t+t1−qε∈0,a with ε>0. Then, by properties of equation (7) and the triangle inequality, we have(16)dFt+t1−qε,Ft=dFt+t1−qε⊖Ft,0¯≤εdFt+t1−qε⊖Ftε,Fqt+εdFqt,0¯,where ε is so small that the H-difference Ft+t1−qε⊖Ft exists. By the differentiability, the right-hand side goes to zero as ε⟶0+, and hence, F is right continuous. The left continuity is proved similarly.Theorem 6.
Letq∈0,1. If F is differentiable and F is q-differentiable, then(17)TqFt=t1−qF′t.The proof is similar to the proof of Theorem8 case (i) in [17] and is omitted.Theorem 7.
Letq∈0,1, and if F,G:0,a⟶ℝℱ are q-differentiable and λ∈ℝ, thenTqF+Gt=TqF+TqG andTqλFt=λTqFtProof.
SinceF is q-differentiable, it follows that Ft+εt1−q⊖Ft exists, i.e., there exists u1t,εt1−q such that(18)Ft+εt1−q=Ft+u1t,εt1−q.
Analogously, sinceG is q-differentiable, there exists v1t,εt1−q such that(19)Gt+εt1−q=Gt+v1t,εt1−q,and we get(20)Ft+εt1−q+Gt+εt1−q=Ft+Gt+u1t,εt1−q+v1t,εt1−q,that is, the H-difference(21)Ft+εt1−q+Gt+εt1−q⊖Ft+Gt=u1t,εt1−q+v1t,εt1−q.
By similar reasoning, we get that there existu2t,εt1−q and v2t,εt1−q such that(22)Ft=Ft−εt1−q+u2t,εt1−q,Gt=Gt−εt1−q+v2t,εt1−q,and so(23)Ft+Gt=Ft−εt1−q+Gt−εt1−q+u2t,εt1−q+v2t,εt1−q,that is, the H-difference(24)Ft+Gt⊖Ft−εt1−q+Gt−εt1−q=u2t,εt1−q+v2t,εt1−q.
We observe that(25)limε⟶0+u1t,εt1−qε=limε⟶0+u2t,εt1−qε=Fqt,limε⟶0+v1t,εt1−qε=limε⟶0+v2t,εt1−qε=Gqt.
Finally, by multiplying (21) and (24) with 1/ε and passing to limit with limε⟶0+, we get that F+G is q-differentiable and TqF+Gt=TqFt+TqGt. The case (ii) is similar to the previous one.


### 3.2. Fuzzy Fractional Integral

Letq∈0,1 and F:0,a⟶ℝℱ be such that Ftα=f1αt,f2αt for all t∈0,a and α∈0,1. Suppose that f1α,f2α∈C0,a,ℝ∩L10,a,ℝ for all α∈0,1 and let(26)Aα≕∫0tf1αx1−qxdx,∫0tf2αx1−qxdx,t∈0,a.Lemma 2.
The familyAα;α∈0,1, given by equation (26), defined a fuzzy number F∈ℝℱ such that Fα=Aα.Proof.
Forα<β, we have f1αx≤f1βx and f2αx≥f2βx. It follows Aα⊇Aβ. Since f10x≤f1αnx≤f11x, we have(27)xq−1fiαnx≤maxaq−1fi0x,aq−1fi1x≕gix,for αn∈0,1 and i=1,2. Obviously, gi is integrable on 0,a. Therefore, if αn↑α, then by Lebesque’s dominated convergence theorem, we have(28)limn⟶∞∫0tfiαnx1−qxdx=∫0tfiαx1−qxdx,i=1,2.
From Theorem1, the proof is complete.Definition 4.
LetF∈C0,a,ℝℱ∩L10,a,ℝℱ define the fuzzy fractional integral for q∈0,1,(29)IqFt=Itq−1Ft=∫0tFx1−qxdx,by`(30)IqFtα=Itq−1Ftα=∫0tFx1−qxdxα=∫0tf1αx1−qxdx,∫0tf2αx1−qxdx,where the integral ∫0tfiα/x1−qxdx for i=1,2 is the usual Riemann improper integral. Also, the following properties are obvious.Lemma 3.
Letq∈0,1 and F,G:0,a⟶ℝℱ be fractional integrable and λ∈ℝ. Then,(i)
IqλFt=λIqFt(ii)
IqF+Gt=IqFt+IqGtProof.
The proof is similar to the proof of Theorem 4.3 cases (i) and (ii) in [2] and is omitted.Theorem 8.
TqIqFt=Ft, for t≥0, where F is any continuous function in the domain of IqProof.
SinceF is continuous, then IqFt is clearly q-differentiable because(31)IqFt=Itq−1Ft,and tq−1Ft is continuous for all t∈0,a; then, by Theorem 5.6 in [2] and Theorem 6, the fractional integral is q-differentiable. Hence,(32)TqIqFtα=t1−qddtIqFtα=t1−qddt∫0tf1αxx1−qdx,t1−qddt∫0tf2αxx1−qdx=t1−qf1αtt1−q,t1−qf2αtt1−q=Ftα.Theorem 9.
Letq∈0,1 and F be q-differentiable in 0,a, and assume that the conformable derivative Fq is integrable over 0,a. Then, for each s∈0,a, we have(33)Fs=F0+IqFq.Proof.
Letq∈0,1 and α∈0,1 be fixed. We shall prove that(34)Fαs=Fα0+IqFαq,where Fαq is the Hukuhara conformable fractional derivative of Fα; then, using Theorems 3 and 6 gives us the following equation.(35)Fαs=Fα0+IqFαq=Fα0+Iqt1−qFα′.By equation (29), we have(36)Fαs=Fα0+Iqt1−qFα′=Fα0+∫0stq−1t1−qFα′.
So,(37)Fαs=Fα0+∫0sFα′,where Fα′ is the Hukuhara derivative of Fα; equation (37) is also true for a fuzzy mapping F:0,a⟶ℝℱ. The equality (34) now follows Theorem 5.7 in [2].


## 3.1. Fuzzy Conformable Fractional Differentiability

Now, we present our new definition, which is the simplest and most natural and efficient definition of fractional derivative of orderq∈0,1.Definition 3 (see[17]).
LetF:0,a⟶ℝℱ be a fuzzy function, and qth order fuzzy conformable fractional derivative of F is defined by(9)TqFt=limε⟶0+Ft+εt1−q⊖Ftε=limε⟶0+Ft⊖Ft−εt1−qε,for all t>0,q∈0,1. Let Fqt stands for TqFt. Hence,(10)Fqt=limε⟶0+Ft+εt1−q⊖Ftε=limε⟶0+Ft⊖Ft−εt1−qε.
IfF is q-differentiable in some 0,a and limt⟶0+Fqt exists, then(11)Fq0=limt⟶0+Fqt,and the limits (in the metric d).Remark 1.
From the definition, it directly follows that ifF is q-differentiable, then the multivalued mapping Fα is q-differentiable for all α∈0,1 and(12)TqFα=Fqtα.where TqFα is denoted from the conformable fractional derivative of Fα of order q.Theorem 3 (see[17]).
LetF:0,a⟶ℝℱ be q-differentiable. Denote Fαt=f1αt,f2αt,α∈0,1. Then, f1αt and f2αt are q-differentiable and(13)Fqtα=f1αqt,f2αqt.Theorem 4.
LetF:0,a⟶ℝℱ is q-differentiable on 0,a. If t1,t2∈0,a with t1≤t2, then there exists λ∈ℝℱ such that Ft2=Ft1+λ.Proof.
For eachs∈t1,t2, there exists δs>0 such that the H-differences Fs+εs1−q⊖Fs and Fs⊖Fs−εs1−q exist for all 0≤ε<δs. Then, we can find a finite sequence t1=s1<s2<⋯<sn=t2 such that the family Isi=si−δsi,si+δsi|i=1,2,…,n covers t1,t2 and Isi∩Isi+1≠∅. Pick xi∈Isi∩Isi+1,i=1,2,…,n−1, such that si<xi<si+1. Then,(14)Fsi+1=Fxi+A1=Fsi+A2+A1=Fsi+λi,i=1,2,…,n−1,for some A1,A2,λi∈ℝℱ. Hence,(15)Ft2=Ft1+∑i=1n−1λi=Ft1+λ.Theorem 5.
IfF:0,a⟶ℝℱ is q-differentiable, then it is continuous.Proof.
Lett,t+t1−qε∈0,a with ε>0. Then, by properties of equation (7) and the triangle inequality, we have(16)dFt+t1−qε,Ft=dFt+t1−qε⊖Ft,0¯≤εdFt+t1−qε⊖Ftε,Fqt+εdFqt,0¯,where ε is so small that the H-difference Ft+t1−qε⊖Ft exists. By the differentiability, the right-hand side goes to zero as ε⟶0+, and hence, F is right continuous. The left continuity is proved similarly.Theorem 6.
Letq∈0,1. If F is differentiable and F is q-differentiable, then(17)TqFt=t1−qF′t.The proof is similar to the proof of Theorem8 case (i) in [17] and is omitted.Theorem 7.
Letq∈0,1, and if F,G:0,a⟶ℝℱ are q-differentiable and λ∈ℝ, thenTqF+Gt=TqF+TqG andTqλFt=λTqFtProof.
SinceF is q-differentiable, it follows that Ft+εt1−q⊖Ft exists, i.e., there exists u1t,εt1−q such that(18)Ft+εt1−q=Ft+u1t,εt1−q.
Analogously, sinceG is q-differentiable, there exists v1t,εt1−q such that(19)Gt+εt1−q=Gt+v1t,εt1−q,and we get(20)Ft+εt1−q+Gt+εt1−q=Ft+Gt+u1t,εt1−q+v1t,εt1−q,that is, the H-difference(21)Ft+εt1−q+Gt+εt1−q⊖Ft+Gt=u1t,εt1−q+v1t,εt1−q.
By similar reasoning, we get that there existu2t,εt1−q and v2t,εt1−q such that(22)Ft=Ft−εt1−q+u2t,εt1−q,Gt=Gt−εt1−q+v2t,εt1−q,and so(23)Ft+Gt=Ft−εt1−q+Gt−εt1−q+u2t,εt1−q+v2t,εt1−q,that is, the H-difference(24)Ft+Gt⊖Ft−εt1−q+Gt−εt1−q=u2t,εt1−q+v2t,εt1−q.
We observe that(25)limε⟶0+u1t,εt1−qε=limε⟶0+u2t,εt1−qε=Fqt,limε⟶0+v1t,εt1−qε=limε⟶0+v2t,εt1−qε=Gqt.
Finally, by multiplying (21) and (24) with 1/ε and passing to limit with limε⟶0+, we get that F+G is q-differentiable and TqF+Gt=TqFt+TqGt. The case (ii) is similar to the previous one.


## 3.2. Fuzzy Fractional Integral

Letq∈0,1 and F:0,a⟶ℝℱ be such that Ftα=f1αt,f2αt for all t∈0,a and α∈0,1. Suppose that f1α,f2α∈C0,a,ℝ∩L10,a,ℝ for all α∈0,1 and let(26)Aα≕∫0tf1αx1−qxdx,∫0tf2αx1−qxdx,t∈0,a.Lemma 2.
The familyAα;α∈0,1, given by equation (26), defined a fuzzy number F∈ℝℱ such that Fα=Aα.Proof.
Forα<β, we have f1αx≤f1βx and f2αx≥f2βx. It follows Aα⊇Aβ. Since f10x≤f1αnx≤f11x, we have(27)xq−1fiαnx≤maxaq−1fi0x,aq−1fi1x≕gix,for αn∈0,1 and i=1,2. Obviously, gi is integrable on 0,a. Therefore, if αn↑α, then by Lebesque’s dominated convergence theorem, we have(28)limn⟶∞∫0tfiαnx1−qxdx=∫0tfiαx1−qxdx,i=1,2.
From Theorem1, the proof is complete.Definition 4.
LetF∈C0,a,ℝℱ∩L10,a,ℝℱ define the fuzzy fractional integral for q∈0,1,(29)IqFt=Itq−1Ft=∫0tFx1−qxdx,by`(30)IqFtα=Itq−1Ftα=∫0tFx1−qxdxα=∫0tf1αx1−qxdx,∫0tf2αx1−qxdx,where the integral ∫0tfiα/x1−qxdx for i=1,2 is the usual Riemann improper integral. Also, the following properties are obvious.Lemma 3.
Letq∈0,1 and F,G:0,a⟶ℝℱ be fractional integrable and λ∈ℝ. Then,(i)
IqλFt=λIqFt(ii)
IqF+Gt=IqFt+IqGtProof.
The proof is similar to the proof of Theorem 4.3 cases (i) and (ii) in [2] and is omitted.Theorem 8.
TqIqFt=Ft, for t≥0, where F is any continuous function in the domain of IqProof.
SinceF is continuous, then IqFt is clearly q-differentiable because(31)IqFt=Itq−1Ft,and tq−1Ft is continuous for all t∈0,a; then, by Theorem 5.6 in [2] and Theorem 6, the fractional integral is q-differentiable. Hence,(32)TqIqFtα=t1−qddtIqFtα=t1−qddt∫0tf1αxx1−qdx,t1−qddt∫0tf2αxx1−qdx=t1−qf1αtt1−q,t1−qf2αtt1−q=Ftα.Theorem 9.
Letq∈0,1 and F be q-differentiable in 0,a, and assume that the conformable derivative Fq is integrable over 0,a. Then, for each s∈0,a, we have(33)Fs=F0+IqFq.Proof.
Letq∈0,1 and α∈0,1 be fixed. We shall prove that(34)Fαs=Fα0+IqFαq,where Fαq is the Hukuhara conformable fractional derivative of Fα; then, using Theorems 3 and 6 gives us the following equation.(35)Fαs=Fα0+IqFαq=Fα0+Iqt1−qFα′.By equation (29), we have(36)Fαs=Fα0+Iqt1−qFα′=Fα0+∫0stq−1t1−qFα′.
So,(37)Fαs=Fα0+∫0sFα′,where Fα′ is the Hukuhara derivative of Fα; equation (37) is also true for a fuzzy mapping F:0,a⟶ℝℱ. The equality (34) now follows Theorem 5.7 in [2].


## 4. Fuzzy Comformable Fractional Differential Equations

We study the fuzzy initial value problem(38)Tqxt=Ft,xt,q∈0,1,x0=x0,where F:0,a×ℝℱ⟶ℝℱ is the continuous fuzzy mapping, and x0 is the fuzzy number. From Theorems 5, 8, and 9, it immediately follows.Theorem 10.
A mapping x:0,a⟶ℝℱ is a solution to problem (38) if and only if it is continuous and satisfies the integral equation:(39)xt=x0+IqFt,xt,for all t∈0,a and q∈0,1.Theorem 11.
LetF:0,a×ℝℱ⟶ℝℱ be continuous, and assume that there exists k>0 such that(40)dFt,x,Ft,y≤kdx,y,for all t∈0,a,x,y∈ℝℱ. Then, problem (38) has a unique solution on 0,a.Proof.
If in problem (38) we consider the conformable derivative xq for all q∈0,1 Theorem 3, then from Theorem 6.1 in [2] and using Definition 4 and Lemma 1,0,a we can prove that there exists an unique solution on 0,a, and the proof is now complete.Remark 2.
In [15], it is observed that if we fuzzify the equivalent ordinary differential equation xq+x=0, then we will get fuzzy differential equations (the equation was fuzzified by adding a forcing term σt in the right-hand side). That is, if we consider fuzzy differential equation xq+x=σt with the same initial condition xt0=x0, we get the result.
Consider the following linear fractional equation:(41)xqt+xt=σt,q∈0,1andt∈0,a,where σ∈C0,a×ℝℱ. Denote xtα=x1αt,x2αt,x0α=x01α,y02α, and σtα=σ1αt,σ2αt.Theorem 12.
Equation (41) has a unique solution in 0,a, and for given initial x0∈ℝℱ, it is given by(42)xt=x0e−tq/q+∫0tsq−1σsesq−tq/qds,t∈0,a,=x0e−tq/q+e−tq/qIqσtetq/q,t∈0,a.Proof.
Equation (41) can be written, levelwise, as(43)x1αqt,x2αqt+x1αt,x2αt=σ1αt,σ2αt,t∈0,a,for every α∈0,1, so that(44)x1αqt+x1αt=σ1αt,x2αqt+x2αt=σ2αt.
Thus, fort∈0,a,(45)x1αetq/qqt=σ1αtetq/q,x2αetq/qqt=σ2αtetq/q,and, therefore, it can be deduced that(46)x1αt=x01αe−tq/q+∫0tsq−1σ1αsesq−tq/qds,x2αt=x02αe−tq/q+∫0tsq−1σ2αsesq−tq/qds.
This proves that, forα∈0,1,(47)xtα=x0αe−tq/q+∫0tsq−1σsαesq−tq/qds,t∈0,a.
So,(48)xtα=x0αe−tq/q+e−tq/qIqσtαetq/q,t∈0,a.


## 5. Conclusion

In this study, for developing and proving some results for fuzzy conformable differentiability and fuzzy fractional integrability of such functions, we provided existence and uniqueness solutions to fuzzy fractional problems for orderq∈0,1 FFDEs, which is interpreted by using the generalized conformable fractional derivatives concept.For future research, we will solve the fractional fuzzy conformable partial differential equations [22, 23] and a class of linear differential dynamical systems [24] by using the proposed method.



---

*Source: 6655450-2021-02-04.xml*