# Solving Higher-Order Fractional Differential Equations by the Fuzzy Generalized Conformable Derivatives

Authors: Atimad Harir; Said Melliani; Lalla Saadia Chadli
Journal: Applied Computational Intelligence and Soft Computing
 (2021)

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## Abstract

In this paper, the generalized concept of conformable fractional derivatives of orderq∈n,n+1 for fuzzy functions is introduced. We presented the definition and proved properties and theorems of these derivatives. The fuzzy conformable fractional differential equations and the properties of the fuzzy solution are investigated, developed, and proved. Some examples are provided for both the new solutions.


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## Body

## 1. Introduction

A closed-form solution for nonlinear fractional differential equations (FDEs) plays a significant role in understanding the qualitative as well as quantitative features of complex physical phenomena. The nonlinearFDEs appear in different sciences and engineering problems such as control theory, signal processing, finance, electricity, mechanics, plasma physics, stochastic dynamical system, economics, and electrochemistry [1–7]. A fuzzy fractional differentiation and fuzzy integration operators have different kinds of definitions that we can mention, the fuzzy Riemann–Liouville definition [8, 9], the fuzzy Caputo definition [9, 10], and so on. Lately, Khalid et al. [11] introduced a new simple definition of the fractional derivative named the conformable fractional derivative, which can redress shortcomings of the other definitions, and this new definition satisfies formulas of derivative of product and quotient of two functions [12, 13]. Harir et al. [14] introduced the fuzzy generalized conformable fractional derivative, which generalized and extended the concept of Hukuhara differentiability for set-valued mappings to the class of fuzzy mapping [15, 16].Our objective of this article is to present a generalized concept of conformable fractional derivative of orderq∈n,n+1,n∈ℕ for fuzzy functions. Then, we have investigated in more detail some new properties of these derivatives and we have proved some useful related theorems. We interpret fuzzy conformable fractional differential equations using this concept. We introduce new definitions of solutions. Two examples are provided.


## 2. Preliminaries

Let us denote byℝℱ=u:ℝ⟶0,1 the class of fuzzy subsets of the real axis satisfying the following properties [13, 17]:(i)
u is normal, i.e., there exists an 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=cx∈ℝ|ux>0 is compact.Then,ℝℱ is called the space of fuzzy numbers. Obviously, ℝ⊂ℝℱ. For 0<α≤1 denoting 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 [10]). If u∈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; 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 is defined as 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:ℝ⟶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 [17]. If u,v∈ℝℱ, then(3)u+vα=u1α+v1α,u2α+v2α,λuα=λuα=λu1α,λu2α,ifλ≥0,λu2α,λu1α,ifλ<0.Definition 1.
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.Theorem 2.
(see [19])(i)
Let we denote(4)0¯=1,t=0,0,t≠0,then0¯∈ℝℱ is a neutral element with respect to +, i.e., u+0¯=0¯+uu∈ℝℱ.(ii)
With respect to0¯, none of u∈ℝℱ\ℝ has opposite in ℝℱ.(iii)
For anya,b∈ℝ with a,b≥0 or a,b≤0 and any u∈ℝℱ, we have a+b⋅u=a⋅u+b⋅u; for general a,b∈ℝ, the above property does not hold.(iv)
For anyλ∈ℝ and any u,v∈ℝℱ, we have λ⋅u+v=λ⋅u+λ⋅v.(v)
For anyλ,ν∈ℝ and any u∈ℝℱ, we have λ⋅ν⋅u=λ⋅ν⋅u.Defined:ℝℱ×ℝℱ⟶ℝ+∪0 by the following 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 [17]:(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 [17] gives us an expression for the limit.Theorem 3.
(see [15]). If dAk,A⟶0 as k⟶∞, then(8)A=∩k≥1m≥k∪¯m.
LetI=0,a⊂ℝ be an interval. We denote by CI,ℝℱ the space of all continuous fuzzy functions on I which is a complete metric space with respect to the metric(9)hu,v=supdut,vt.


## 3. Generalized Fuzzy Conformable Fractional Derivatives

Definition 2.
(see [14]). Let F:I⟶ℝℱ be a fuzzy function. qth order “fuzzy conformable fractional derivative” of F is defined by(10)TqFt=limε⟶0+Ft+εt1−q⊖Ftε=limε⟶0+Ft⊖Ft−εt1−qε,for all t>0,q∈0,1. If F is q-differentiable in some I, and limt⟶0+TqFt exists, then(11)TqF0=limt⟶0+TqFt,and the limits exist (in the metric d).Remark 1.
(see [14]). From the definition, it directly follows that if F is q-differentiable, then the multivalued mapping Fα is q-differentiable for all α∈0,1 and(12)TqFα=TqFtα.
Here,TqFα is denoted as the conformable fractional derivative of Fα of order q. The converse result does not hold, since the existence of Hukuhara differences xα⊖yα,α∈0,1, does not imply the existence of H-difference x⊖y.We consider the following definition [14].Definition 3.
LetF:I⟶ℝℱ be a fuzzy function and q∈0,1. One says, F is q1-differentiable at point t>0 if there exists an element TqFt∈ℝℱ such that for all ε>0 sufficiently near to 0, there exist Ft+εt1−q⊖Ft,Ft⊖Ft−εt1−q and the limits (in the metric d)(13)limε⟶0+Ft+εt1−q⊖Ftε=limε⟶0+Ft⊖Ft−εt1−qε=TqFt.
F is q2-differentiable at t>0 if for all ε<0 sufficiently near to 0, there exist Ft+εt1−q⊖Ft,Ft⊖Ft−εt1−q(14)limε⟶0−Ft+εt1−q⊖Ftε=limε⟶0−Ft⊖Ft−εt1−qε=TqFt.
IfF is qn-differentiable at t>0, we denote its q-derivatives, for n=1,2.Definition 4.
LetF:I⟶ℝℱ be a fuzzy function and now we introduce definitions and theorems for q∈n,n+1 for some natural number n. For the sake of convenience, we concentrate on q∈1,2 case be n-differentiable at t, where t>0. Then the fuzzy conformable fractional derivative of f of order q is defined by(15)TqFt=limε⟶0+Fq−1t+εtq−q⊖Fq−1tε=limε⟶0+Fq−1t⊖Fq−1t−εtq−qε,where q∈n,n+1 and q is the smallest integer greater than or equal to q and the limits exist (in the metric d).Theorem 4.
LetF:I⟶ℝℱ and q∈1,2 and n,m=1,2. If F is n,m-differentiable and F is qn,m-differentiable, then(16)Tqn,mFt=t2−qDn,m2Ft.Remark 2 (see [20]).
F is n,m-differentiable on I, if Dn1 exists on I and it is m-differentiable on I. The second derivatives of F are denoted by Dn,m2Ft for n,m=1,2.Proof.
We present the details only forn=m=1, since the other case is analogous. Let h=εt2−q in Definition 4, then ε=tq−2h. Therefore, if ε>0 and α∈0,1, we have(17)D11Ft+εt2−q⊖D11Ftα=f1α′t+εt2−q−f1α′t,f2α′t+εt2−q−f2α′t.
Dividing byε, we have(18)D11Ft+εt2−q⊖D11Ftαε=f1α′t+εt2−q−f1α′tε,f2α′t+εt2−q−f2α′tε,and passing to the limit(19)limε⟶0+D11Ft+εt2−q⊖D11Ftαε=limε⟶0+f1α′t+εt2−q−f1α′tε,f2α′t+εt2−q−f2α′tε=limh⟶0+f1α′t+h−f1α′ttq−2h,f2α′t+h−f2α′ttq−2h=t2−qlimh⟶0+f1α′t+h−f1α′th,f2α′t+h−f2α′th=t2−qf1α″t,f2α″t.
Similarly, we obtain(20)D11Ft⊖D11Ft−εt2−qαε=f1α′t−f1α′t−εt2−qε,f2α′t−f2α′t−εt2−qε,and passing to the limit and ε=tq−2h gives Tq1,1Ft=t2−qf1α″t,f2α″t.Theorem 5.
Let0<q,p<1, and F be a fuzzy function 2 times differentiable on an open real interval I, then the fuzzy conformable derivative obeys to the following(21)TpnTqmFt≠TqmTpnFt,where n,m=1,2.Proof.
Letn,m=1,2. By using Theorem 7 and Theorem 8 in [14], we have the following relation;(22)TpnTqmFt=Tpnt1−qDm1Ft=Tpnt1−qDm1Ft+t1−qTpnDm1Ft=t1−p1−qt−qDm1Ft+t1−qt1−pDn1Dm1Ft=t1−p−q1−qDm1Ft+t2−q−pDm,n2Ft=t1−p−q1−qDm1Ft+tDm,n2Ft.
On the other hand,(23)TqmTpnFt=Tqmt1−pDn1Ft=Tqmt1−pDn1Ft+t1−pTqmDn1Ft=t1−q1−pt−pDn1Ft+t1−pt1−qDm1Dn1Ft=t1−p−q1−pDn1Ft+t2−q−pDn,m2Ft=t1−p−q1−pDn1Ft+tDn,m2Ft.
It follows that(24)t1−p−q1−qDm1Ft+tDm,n2Ft≠t1−p−q1−pDn1Ft+tDn,m2Ft,for all n,m=1,2. The proof is complete.Lemma 2.
Let0<q<1 and F be a fuzzy function 2 times differentiable on an open real interval I, then the fuzzy conformable derivative obeys the following(25)TpnDm1Ft≠Dm1TpnFt,where n,m=1,2.Proof.
Letn,m=1,2. By using Theorems 7 and 8 in [14], we have the following relation:(26)TpnDm1Ft=TpnDm1Ft=t1−pDn1Dm1Ft=t1−pDm,n2Ft.
On the other hand,(27)Dm1TpnFt=Dm1t1−pDn1Ft=Dm1t1−pDn1Ft+t1−pDm1Dn1Ft=1−pt−pDn1Ft+t1−pDm1Dn1Ft=t−p1−pDn1Ft+t1−pDn,m2Ft.
It follows that(28)t−p1−pDn1Ft+t1−pDn,m2Ft≠t1−pDm,n2Ft,for all n,m=1,2. The proof is complete.Theorem 6.
LetF:I⟶ℝℱ,1<q≤2 and p=q−1. Then the fuzzy conformable fractional derivative of order q, where Dn1Ft exists, is defined by(29)Tqn,mFt=TpmDn1Ft,where n,m=1,2.Proof.
We present the details only for the casen=m=1, since the other case is analogous. By using Theorem 8 in [14] and Theorem 2.2 in [20], we have(30)Tp1D11Ft=t1−pD11D11Ft=t1−pD1,12Ft=t1−q−1D1,12Ft=t2−qD1,12Ft,and of Theorem 4, it is that(31)Tq1,1Ft=t2−qD1,12Ft.Remark 3.
F is qn,m-differentiable on I, if Dn1 exists on I and it is pm-differentiable on I and pm=qn,m−1. The q-differentiable (conformable fractional derivatives of order q∈0,1) of F is denoted by Tqn,mFt for n,m=1,2.Theorem 7.
Let1<q≤2 and p=q−1. Then the fuzzy conformable fractional derivative of F order q, where D11F:I⟶ℝℱ or D21F:I⟶ℝℱ exists, where Ftα=f1αt,f2αt, is defined by(i)
IfD11Fisp1-differentiable, then f1α′t and f2α′t are p-differentiable and(32)Tq1,1Ftα=Tp1D11Ftα=Tpf1α′t,Tpf2α′t.(ii)
If D11F is p2-differentiable, then f1α′t and f2α′t are p-differentiable and(33)Tq1,2Ftα=Tp2D11Ftα=Tpf2α′t,Tpf1α′t.(iii)
IfD21F is p1-differentiable, then f1α′t and f2α′t are p-differentiable and(34)Tq2,1Ftα=Tp1D21Ftα=Tpf2α′t,Tpf1α′t.(iv)
IfD21F is p2-differentiable, then f1α′t and f2α′t are p-differentiable and(35)Tq2,2Ftα=Tp2D21Ftα=Tpf1α′t,Tpf2α′t.Proof.
We present the details only forn=m=1, since the other cases are analogous. If ε>0 and α∈0,1, we have(36)D11Ft+εt1−p⊖D11Ftα=f1α′tt+εt1−p−f1α′tt,f2α′tt+εt1−p−f2α′tt.
Dividing byε, we have(37)D11Ft+εt1−p⊖D11Ftα=f1α′tt+εt1−p−f1α′tt,f2α′tt+εt1−p−f2α′tt.
Dividing byε, we have(38)D11Ft+εt1−p⊖D11Ftαε=f1α′t+εt1−p−f1α′tε,f2α′t+εt1−p−f2α′tε.
Similarly, we obtain(39)D11Ft⊖D11Ft−εt1−pαε=f1α′t−f1α′t−εt1−pε,f2α′t−f2α′t−εt1−pε,and passing to the limit, we have(40)Tp1D11Ftα=Tpf1α′t,Tpf2α′t,and using Theorem 6 gives the theorem.


## 4. Fuzzy Conformable Fractional Differential Equations of Orderq

In this section, we study the fuzzy conformable fractional differential equations of orderq∈1,2,p∈0,1:(41)dqdtqyt+adpdtpyt+byt=gt,t≥0,y0=σ0,y′0=σ1,where a,b>0,σ0,σ1∈ℝℱ, and gt is a continuous fuzzy function on some interval I. We give the following definition for the solutions of (41).Definition 5.
Lety:I⟶ℝℱ,1<q≤2,0<p≤1 and n,m∈1,2. y is a solution (n, m), for problem (41) on I, if Dn1y,Tpny,Tqn,my (or Dn1y,Tpmy and TpmDn1ypm=qn,m−1) exist on I and(42)Tqn,myt+aTpnyt+byt=gt,t≥0,y0=σ0,Dn1y0=σ1.Remark 4.
From Theorem4 and Theorem 8 in [14], it directly follows that y is a solution (n, m), for problem (41) on I, if Dn1y and Dn,m2y exist on I, and(43)t2−qDn,m2yt+at1−pDnyt+byt=gt,t≥0,y0=σ0,Dn1y0=σ1,where q∈1,2,p∈0,1 and n,m∈1,2.Therefore, since the fuzzy conformable derivatives of fuzzy processes(44)Tpytα=Tpy1αt,Tpy2αt,p∈0,1,α∈0,1,Tqytα=Tqy1αt,Tqy2αt,q∈1,2,α∈0,1,provided these two intervals define fuzzy numbers Tpyt,p∈0,1 and Tqyt,q∈1,2 in ℝℱ, otherwise we apply Definition 2 and Theorem 7, then we have one of the following cases for q∈1,2 and p∈0,1:(i)
System (1, 1)(45)Tqy1αt+aTpy1αt+by1αt=g1αty1α0=σ01α,y1α′0=σ11α,Tqy2αt+aTpy2αt+by2αt=g2αty2α0=σ02α,y2α′0=σ12α.(ii)
System (1, 2)(46)Tqy2αt+aTpy1αt+by1αt=g1αty10=σ01α,y1α′0=σ11α,Tqy1αt+aTpy2αt+by2αt=g2αty20=σ02α,y2α′0=σ12α.(iii)
System (2, 1)(47)Tqy2αt+aTpy2αt+by1αt=g1αty10=σ01α,y2α′0=σ11α,Tqy1αt+aTpy1αt+by2αt=g2αty20=σ02α,y1α′0=σ12α.(iv)
System (2, 2)(48)Tqy1αt+aTpy2αt+by1αt=g1αty10=σ01α,y2α′0=σ11α,Tqy2αt+aTpy1αt+by2αt=g2αty20=σ02α,y1α′0=σ12α,where σ0α=σ01α,σ02α,σ1α=σ11α,σ12α and gtα=g1αt,g2αt.Theorem 8.
Letn,m∈1,2 and y=y1,y2 be a solution (n, m) for problem (41) on I. Then y1 and y2 solve the associated system (n, m).Proof.
By using Theorem4 and Remark 4 and suppose y is the solution (n, m) of problem (41), according to Definition 5, then Tpn,p∈0,1 and Tqn,m,q∈1,2 or t1−pDn1 and t2−qDn,m2 exist and satisfy problem (41). By Theorem 6 in [14], (21) and (25) and substituting y1,y2 and their conformable fractional derivatives in problem (41), we get the system (n, m) corresponding to solution (n, m). This completes the proof.Theorem 9.
Letn,m∈1,2,p∈0,1 and y1α and y2α solve the system (n, m) on I, for all α∈0,1. Let ytα=y1αt,y2αt. If y has valid cut sets on I and TpmDn1 exists, then y is a solution (n, m) for fuzzy problem (41).Proof.
Letq∈1,2 and ytα=y1αt,y2αt is q-differentiable fuzzy function, let p∈0,1, so by using Theorems 6 and 7 and Theorem 6 in [14] and Remark 4, we can compute Dn1y,Tpny and TpmDn1y,pm=qn,m−1 according to y1α′,y2α′,t2−qy1α′′,t2−qy2α′′. Due to the fact that y1α and y2α solve system (n, m), from Definition 5, then y is a solution (n, m) for equation (41).


## 5. Examples

Example 1.
We consider a conformable fractional ordinary differential equation [13]:(49)d3/2ytdt3/2=k,t≥0,y0=σ0,y′0=σ1,where k=σ0=σ1 are the triangular fuzzy numbers having α-cuts α−1,1−α. If y is solution (1, 1) for problem (49) and p=1/2, then D11ytα=y1α′t,y2α′t and(50)T3/21,1ytα=T1/21y1α′t,y2α′tα=d3/2dt3/2y1αt,d3/2dt3/2y2αtα,and they satisfy system (1, 1) associated with equation (41). Using Theorems 6 and 4, so the conformable fractional system (1, 1) has only the following solution:(51)y1αt=α−143t3/2+t+1,y2αt=1−α43t3/2+t+1.
Then,yt=α−1,1−α4/3t3/2+t+1 has valid α-cuts for t≥0. By Theorem 2.3 in [20], yt is a conformable fractional derivative of order 3/21,1 for t≥0. So yt defines a solution (1, 1) for t≥0. For solution (1, 2), we deduce(52)y1αt=α−1−43t3/2+t+1,y2αt=1−α−43t3/2+t+1,where yt has valid α-cuts for t∈0,0.25 and is a conformable fractional derivative of order 3/21,2 for t∈0,0.25. Hence, yt gives us a solution on t∈0,0.25. For solution (1, 2), we get(53)y1αt=α−1−43t3/2−t+1,y2αt=1−α−43t3/2−t+1,where yt has valid α-cuts for t∈0,0.51. We can see yt is a solution (1, 2) on 0,0.51. Finally, system (2, 2) gives(54)y1αt=α−143t3/2−t+1,y2αt=1−α43t3/2−t+1,where yt has valid α-cuts for all 0≤t≤0.25 and defines a solution (2, 2) on t∈0,0.25. Then, we have an example of a fuzzy conformable fractional ordinary differential equation with four solutions.Example 2.
Given a conformable fractional ordinary differential equation [13, 21](55)d3/2ytdt3/2=d1/2ytdt1/2,t≥0,y0=σ0,y′0=σ1,where σ0andσ1 are the triangular fuzzy numbers having α-cuts(56)σ0α=α−1,1−α,σ1α=α,2−α,for allα∈0,1.
To find solution (1, 1), we have(57)y1αt=−1+αet,y2αt=−1+2−αet,where yt has valid α-cuts for t≥0 and yt=−1+σ1et. From Theorem 2.3 in [20], yt is a conformable fractional derivative of order 3/21,1 for t≥0. So yt defines a solution (1, 1) for t≥0. For solution (1, 2), we deduce(58)y1αt=−1+et+α−1e−t,y2αt=−1+et+1−αe−t.
We see thatyt has valid alpha-cut and yt=−1+et+σ0e−t. From Theorem 2.3 in [20], yt is a conformable fractional derivative of order T3/22,2 for t≥0. Since system (1, 2) has only the above solution, then solution (1, 2) does not exist. For solution (2, 1), we get(59)y1αt=−1+αet,y2αt=−1+2−αet.
yt has valid α-cuts and yt is a conformable fractional derivative of order 3/21,1 for t. Then, solution (2, 1) does not exist. For solution (2, 2), we deduce(60)y1αt=−1+et+α−1e−t,y2αt=−1+et+1−αe−t.
We see thatyt has valid α-cut and yt=−1+et+σ0e−t. From Theorem 2.3 in [20], yt is a conformable fractional derivative of order 3/22,2 for all t≥0. Then yt defines a solution for t≥0. Then, we have a fuzzy conformable fractional ordinary differential equation and two solutions.


## 6. Conclusion

By using the concept of conformable generalized derivative and its extension to fractional derivatives of orderq∈0,2, we show that we have several possibilities or types to define fractional derivatives of order q∈0,2 of fuzzy-number-valued functions. Then, we propose a new method to solve fuzzy fractional differential equations based on the selection of conformable derivative types covering all former solutions. With these ideas, the selection of conformable derivative type in each step of deprivation plays a crucial role.For future research, we will solve the fractional fuzzy conformable partial differential equations [22, 23] by using the proposed method.



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