# Series Type Solution of Fuzzy Fractional Order Swift–Hohenberg Equation by Fuzzy Hybrid Sumudu Transform

Authors: Abd Ullah; Aman Ullah; Shabir Ahmad; Mahmood Ul Haq; Kamal Shah; Nabil Mlaiki
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

There are some confusion and complexity in our everyday lives, as we live in an uncertain environment. In such type of environment, an accurate calculation of the data and finding a solution to a problem is not an easy job. So, fuzzy differential equations are the better tools to model problems in the fuzzy domain. Modeling the real-world phenomenon more accurately requires such operators. Therefore, we investigate the fractional-order Swift–Hohenberg equation in the fuzzy concept. We study this equation under the fuzzy Caputo fractional derivative. We use the fuzzy Sumudu transform to find out the semianalytical solution of the considered equation. To deal with the nonlinear term of the problem, we also use the Adomian decomposition method. To confirm the accuracy of the proposed procedure, we give two test problems. Lastly, we plot the numerical results for various fractional orders, and uncertainty belongs to [0, 1].


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## Body

## 1. Introduction

Over the past few decades, we have seen significant development in fractional calculus. This area of mathematics has been used widely by the researchers. Modeling such as biological population models, the predator-prey models, and infectious diseases models are generalized to fractional order. Fractional operators are much better to explain the physical phenomena more accurately compare to ordinary operators. The Swift–Hohenberg equation (named after Swift and Hohenberg) is one of the models that is studied in fractional order as well. The Swift–Hohenberg equation (S-H equation) is a PDE that deals with the study of fluid velocity and convection temperature dynamics. S-H equations play a significant role in different branches of physics [1–3]. In the investigation of pattern formation, this equation also plays an important role [4]. The details about the Swift–Hohenberg equation’s physics can be accessed in [5–7]. The long-term behavior of the Swift–Hohenberg equation’s solution has determined by Peletier and Rottschafer [8]. Khan et al. [9] also resolved the said equation with the Cauchy–Dirichlet condition. The nonlinear fractional order S-H equation is given by(1)Dtαϑδ,t+Dδ4+2Dδ2ϑδ,t+1−gϑδ,t+ϑ3δ,t=0,ϑδ,0=f0δ,where δ represents the spatial coordinate and g represents real bifurcation parameter.In mathematics, engineering, and physics, nonlinear differential equations have great significance. It is not always possible to determine the exact solution to nonlinear differential equations. Some new approaches have been developed to solve nonlinear physical problems, including the method of homotopy perturbation, the method of Adomian decomposition, and the method of homotopy analysis [10–13]. Recently, some other methods are also studied, which includes, the Fourier spectral method [14] and He’s variational method [15]. Also, Wang et al. have introduced He’s fractional derivative in [16]. In [17], the authors have studied Elzaki transform and He’s Homotopy perturbation method, and they have compared the results with the Sumudu transform method. The deterministic model of such physical problems cannot always be established because there are some ambiguity and vagueness in our environment. In such type of environment, an accurate calculation of the data and modeling a problem is not an easy job. In 1965, Zadeh introduced the notion of fuzzy sets in order to avoid such a challenge [18]. Thereafter, the use of this fuzzy set in modeling expanded increasingly. The construction and implementation of fuzzy differential equations have thus increased rapidly over the last few years. Fuzzy differential equations were first constructed by Kaleva [19] and Seikkala [20]. Agarwal has worked with others on the concept of fuzzy fractional differential equations [21]. The idea of integration of fuzzy functions was introduced by Dubois and Prade [22]. The fuzzy FDEs under the Caputo–Katugampola operator and Caputo gH differentiability have been studied by Hoa [23, 24]. The applications of fuzzy differential equations in various applied sciences are listed in [25–29].We will use the Sumudu transform as it holds a scale preserving property which results in the original function to be similar with the transformed function. Also, this transform over Laplace transform has advantage that if the solution of the differential equations exists by Laplace transform, then it is not necessarily existed by the Sumudu transform; however, if the solution of the differential equations exists by the Sumudu transform, then it is necessarily existed by the Laplace transform. Since the physical phenomena are well handled by fuzzy fractional operators, which motivated us to study the Swift–Hohenberg (1) in the fuzzy notion, under the Caputo fractional operator, the equation is given as(2)Dtαϑ˜δ,t⊕Dδ4+2⊙Dδ2ϑ˜δ,t⊕1−g⊙ϑ˜δ,t⊕ϑ˜3δ,t=0,ϑ˜δ,0=l˜r⊙f0δ,where α∈0,1, ϑ˜ is a fuzzy function, l˜r=l¯r,l¯r is a fuzzy number, and r∈0,1.


## 2. Preliminaries

This section is devoted to the basic notions of fuzzy set theory and fuzzy calculus that are necessary for the paper. We consider 1-gH differentiability. More details can be found in [30].Definition 1.
(see [18]). A fuzzy set ϑ˜:ℝ⟶0,1 is called a fuzzy number if it satisfies the following:(i)
Closure of the setm∈ℝ,ϑ˜m>0 is compact(ii)
ϑ˜ is normal for somez0∈ℝ;ϑ˜z0=1(iii)
ϑ˜ is fuzzy convex ϑ˜bx+1−by≥ϑ˜x∧ϑ˜yfor allb∈0,1,x,y∈ℝ(iv)
ϑ˜ is upper semicontinuous on ℝ
LetF represent the set of all fuzzy numbers.Definition 2.
(see [18]). The r-level set of ϑ˜∈F, denoted by ϑ˜r, is defined as(3)ϑ˜r=a∈ℝ:ϑa≥r˜if0<r≤1,clsuppϑ˜ifr=0.
It is clear thatr-level set is a closed and bonded interval, and ϑ˜r=ϑr¯,ϑ¯r for r∈0,1. For sake of simplicity, we write ϑ˜=ϑ¯r,ϑ¯r.Definition 3.
(see [31]). The parametric form of ϑ˜∈F is represented by ϑ¯r,ϑ¯r, for r∈0,1, having the following properties:(i)
ϑ¯r≤ϑ¯r(ii)
ϑ¯r is a nonincreasing and ϑ¯r is a nondecreasing function(iii)
Bothϑ¯r and ϑ¯r are right continuous at 0 and left continuous over 0,1
Letϑ1˜,ϑ2˜∈F be represented by ϑ1˜r=ϑ1¯r,ϑ¯r, ϑ2˜r=ϑ2r¯,ϑ2¯r, where 0≤r≤1, and k is any nonzero scalar. The basic fuzzy operators, such as the sum ϑ1˜⊕ϑ2˜, the subtraction ϑ1˜⊖ϑ2˜, the product k⊙ϑ1˜, and Hausdorff distance ρϑ1˜,ϑ2˜, are defined as follows:(i)
ϑ1˜⊕ϑ2˜r=ϑ1˜r+ϑ2˜r=ϑ1¯r+ϑ2¯r,ϑ1¯r+ϑ2¯r(ii)
k⊙ϑ1˜r=k⊙ϑ1˜r=kϑ1¯r,kϑ1¯rk≥0,kϑ1¯r,kϑ1¯rk<0(iii)
ϑ1˜⊖ϑ2˜r=ϑ1˜r+−1⊙ϑ2˜r=ϑ1¯r−ϑ2¯r,ϑ1¯r−ϑ2¯r(iv)
ρϑ1˜,ϑ2˜=supr∈0,1maxϑ1¯r−ϑ2¯r,ϑ1¯r−ϑ2¯rDefinition 4.
(see [30]). Let τ1,τ2∈F. The Hukahara difference between τ1andτ2 is denoted by τ1⊖Hτ2, if there exists τ3 such that τ1=τ2⊕τ3. Also τ1⊖Hτ2≠τ1+−τ2.Definition 5.
(see [30]). Let τ1,τ2∈F. The generalized Hukahara difference between τ1andτ2 is denoted by τ1⊖gHτ2, if there exists τ3 such that(4)τ1⊖gHτ2=τ3⇔iτ1=τ2⊕τ3or,iiτ2=τ1⊕−1τ3.
If casei exists, then there will be no need to consider the other case. In this paper, we consider case i only.Definition 6.
(see [32]). Let ϑ˜:℘1,℘2⟶F, where ℘1,℘2∈ℝ, and let z0∈℘1,℘2 and h be such that z0+h∈℘1,℘2. Then, the generalized Hukuhara derivative (gH-derivative for short) of ϑ˜ at z0 is defined as(5)ϑ˜gHz0=limh⟶0ϑ˜z0+h⊖gHϑ˜z0h.
Ifϑ˜gHz0 satisfying (5) exists, then we say ϑ˜ is gH differentiable at z0.
LetCF0,b be the space of all continuous fuzzy valued functions, and LF0,b be the space of all Lebesgue integrable fuzzy valued functions.Theorem 1 (see [30]).
Ifϑ˜∈CF0,1∩LF0,1, then the fuzzy fractional integral of order 0<α≤1 of a function ϑ˜ is given by(6)Iαϑ˜z,r=Iαϑ¯z,r,Iαϑ¯z,r,r∈0,1.
Also,(7)Iαϑ¯z,r=1Γα∫0zz−δα−1ϑ¯δ,rdδ,z>0,Iαϑ¯z,r=1Γα∫0zz−δα−1ϑ¯δ,rdδ,z>0.Theorem 2 (see [30]).
Letϑ˜∈LF0,b and α∈0,1. If ϑ˜ is gH differentiable at z∈0,b, then(8)DCαϑ˜z=1Γ1−α∫0zz−t−αϑ˜gHtdt,∀z∈0,b.Theorem 3 (see [30]).
Letϑ˜∈CF0,b∩LF0,b and α∈0,1 be such that the gH-derivative ϑ˜gHz exists on 0,b. We have(9)DCαϑz˜r=DCαϑ¯z,r,DCαϑ¯z,r,∀z∈0,b,where(10)DCαϑ¯z,r=1Γ1−α∫0zz−t−αddtϑ¯t,rdt,DCαϑ¯z,r=1Γ1−α∫0zz−t−αddtϑ¯t,rdt.Definition 7.
(see [33]). Let ϑ˜ be continuous fuzzy valued function. We assume that ϑ˜uz⊙e−z is an improper fuzzy Riemann integrable on R+∪0; then, the fuzzy Sumudu transform is represented as(11)G˜u=Sϑ˜zu=∫0∞ϑ˜uz⊙e−vdv.
In parametric form, forr∈0,1, we have(12)∫0∞ϑ˜uz,r⊙e−zdz=∫0∞ϑ¯uz,re−zdz,∫0∞ϑ¯uz,re−zdz.
Hence,(13)Sϑ˜z,ru=Sϑ¯z,r,Sϑ¯z,r.Theorem 4.
(see [31]). Let ϑ∈CF0,b∩LF0,b; then, the Sumudu transform of fuzzy Caputo fractional derivative is given as(14)SDαϑ˜z,r=s−αSϑ˜z,r⊖gHϑ˜0,r.In parametric form, we have(15)SDαϑ˜z,r=SDαϑ¯z,r,SDαϑ¯z,r,where(16)SDαϑ¯z,r=s−αSϑ¯z,r−ϑ¯0,r,SDαϑ¯z,r=s−αSϑ¯z,r−ϑ¯0,r.


## 3. Solution of Fuzzy Fractional Order Swift–Hohenberg Equation

Here, we will discuss the procedure of the fuzzy Sumudu transform method with the Adomian decomposition method for solving the given equation under type1−gH differentiability.(17)Dtαϑ˜δ,t⊕Dδ4+2⊙Dδ2ϑ˜δ,t⊕1−g⊙ϑ˜δ,t⊕ϑ˜3δ,t=0,ϑ˜δ,0=l˜r⊙f0δ,where α∈0,1, ϑ˜ is a fuzzy function, g represents real bifurcation parameter, Dδ4 and Dδ2 are the 4th order and 2nd order derivatives w.r.t δ, f0 is a function of δ, l˜r=l¯r,l¯r is a fuzzy number, and r∈0,1. Since the given equation is nonlinear, we therefore use the decomposition in term of Adomian polynomials. Let us consider the terms ϑδ,t and ϑ3δ,t in the crisp case, which can be depicted as(18)ϑδ,t=∑n=0∞ϑnδ,t,ϑ3δ,t=∑n=0∞Gn,where(19)Gn=1n!dndλn∑h=0∞λhϑh3λ=0,Based on the above work, the functionsϑ˜δ,t and ϑ˜3δ,t in the fuzzy concept can be decomposed as(20)ϑ˜δ,t=∑n=0∞ϑ˜nδ,t=ϑ¯δ,t,r=∑n=0∞ϑ¯nδ,t,r,ϑ¯δ,t,r=∑n=0∞ϑ¯nδ,t,r,ϑ˜3δ,t=∑n=0∞G˜n=ϑ¯3δ,t,r=∑n=0∞G¯n,ϑ¯3δ,t,r=∑n=0∞G¯n,where(21)G¯n=1n!dndλn∑h=0∞λhϑ¯h3λ=0,G¯n=1n!dndλn∑h=0∞λhϑ¯h3λ=0.Applying fuzzy Sumudu transform to (17), we obtain(22)SDtαϑ˜δ,t⊕Dδ4+2⊙Dδ2ϑ˜δ,t⊕1−g⊙ϑ˜δ,t⊕ϑ˜3δ,t=0,1sαSϑ˜δ,t⊖1sα⊙l˜r⊙f0δ⊕SDδ4+2⊙Dδ2ϑ˜δ,t⊕1−g⊙ϑ˜δ,t⊕ϑ˜3δ,t=0.The equivalent form of the above equation is(23)Sϑ¯δ,t,r=l¯rf0δ+sαS−Dδ4+2Dδ2ϑ¯δ,t,r−1−gϑ¯δ,t,r−ϑ¯3δ,t,r,Sϑ¯δ,t,r=l¯rf0δ+sαS−Dδ4+2Dδ2ϑ¯δ,t,r−1−gϑ¯δ,t,r−ϑ¯3δ,t,r.Substituting (20) in (23), we obtain(24)S∑n=0∞ϑ¯nδ,t,r=l¯rf0δ+sαS−Dδ4+2Dδ2∑n=0∞ϑ¯nδ,t,r−1−g∑n=0∞ϑ¯nδ,t,r−∑n=0∞Gn¯S∑n=0∞ϑ¯nδ,t,r=l¯rf0δ+sαS−Dδ4+2Dδ2∑n=0∞ϑ¯nδ,t,r−1−g∑n=0∞ϑ¯nδ,t,r−∑n=0∞Gn¯Comparing terms on both sides, we get(25)Sϑ0δ,t,r=l¯rf0δ,Sϑ0δ,t,r=l¯rf0δ,Sϑ1¯δ,t,r=sαS−Dδ4+2Dδ2ϑ¯0δ,t,r−1−gϑ¯0δ,t,r−F0¯,Sϑ1δ,t,r¯=sαS−Dδ4+2Dδ2ϑ¯0δ,t,r−1−gϑ¯0δ,t,r−F0¯,Sϑ2¯δ,t,r=sαS−Dδ4+2Dδ2ϑ¯1δ,t,r−1−gϑ¯1δ,t,r−G1¯,Sϑ2¯δ,t,r=sαS−Dδ4+2Dδ2ϑ¯1δ,t−1−gϑ¯1δ,t,r−G1¯,⋮Sϑn¯δ,t,r=sαS−Dδ4+2Dδ2ϑ¯n−1δ,t,r−1−gϑ¯n−1δ,t,r−Gn−1¯,Sϑnδ,t,r¯=sαS−Dδ4+2Dδ2ϑ¯n−1δ,t,r−1−gϑ¯n−1δ,t,r−Gn−1¯.Using the inverse Sumudu transform, we get(26)ϑ0¯δ,t,r=l¯rf0δ,ϑ0¯δ,t,r=l¯rf0δ,ϑ1¯δ,t,r=S−1sαS−Dδ4+2Dδ2ϑ¯0δ,t,r−1−gϑ¯0δ,t,r−F0¯,ϑ1¯δ,t,r=S−1sαS−Dδ4+2Dδ2ϑ¯0δ,t,r−1−gϑ¯0δ,t,r−F0¯,ϑ2¯δ,t,r=S−1sαS−Dδ4+2Dδ2ϑ¯1δ,t,r−1−gϑ¯1δ,t,r−G1¯,ϑ2¯δ,t,r=S−1sαS−Dδ4+2Dδ2ϑ¯1δ,t−1−gϑ¯1δ,t,r−G1¯,⋮ϑn¯δ,t,r=S−1sαS−Dδ4+2Dδ2ϑ¯n−1δ,t−1−gϑ¯n−1δ,t,r−Gn−1¯,ϑn¯δ,t,r=S−1sαS−Dδ4+2Dδ2ϑ¯n−1δ,t,r−1−gϑ¯n−1δ,t,r−Gn−1¯.The semianalytical solution of the equation is(27)ϑ¯δ,t,r=ϑ0¯δ,t,r+ϑ1¯δ,t,r+⋯,ϑ¯δ,t,r=ϑ0¯δ,t,r+ϑ1¯δ,t,r+⋯.


### 3.1. Convergence and Error Analysis

In this section, we discuss the convergence and error analysis of (17).Theorem 5.
Letϑ˜pδ,t and ϑ˜nδ,t be the elements of a Banach space H, and the exact solution of (5) be ϑ˜δ,t. The series solution ∑p=0∞ϑ˜pδ,t converges to ϑ˜δ,t, if ϑ˜pδ,t≤ξϑ˜p−1δ,t, for ξ∈0,1, that is(28)for anyϑ>0∃Ma positive number∋ϑ˜p+nδ,t≤ϑ,∀p,n>M.Proof.
Let∑p=0∞ϑ˜pδ,t be a sequence defined as(29)S0δ,t=ϑ˜0δ,t,S1δ,t=ϑ˜0δ,t+ϑ˜1δ,t,S2δ,t=ϑ˜0δ,t+ϑ˜1δ,t+ϑ˜2δ,t,S3δ,t=ϑ˜0δ,t+ϑ˜1δ,t+ϑ˜2δ,t+ϑ˜3δ,t,⋮Spδ,t=ϑ˜0δ,t+ϑ˜1δ,t+ϑ˜2δ,t+⋯+ϑ˜pδ,t.
Now, we consider(30)Sp+1δ,t−Spδ,t=ϑ˜p+1δ,t≤ξϑ˜pδ,t≤ξ2ϑ˜p−1δ,t≤ξ3ϑ˜p−2δ,t⋮≤ξp+1ϑ˜0δ,t.
Next, for anyn, p (elements of ℕ)(31)Spδ,t−Snδ,t=ϑ˜p+nδ,t=Spδ,t−Sp−1δ,t+Sp−1δ,t−Sp−2δ,t+Sp−2δ,t−Sp−3δ,t+⋯+Sn+1δ,t−Snδ,t≤Spδ,t−Sp−1δ,t+Sp−1δ,t−Sp−2δ,t+⋯+Sn+1δ,t−Snδ,t≤ξpϑ˜0δ,t+ξp−1ϑ˜0δ,t+⋯+ξp+1ϑ˜0δ,t=ϑ˜0δ,tξp+ξp−1+⋯+ξp+1=ϑ˜0δ,t1−ξp−n1−ξξn+1.
Since0<ξ<1, and ϑ˜0δ,t is bounded, so by letting ϑ=1−ξ/1−ξp−nξn+1ϑ˜0δ,t, we obtain (28). Thus, ∑p=0∞ϑ˜pδ,t is a Cauchy sequence in H, which implies that there exists ϑ˜δ,t∈H such that limp⟶∞ϑ˜pδ,t=ϑ˜δ,t, which completes the proof.Theorem 6.
Let∑i=0jϑ˜iδ,t be finite and approximate solution of ϑ˜δ,t. If ϑ˜i+1δ,t≤ξδ,t, for ξ∈0,1, then the maximum absolute error is(32)ϑ˜δ,t−∑i=0jϑ˜iδ,t<ξj+11−ξϑ˜0δ,t.Proof.
We consider(33)ϑ˜δ,t−∑i=0jϑ˜iδ,t=∑i=j+1∞ϑ˜iδ,t≤∑i=j+1∞ϑ˜iδ,t≤∑i=j+1∞ξjϑ˜0δ,t≤ξj+11+ξ+ξ2+⋯ϑ˜0δ,t≤ξj+11−ξϑ˜0δ,t,
which completes the proof.


## 3.1. Convergence and Error Analysis

In this section, we discuss the convergence and error analysis of (17).Theorem 5.
Letϑ˜pδ,t and ϑ˜nδ,t be the elements of a Banach space H, and the exact solution of (5) be ϑ˜δ,t. The series solution ∑p=0∞ϑ˜pδ,t converges to ϑ˜δ,t, if ϑ˜pδ,t≤ξϑ˜p−1δ,t, for ξ∈0,1, that is(28)for anyϑ>0∃Ma positive number∋ϑ˜p+nδ,t≤ϑ,∀p,n>M.Proof.
Let∑p=0∞ϑ˜pδ,t be a sequence defined as(29)S0δ,t=ϑ˜0δ,t,S1δ,t=ϑ˜0δ,t+ϑ˜1δ,t,S2δ,t=ϑ˜0δ,t+ϑ˜1δ,t+ϑ˜2δ,t,S3δ,t=ϑ˜0δ,t+ϑ˜1δ,t+ϑ˜2δ,t+ϑ˜3δ,t,⋮Spδ,t=ϑ˜0δ,t+ϑ˜1δ,t+ϑ˜2δ,t+⋯+ϑ˜pδ,t.
Now, we consider(30)Sp+1δ,t−Spδ,t=ϑ˜p+1δ,t≤ξϑ˜pδ,t≤ξ2ϑ˜p−1δ,t≤ξ3ϑ˜p−2δ,t⋮≤ξp+1ϑ˜0δ,t.
Next, for anyn, p (elements of ℕ)(31)Spδ,t−Snδ,t=ϑ˜p+nδ,t=Spδ,t−Sp−1δ,t+Sp−1δ,t−Sp−2δ,t+Sp−2δ,t−Sp−3δ,t+⋯+Sn+1δ,t−Snδ,t≤Spδ,t−Sp−1δ,t+Sp−1δ,t−Sp−2δ,t+⋯+Sn+1δ,t−Snδ,t≤ξpϑ˜0δ,t+ξp−1ϑ˜0δ,t+⋯+ξp+1ϑ˜0δ,t=ϑ˜0δ,tξp+ξp−1+⋯+ξp+1=ϑ˜0δ,t1−ξp−n1−ξξn+1.
Since0<ξ<1, and ϑ˜0δ,t is bounded, so by letting ϑ=1−ξ/1−ξp−nξn+1ϑ˜0δ,t, we obtain (28). Thus, ∑p=0∞ϑ˜pδ,t is a Cauchy sequence in H, which implies that there exists ϑ˜δ,t∈H such that limp⟶∞ϑ˜pδ,t=ϑ˜δ,t, which completes the proof.Theorem 6.
Let∑i=0jϑ˜iδ,t be finite and approximate solution of ϑ˜δ,t. If ϑ˜i+1δ,t≤ξδ,t, for ξ∈0,1, then the maximum absolute error is(32)ϑ˜δ,t−∑i=0jϑ˜iδ,t<ξj+11−ξϑ˜0δ,t.Proof.
We consider(33)ϑ˜δ,t−∑i=0jϑ˜iδ,t=∑i=j+1∞ϑ˜iδ,t≤∑i=j+1∞ϑ˜iδ,t≤∑i=j+1∞ξjϑ˜0δ,t≤ξj+11+ξ+ξ2+⋯ϑ˜0δ,t≤ξj+11−ξϑ˜0δ,t,
which completes the proof.


## 4. Numerical Examples

Here, we solve two examples using the above method and we simulate their solutions through 2D and 3D plots.Example 1.
We consider the following fuzzy fractional order Swift–Hohenberg equation:(34)Dtαϑ˜δ,t⊕Dδ4+2⊙Dδ2ϑ˜δ,t⊕1−g⊙ϑ˜δ,t⊕ϑ˜3δ,t=0,ϑ˜δ,0=l˜r⊙110⊙sinδ,δ∈0,πforl˜r=r−1,1−r.
Applying fuzzy Sumudu transform to (34), we obtain(35)SDtαϑ˜δ,t⊕Dδ4+2⊙Dδ2ϑ˜δ,t⊕1−g⊙ϑ˜δ,t⊕ϑ˜3δ,t=0,1sαSϑ˜δ,t⊖1sα⊙l˜r⊙110⊙sinδ⊕SDδ4+2⊙Dδ2ϑ˜δ,t⊕1−g⊙ϑ˜δ,t⊕ϑ˜3δ,t=0.
The equivalent form of the above equation is(36)Sϑ¯δ,t,r=r−1110sinδ+sαS−Dδ4+2Dδ2ϑ¯δ,t−1−gϑ¯δ,t,r−ϑ¯3δ,t,r,Sϑ¯δ,t,r=1−r110sinδ+sαS−Dδ4+2Dδ2ϑ¯δ,t,r−1−gϑ¯δ,t,r−ϑ¯3δ,t,r.
Substituting (20) in (36), we obtain(37)S∑n=0∞ϑ¯nδ,t,r=r−1110sinδ+sαS−Dδ4+2Dδ2∑n=0∞ϑ¯nδ,t,r,−1−g∑n=0∞ϑ¯nδ,t,r−∑n=0∞Gn¯,S∑n=0∞ϑ¯nδ,t,r=1−r110sinδ+sαS−Dδ4+2Dδ2∑n=0∞ϑ¯nδ,t,r−1−g∑n=0∞ϑ¯nδ,t,r−∑n=0∞Gn¯.
Comparing terms on both sides, we get(38)Sϑ0δ,t,r¯=r−1110sinδ,Sϑ0δ,t,r¯=1−r110sinδ,ϑ0¯δ,t,r=r−1110sinδ,ϑ0¯δ,t,r=1−r110sinδ,Sϑ1δ,t,r¯=sαS−Dδ4+2Dδ2ϑ¯0δ,t,r−1−gϑ¯0δ,t,r−F0¯,Sϑ1δ,t,r¯=sαS−Dδ4+2Dδ2ϑ¯0δ,t,r−1−gϑ¯0δ,t,r−F0¯.
From (21), we get(39)F0=r−131103sin3δ,F0¯=1−r31103sin3δ,Sϑ1δ,t,r¯=sαS−Dδ4+2Dδ2r−1110sinδ−1−gr−1110sinδ−r−131103sin3δ,Sϑ1¯δ,t,r=sαS−Dδ4+2Dδ21−r110sinδ−1−g1−r110sinδ−1−r31103sin3δ,Sϑ1¯δ,t,r=sαS−r−1110sinδ+2r−1110sinδ−1−gr−1110sinδ−r−131103sin3δ,Sϑ1¯δ,t,r=sαS−1−r110sinδ+21−r110sinδ−1−g1−r110sinδ−1−r31103sin3δ,Sϑ1¯δ,t,r=sαSr−1110sinδg−r−121102sin2δ,Sϑ1¯δ,t,r=sαS1−r110sinδg−1−r21102sin2δ,ϑ1¯δ,t,r=tαΓα+1r−1110sinδg−r−121102sin2δ,ϑ1¯δ,t,r=tαΓα+11−r110sinδg−1−r21102sin2δ.
The next term is(40)Sϑ2¯δ,t,r=sαS−Dδ4+2Dδ2ϑ¯1δ,t,r−1−gϑ¯1δ,t,r−G1¯,Sϑ2¯δ,t,r=sαS−Dδ4+2Dδ2ϑ¯1δ,t,r−1−gϑ¯1δ,t,r−G1¯.
From (21), we get(41)G1¯=tαΓα+13r−121102sin2δgr−1110sinδ−r−131103sin3δ,G1¯=tαΓα+131−r21102sin2δg1−r110sinδ−1−r31103sin3δ.
After solving (40) and some simplification, we have(42)ϑ2¯δ,t,r=t2αΓ2α+11105r−1sinδ6600r−12cos2δ+10000g2=−2300r−12sin2δ−400gr−12sin2δ+3r−14sin4δ−2300r−12sin2δ−400gr−12sin2δ+3r−14sin4δϑ2¯δ,t,r=tαΓα+111051−rsinδ66001−r2cos2δ+10000g2=−23001−r2sin2δ−400g1−r2sin2δ+31−r4sin4δ.
Similarly, we can find the other terms. The semianalytical solution of the equation is(43)ϑ¯δ,t,r=r−1110sinδ1+tαΓα+1g−r−121102sin2δ+t2αΓ2α+11105r−1sinδ6600r−12cos2δ+10000g2−2300r−12sin2δ−400gr−12sin2δ+3r−14sin4δ+⋯,ϑ¯δ,t,r=1−r110sinδ1+tαΓα+1g−1−r21102sin2δ+t2αΓ2α+111051−rsinδ66001−r2cos2δ+10000g2−23001−r2sin2δ−400g1−r2sin2δ+31−r4sin4δ+⋯.
We have simulated the obtained results in 2D and 3D graphs. Figure1 shows the lower and upper solution for α=0.4 and α=1. The graph represents the fuzzy triangular number. Figure 2 represents the surface plot of the obtained solution of the proposed equation. As fractional order α increases, the solution curves tends to integer order solution, i.e., the graph at α=1 represents the solution at integer order. Also, the diameter of the fuzzy triangular number increases as fractional order increases. Thus, the fractional operator has great impact on the solution of the proposed equations. Figure 3 elaborates the evolution of the obtained results at fractional orders α=0.4, α=0.9, and various values of r∈0,1. In view of Theorem 6, we see that(44)ϑ˜δ,t−∑i=0jϑ˜iδ,t≤r−1110sinδ,≤ξj+1101−ξ,assinδ≤1.
Sinceξj+1/101−ξ<1. Therefore, we have(45)ξ=exp1j+1,j=1,2….
Atj⟶∞, the value of ξ⟶1.Figure 1
2D rious fractional orders, and uncertaintyr∈0,1.Figure 2
3D representation of the solution of Example1 at various fractional orders, and uncertainty r∈0,1.Figure 3
Simulation of the solution of Example1 at various fractional orders, and uncertainty r∈0,1.Example 2.
We consider the following fuzzy fractional order Swift–Hohenberg equation:(46)Dtαϑ˜δ,t⊕Dδ4+2⊙Dδ2ϑ˜δ,t⊕1−g⊙ϑ˜δ,t⊕ϑ˜3δ,t=0,ϑ˜δ,0=l˜r⊙eδ,forl˜r=r−1er,1−rer.
Applying the fuzzy Sumudu transform to (46), we obtain(47)SDtαϑ˜δ,t⊕Dδ4+2⊙Dδ2ϑ˜δ,t⊕1−g⊙ϑ˜δ,t⊕ϑ˜3δ,t=0,1sαSϑ˜δ,t⊖1sα⊙l˜r⊙eδ⊕SDδ4+2⊙Dδ2ϑ˜δ,t⊕1−g⊙ϑ˜δ,t⊕ϑ˜3δ,t=0.
The equivalent form of the above equation is(48)Sϑ¯δ,t,r=r−1ereδ+sαS−Dδ4+2Dδ2ϑ¯δ,t,r−1−gϑ¯δ,t,r−ϑ¯3δ,t,r,Sϑ¯δ,t,r=1−rereδ+sαS−Dδ4+2Dδ2ϑ¯δ,t,r−1−gϑ¯δ,t,r−ϑ¯3δ,t,r.
Substituting (20) in (48), we obtain(49)S∑n=0∞ϑ¯nδ,t,r=r−1ereδ+sαS−Dδ4+2Dδ2∑n=0∞ϑ¯nδ,t,r−1−g∑n=0∞ϑ¯nδ,t,r−∑n=0∞Gn¯,S∑n=0∞ϑ¯nδ,t,r=1−rereδ+sαS−Dδ4+2Dδ2∑n=0∞ϑ¯nδ,t,r−1−g∑n=0∞ϑ¯nδ,t,r−∑n=0∞Gn¯.
Comparing terms on both sides, we get(50)Sϑ0¯δ,t,r=r−1ereδ,Sϑ0¯δ,t,r=1−rereδ,ϑ0¯δ,t,r=r−1ereδ,ϑ0¯δ,t,r=1−rereδ,Sϑ1¯δ,t,r=sαS−Dδ4+2Dδ2ϑ¯0δ,t,r−1−gϑ¯0δ,t,r−F0¯,Sϑ1¯δ,t,r=sαS−Dδ4+2Dδ2ϑ¯0δ,t,r−1−gϑ¯0δ,t,r−F0¯.
From (21), we get(51)F0¯=r−13e3re3δ,F0¯=1−r3e3re3δ,Sϑ1¯δ,t,r=sαS−Dδ4+2Dδ2r−1ereδ−1−gr−1ereδ−r−13e3re3δ,Sϑ1¯δ,t,r=sαS−Dδ4+2Dδ21−rereδ−1−g1−rereδ−1−r3e3re3δ,Sϑ1¯δ,t,r=sαS−r−1ereδ−2r−1ereδ−1−gr−1ereδ−r−13e3re3δ,Sϑ12δ,t,r¯=sαS−1−rereδ−21−rereδ−1−g1−rereδ−1−r3e3re3δ,Sϑ1¯δ,t,r=sαSr−1ereδ−3−1−g−r−12e2re2δ,Sϑ1δ,t,r¯=sαS1−rereδ−3−1−g−1−r2e2re2δ,ϑ1¯δ,t,r=tαΓα+1r−1ereδg−4−r−12e2re2δ,ϑ1¯δ,t,r=tαΓα+11−rereδg−4−1−r2e2re2δ.
The next term is(52)Sϑ2¯δ,t,r=sαS−Dδ4+2Dδ2ϑ¯1δ,t,r−1−gϑ¯1δ,t,r−G1¯,Sϑ2¯δ,t,r=sαS−Dδ4+2Dδ2ϑ¯1δ,t,r−1−gϑ¯1δ,t,r−G1¯.
From (21), we get(53)G1¯=3r−12e2re2δtαΓα+1r−1gereδ−r−13e3re3δ,G1¯=31−r2e2re2δtαΓα+11−rgereδ−1−r3e3re3δ.
After solving (52) and some simplification, we have(54)ϑ2¯δ,t,r=t2αΓ2α+12gr−1ereδ+8r−1ereδ−50r−13e3re3δ−g2r−1ereδ−4gr−13e3re3δ+3gr−15e5re5δ,ϑ2¯δ,t,r=t2αΓ2α+12g1−rereδ+81−rereδ−501−r3e3re3δ−g21−rereδ−4g1−r3e3re3δ+3g1−r5e5re5δ.
Similarly, we can find the other terms. The semianalytical solution of the equation is(55)ϑ¯δ,t,r=r−1ereδ1+tαΓα+1g−4−r−12e2re2δ+t2αΓ2α+12gr−1ereδ+8r−1ereδ−50r−13e3re3δ−g2r−1ereδ−4gr−13e3re3δ+3gr−15e5re5δ+⋯,ϑ¯δ,t,r=1−rereδ1+tαΓα+1g−4−1−r2e2re2δ+t2αΓ2α+12g1−rereδ+81−rereδ−5,01−r3e3re3δ−g21−rereδ−4g1−r3e3re3δ+3g1−r5e5re5δ+⋯.
We have simulated the obtained results in 2D and 3D graphs. Figure4 shows the lower and upper solution for α=0.4,0.6,0.8,1. Figure 5 represents the surface plot of the obtained solution of the proposed equation. As fractional order α increases, the solution curves tends to integer order solution, i.e., the graph at α=1 represents the solution at integer order. Also, the diameter of the fuzzy triangular number increases as fractional order increases. Thus, the fractional operator has great impact on the solution of the proposed equations. Figure 6 elaborates the evolution of the obtained results at fractional orders α=0.4,0.6,0.8 and various values of r∈0,1.Figure 4
2D representation of the solution of Example2 at various fractional orders, and uncertainty r∈0,1.Figure 5
3D representation of the solution of Example2 at various fractional orders, and uncertainty r∈0,1.Figure 6
Simulation of the solution of Example2 at various fractional orders, and uncertainty r∈0,1.


## 5. Conclusion

In this paper, we have considered the Swift–Hohenberg equation of fractional order in the fuzzy concept. We have studied this equation under the fuzzy Caputo fractional derivative. To find the semianalytical solution of the proposed nonlinear equation, we have used the fuzzy Sumudu transform method coupled with the Adomian decomposition method. To confirm the accuracy of the proposed procedure, we have provided two test problems. Lastly, the numerical results for various fractional orders and uncertainty in [0, 1] have been visualized through 2D and 3D graphs.



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