# Analysis of Fuzzy Kuramoto-Sivashinsky Equations under a Generalized Fuzzy Fractional Derivative Operator

Authors: Noufe H. Aljahdaly; Muhammad Naeem; Noorolhuda Wyal
Journal: Journal of Function Spaces
 (2022)

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## Abstract

This paper evaluates a semianalytical strategy combined with a novel fuzzy integral transformation and an iterative method inside the fuzziness concept known as the new iterative transform method. Additionally, we apply the abovementioned technique to the fractional fuzzy Kuramoto-Sivashinsky equations withgH-differentiability by employing various initial conditions. Numerous algebraic properties of the fuzzy fractional derivative Atangana-Baleanu operator are illustrated concerning the Shehu transformation to demonstrate their utility. Additionally, a general technique for Atangana-Baleanu fuzzy fractional derivatives is proposed in the sense of Caputo. It is important to note that the purpose of the suggested fuzziness technique is to establish the efficiency and accuracy of analytical solution to nonlinear fuzzy fractional partial differential equations that emerge in complex and physical structures.


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## Body

## 1. Introduction

Physical models of real-world phenomena generally involve considerable uncertainty due to a wide range of variables. Fuzzy models appears to be an effective strategy for modelling the uncertainty introduced by imprecision and ambiguity. We apply it here to disciplines where data is unclear, such as medicine, the environmental, social and economic sciences, and applied sciences. Zadeh contributed fuzziness to the set theory in 1965 to address these concerns. Over the last two centuries, fractional calculus has grown in popularity due to its numerous applications in applied sciences. Hundreds of fuzzy problems rather than stochastic uncertainty in the behavior of the described scheme process can be found. Numerous researchers have been interested in investigating the foundation theoretical fuzzy problems in recent years [1]. Fractional fuzzy differential equations (FFDE) are particularly valuable in scientific modelling and technical examples such as models of population, system of weapon evaluation, civil engineering, and electrohydraulic modelling. Fractional calculus, in conjunction with the theory of fuzzy implementation, is a major tool for resolving uncertainty and dealings with ambiguous or subjective status in Maths and providing more broadly findings. Handled with several real scenarios, for instance, the mean golden [2], medicine [3], gravity and quantum optics [4], systems of practical [5], and engineering phenomenon. Zadeh [6] became familiar with fuzzy sets for the first time. The work on the definition of fuzzy numbers and their applications in fuzzy control [7] and reasoning approximate issues [8, 9] was then completed. It is challenging to accurately depict numerous scenarios in data analysis utilizing real numbers. Furthermore, Dubois and Prade [10, 11], Mizumoto and Tanaka [12, 13], and the elementary principles on noncrisp sets, as well as details on fuzzy differentials and fuzzy differential equations, can be found in [14–16]. The ideas propose a generalization of differential equations. This is a novel idea that has piqued the interest of many researchers. Real-world settings involving fractional-order differential equations are extremely important; implementations can be found in chemistry, engineering, physics, and other fields [17–19].Fuzzy partial differential equations arose from the necessity to replicate many real-world difficulties while accounting for uncertainty data. As we will see later, partially differential equations have essential implications in several domain-applied sciences. Heat transfer is a critical topic of research in mechanic and aeronautical engineering because it affects a wide variety of equipment, mechanical systems, and aerospace engineering domains [20]. Numerous engineering problems, by definition, fall into this category, and scientists must approach them numerically; for more information, see [21, 22].Furthermore, it is difficult to find a correct solution for the nonlinear equations that illustrate these events but necessary for analyzing the relevant behaviors. In the cited phenomena, in order to describe the behavior of nonlinear systems in real life, analytical and numerical solutions for fractional differential equations are essential; plasma instabilities, chemical reaction-diffusion, flame front propagation, viscous flow issues, and magnetized plasmas are the physical and chemical phenomena described by the Kuramoto-Sivashinsky equation [23, 24]. Several researchers applied many methods to analyze the Kuramoto-Sivashinsky equation like, the homotopy analysis technique [25], finite-difference discretization [26], Lattice Boltzmann technique [27], Chebyshev spectral collocation technique [28], cubic B-spline finite difference-collocation technique [29], and He’s variational iteration technique [30, 31].The structure of the remaining paper is summarized as follows. Section2 defines some basic definitions related to our present work. The general methodology for solving the fractional fuzzy partial differential equation is provided in Section 3. Section 4 presents the main results, numerical simulations, and graphical representations. The conclusion along with future research directions is drawn in Section 5.


## 2. Preliminary Concepts

Definition 1.
We say thatU:R↦0,1 is a set of fuzzy; then, it is recognized to be a fuzzy set if it continues to hold the subsequent presumptions [32–36]:
(1)
U is normal (for some η0∈R;Uℓ0=1)(2)
U is upper semicontinuous(3)
Uℓ1ξ+1−ξℓ2≥Uℓ1∧Uℓ2∀ξ∈0,1,ℓ1,ℓ2∈R, i.e., U is convex(4)
clℓ∈R,Uℓ>0 is compactDefinition 2.
Assume that a fuzzy setU is the P level set discussed as [32–36]
(1)UP=Ψ∈R:UΨ≥1,where P∈0,1 and Ψ∈R.Definition 3.
The parameter version of a fuzzy set is signified asU¯P,U¯P so that P∈0,1 fulfils the successive presumptions [32–36]:
(1)
U¯P is right continuous, nonincreasing, bounded over 0,1, and right continuous at 0(2)
UP is left continuous, nondecreasing, bounded over 0,1, and left continuous at 0(3)
UP≤U¯PDefinition 4.
ForP∈0,1 and Y to be scalar, suppose that there are two fuzzy sets α~1=α1¯,α1¯,α~2=α2¯,α2¯; then the addition, subtraction, and scalar multiplication are stated as [32–35]
(1)
α~1⊕α~2=α1¯P+α2¯P,α1¯P+α2¯P(2)
α~1!α~2=α1¯P−α2¯P,α1¯P−α2¯P(3)
Yeα~1=Yα1¯,Yα1¯Y≥0,Yα1¯,Yα1¯Y<0Definition 5.
Supposing that a fuzzy modellingΘ:E~×E~↦R having fuzzy two sets α~1=α1¯,α1¯,α~2=α2¯,α2¯, then, the Θ distance between α~1 and α~2 is defined as [32–36]
(2)Θα~1,α~2=supP∈0,1maxα1¯P−α2¯P,α1¯P−α2¯P.Definition 6.
Recognize a fuzzy modellingΞ:R↦E~; if for any ε>0, there exists δ>0 and a fixed value of μ0∈a1,a2, we have [32–36]:
(3)ΘΞμ,Ξμ0<ε;wheneverμ−μ0<δ.
Then,Ξ is known to be continuous.Definition 7.
Suppose thatΞ:b1,b2↦E~ and τ0∈b1,b2. Then, Ξ is said to be strongly generalized differentiable at τ0 if Ξ′τ0∈E~ exists such that [32–35]
(i)
Ξ′τ0=limℏ↦0Ξτ0+ℏ!gHΞτ0/h=limℏ↦0Ξτ0!gHΞτ0−ℏ/h(ii)
Ξ′τ0=limℏ↦0Ξτ0!gHΞτ0+ℏ/−ℏ=limℏ↦0Ξτ0−ℏ!gHΞτ0/−ℏTheorem 8.
Consider a fuzzy value termΞ:R↦E¯ such that Ξτ0;P=Ξτ0;P,Ξ¯¯τ0;P and P∈0,1. Then,
(I)
Ξτ0;P and Ξτ0;P are differentiable, if Ξ is (1)-differentiable, and
(4)Ξ′τ0P=Ξ¯¯′τ0;P,Ξ¯′τ0;P(II)
Ξτ0;P and Ξ¯τ0;P are differentiable, if Ξ is (2)-differentiable, and
(5)Ξ′τ0P=Ξ¯′τ0;P,Ξ′τ0;PDefinition 9.
Assume that a fuzzy mappingΨgHr=Ψr∈ℂF0,s∩LF0,s. Then, the fuzzy gH-fractional Caputo differentiability of fuzzy valued mapping Ψ is defined as
(6)DgHcθΨÿ=Ja1r−θ⨀Ψrτ=1Γr−ϑ⨀∫a1ÿÿ−ℓr−θ−1⨀Ψrℓdℓ,θ∈r−1,r,r∈ℕ,ÿ>a1.
Therefore, the parameterized versions ofΨ=Ψ¯Pÿ,Ψ¯Pÿ,P∈0,1 and ÿ0∈0,s and CFD in a fuzzy sense is stated as
(7)Di−gHϑΨÿ0P=Di−gHϑΨÿ0,Di−gHϑΨ¯ÿ0,P∈0,1,where r=P:
(8)Di−gHϑΨ¯ÿ0=1Γr−ϑ∫0ÿÿ−xr−ϑ−1drdxrΨ¯i−gHxdxÿ=ÿ0′,Di−gHϑΨ¯ÿ0=1Γr−θ∫0ÿÿ−xr−θ−1drdxrΨ¯i−gHxdxÿ=ÿ0.Definition 10.
Assume that a fuzzy mappingΨ~ÿ∈ℍ~10,T and ϑ∈0,1; then, the fuzzy gH-fractional Atangana-Baleanu differentiability of fuzzy-valued mapping is defined as
(9)gHϑΨÿ=Bϑ1−ϑ⨀∫0ÿΨ¯′x⨀Eθ−ϑÿ−xθ1−ϑdx.Remark 11.
In (33), Ψ satisfy the presumption of the decreasing diameter Ψ and increasing diameter Ψ of a fuzzy modelling Ψ. If v=1, then, the fuzzy Shehu transformation is simplified to fuzzy Laplace transformation. Applying the evidence of Salahshour et al. [30], we get
(10)∫0+∞exp−ζvÿ⨀Ψ¯ÿdÿ=∫0+∞exp−ξvÿΨÿ;Pdÿ,∫0+∞exp−ξvÿΨ¯ÿ;Pdÿ.
Moreover, when we consider the classic Shehu transformation, we get(11)SΨ¯ÿ;P=∫0+∞exp−ξvÿΨÿ;Pdÿ,and
(12)SΨ¯ÿ;P=∫0+∞exp−ζvÿΨ¯ÿ;Pdÿ.
The aforementioned ideas can then be expressed as(13)SΨ¯ÿ=SΨ¯ÿ;P,SΨ¯ÿ;P=S¯ζ¯,v,S¯ζ¯,v.
The fuzzy Shehu transformation of the Caputo generalized Hukuhara derivative will then be definedgcHθDÿθΨÿ.Definition 12.
Suppose that there is an integrable fuzzy-valued mappingDgHcÿθΨ¯ÿ and Ψÿ is the primitive of HgcÿθΨ¯ÿ on 0,+∞; then, the CFD of order θ is defined as
(14)SHgcDÿθΨ¯ÿ=ζ¯Vθ⨀SΨ¯ÿ!∑x=0r−1ξ¯Vθ−κ−1⨀Ψ¯κ0,θ∈r−1,r.
Again, using the fact of Maitama et al. [35], we have
(15)ζ¯Vθ⨀SΨ¯ÿ!∑k=0r−1ξVθ−κ−1⨀f¯κ0=ξVθSΨ¯ÿ;P−∑k=0r−1ξVθ−κ−1⨀Ψ¯κ0;P,ξVθSΨ¯ÿ;P−∑κ=0r−1ξVθ−κ−1Ψ¯κ0;P.The fractional derivative of the ABC operator was defined in the sense of Shehu by Bokhari et al. Furthermore, in the context of a fuzzy Shehu transform, we extend the concept of the fuzzyABC fractional derivative as follows:Definition 13.
ConsiderΨ∈ℂF0,s∩LF0,s such that Ψ~ÿ=Ψ¯ÿ,P,Ψ¯ÿ,P, P∈0,1; then, the Shehu transformation of the fuzzy ABC of order θ∈0,1 is describe as follows:
(16)SgHDÿθΨ¯ÿ=Bϑ1−θ+θv/ξθ⨀V¯v,ξ!vξΨ¯0.
Moreover, utilizing the evidence of Porshokouhi and Ghanbari [30], we get
(17)Bθ1−θ+θv/ξθ⨀V¯v,ξ¯!ξvΨ~0=Bθ1−θ+ϑv/ξV¯v,ξ¯;P−vξΨ¯0;P,Bϑ1−θ+θv/ξθ∇v,ξ¯;P−vξΨ0;P.


## 3. General Implementation of the Proposed Method

In this section, we investigated the proposed method for an analytical solution. For this, we implemented the Shehu transform of the fractional Atangana-Baleanu operator along with the iterative transformation technique as(18)SABCD℘ςU~ψ,℘=SDψ2U~ψ,℘+Dψ3U~ψ,℘+k~rFψ,℘,where ς∈0,1; therefore, the Shehu transformation of (18) is
(19)Bς1−ς+ςσ/ωςSU~ψ,℘−Bς1−ς+ςσ/ωςvωU~ψ,0=SDψ2U~ψ,℘+Dψ3U~ψ,℘+k~rFψ,℘,using the initial condition, we obtain as
(20)SU~ψ,℘=gψω+1−ς+ςσ/ωςBςSDψ2U~ψ,℘+Dψ3U~ψ,℘+k~rFψ,℘.Decompose the result asU~ψ,℘=∑n=0∞U~nψ,℘; then, (20) apply that
(21)S∑n=0∞U~nψ,℘=gψω+1−ς+ςσ/ωςBςSDψ2∑n=0∞U~nψ,℘+Dψ3∑n=0∞U~nψ,℘+k~rFψ,℘.Aspects of the outcome may be seen as comparisons(22)SU~0ψ,℘=gψω+1−ς+ςσ/ωςBςSk~rFψ,℘,SU~1ψ,℘=1−ς+ςσ/ωςBςSDψ2U~0ψ,℘+Dψ3U~0ψ,℘,SU~2ψ,℘=1−ς+ςσ/ωςBςSDψ2U~1ψ,℘+Dψ3U~1ψ,℘,⋮SU~n+1ψ,℘=1−ς+ςσ/ωςBςSDψ2U~nψ,℘+Dψ3U~nψ,℘.Applying the Shehu inverse transformation, we achieve(23)U¯0ψ,℘=gψ+S−11−ς+ςσ/ωςBςSk¯rFψ,℘,U¯0ψ,℘=gψ+S−11−ς+ςσ/ωςBςSk¯rFψ,℘,U¯1ψ,℘=S−11−ς+ςσ/ωςBςSDψ2U¯0ψ,℘+Dψ3U¯0ψ,℘,U¯1ψ,℘=S−11−ς+ςσ/ωςBςSDψ2U¯0ψ,℘+Dψ3U¯0ψ,℘,U¯2ψ,℘=S−11−ς+ςσ/ωςBςSDψ2U¯1ψ,℘+Dψ3U¯1ψ,℘,U¯2ψ,℘=S−11−ς+ςσ/ωςBςSDψ2U¯1ψ,℘+Dψ3U¯1ψ,℘,⋮U¯n+1ψ,℘=S−11−ς+ςσ/ωςBςSDψ2U¯nψ,℘+Dψ3U¯nψ,℘,U¯n+1ψ,℘=S−11−ς+ςσ/ωςBςSDψ2U¯nψ,℘+Dψ3U¯nψ,℘.Thus, the result becomes(24)U¯ψ,℘=U¯0ψ,℘+U¯1ψ,℘+U¯2ψ,℘+⋯,U¯ψ,℘=U¯0ψ,℘+U¯1ψ,℘+U¯2ψ,℘+⋯.The solution to equation (24) in the series form is obtained as follows.


## 4. Numerical Problems

Example 14.
Consider the fuzzy fractional Kuramoto-Sivashinsky equation defined asτ=σ=1 and ω=4:
(25)ABCD℘ςU~ψ,℘+U~∂U~∂ψ+τ∂2U~∂ψ2+σ∂3U~∂ψ3+ω∂4U~∂ψ4=0,with the initial condition
(26)U~0ψ,0=k~11+15tanh−12ψ−15tanh2−12ψ−15tanh3−12ψ.
Using the scheme of equation (23), we obtain
(27)U¯0ψ,℘=k¯r11+15tanh−12ψ−15tanh2−12ψ−15tanh3−12ψ,U¯0ψ,℘=k¯r11+15tanh−12ψ−15tanh2−12ψ−15tanh3−12ψ,U¯1ψ,℘=−k¯r60−2+coshψ−sinhψcosh2ψ+2coshψ+11Bςς℘ςΓς+1+1−ς,U¯1ψ,℘=−k¯r60−2+coshψ−sinhψcosh2ψ+2coshψ+11Bςς℘ςΓς+1+1−ς,U¯2ψ,℘=k¯r60cosh2ψ−coshψ−sinhψcoshψ−2+5sinhψcosh3ψ+3cosh2ψ+3coshψ+11B2ςς2℘2ςΓ2ς+1+2ς1−ς℘ςΓς+1+1−ς2,U¯2ψ,℘=k¯r60cosh2ψ−coshψ−sinhψcoshψ−2+5sinhψcosh3ψ+3cosh2ψ+3coshψ+11B2ςς2℘2ςΓ2ς+1+2ς1−ς℘ςΓς+1+1−ς2,U¯3ψ,℘=k¯rcosh2ψ−13coshψ−sinhψcoshψ+16+5sinhψcosh3ψ+3cosh2ψ+3coshψ+11B3ςς3℘3θΓ3ς+1+3ς21−ς℘2ςΓ2ς+1+3ς1−ς2℘θΓς+1,U¯3ψ,℘=k¯rcosh2ψ−13coshψ−sinhψcoshψ+16+5sinhψcosh3ψ+3cosh2ψ+3coshψ+11B3ςς3℘3θΓ3ς+1+3ς21−ς℘2ςΓ2ς+1+3ς1−ς2℘θΓς+1.
The series solution is obtained using equation (24); therefore, we write
(28)U~ψ,℘=U~0ψ,℘+U~1ψ,℘+U~2ψ,℘+U~3ψ,℘+U~4ψ,℘+⋯.
The upper and bottom portion forms can be written as(29)U¯ψ,℘=U¯0ψ,℘+U¯1ψ,℘+U¯2ψ,℘+U¯3ψ,℘+U¯4ψ,℘+⋯,U¯ψ,℘=U¯0ψ,℘+U¯1ψ,℘+U¯2ψ,℘+U¯3ψ,℘+U¯4ψ,℘+⋯.U¯ψ,℘=k¯r11+15tanh−12ψ−15tanh2−12ψ−15tanh3−12ψ−k¯r60−2+coshψ−sinhψcosh2ψ+2coshψ+11Bςς℘ςΓς+1+1−ς+k¯r60cosh2ψ−coshψ−sinhψcoshψ−2+5sinhψcosh3ψ+3cosh2ψ+3coshψ+11Bςς℘ςΓς+1+1−ς+k¯r60cosh2ψ−coshψ−sinhψcoshψ−2+5sinhψcosh3ψ+3cosh2ψ+3coshψ+11B2ςς2℘2ςΓ2ς+1+2ς1−ς℘ςΓς+1+1−ς2+k¯rcosh2ψ−13coshψ−sinhψcoshψ+16+5sinhψcosh3ψ+3cosh2ψ+3coshψ+11B3ςς3℘3θΓ3ς+1+3ς21−ς℘2ςΓ2ς+1+3ς1−ς2℘θΓς+1+⋯,U¯ψ,℘=k¯r11+15tanh−12ψ−15tanh2−12ψ−15tanh3−12ψ−k¯r60−2+coshψ−sinhψcosh2ψ+2coshψ+11Bςς℘ςΓς+1+1−ς+k¯r60cosh2ψ−coshψ−sinhψcoshψ−2+5sinhψcosh3ψ+3cosh2ψ+3coshψ+11B2ςς2℘2ςΓ2ς+1+2ς1−ς℘ςΓς+1+1−ς2+k¯rcosh2ψ−13coshψ−sinhψcoshψ+16+5sinhψcosh3ψ+3cosh2ψ+3coshψ+11B3ςς3℘3θΓ3ς+1+3ς21−ς℘2ςΓ2ς+1+3ς1−ς2℘ςΓς+1+⋯.
The exact solution of equation (25) is given as
(30)U~ψ,℘=k~11+15tanh−12ψ+℘−15tanh2−12ψ+℘−15tanh3−12ψ+℘.Figure1 represents the three-dimensional upper and lower fuzzy branch plots of the approximate series result and the different fractional order of ς. Figure 2 represents the two-dimensional upper and lower fuzzy branch plots of the approximate series result and the different fractional order of ς.Example 15.
Consider the fuzzy fractional Kuramoto-Sivashinsky equation defined asτ=2, σ=1, and ω=0:
(31)ABCD℘ςU~ψ,℘+U~∂U~∂ψ+τ∂2U~∂ψ2+σ∂3U~∂ψ3+ω∂4U~∂ψ4=0,with the initial condition
(32)U~0ψ,0=k~−41811−270361418tanh41838ψ+330361418tanh341838ψ.
Using the scheme of equation (23), we obtain
(33)U¯0ψ,℘=k¯r−41811−270361418tanh41838ψ+330361418tanh341838ψ,U¯0ψ,℘=k¯r−41811−270361418tanh41838ψ+330361418tanh341838ψ,U¯1ψ,℘=k¯r−90418−152cosh341838ψ+209cosh41838ψ+88sinh41838ψcosh241838ψ−242sinh41838ψ16859cosh5418/38ψ1Bςς℘ςΓς+1+1−ς,U¯1ψ,℘=k¯r−90418−152cosh341838ψ+209cosh41838ψ+88sinh41838ψcosh241838ψ−242sinh41838ψ16859cosh5418/38ψ1Bςς℘ςΓς+1+1−ς,U¯2ψ,℘=k¯r180418288574sinh41838ψcosh441838ψ−2495625sinh41838ψcosh241838ψ+2635380sinh41838ψ+444752cosh541838ψ−436810cosh341838ψ−63536cosh741838ψ+73264sinh41838ψcosh641838ψ12476099cosh9418/38ψ1B2ςς2℘2ςΓ2ς+1+2ς1−ς℘ςΓς+1+1−ς2,U¯2ψ,℘=k¯r180418288574sinh41838ψcosh441838ψ−2495625sinh41838ψcosh241838ψ+2635380sinh41838ψ+444752cosh541838ψ−436810cosh341838ψ−63536cosh741838ψ+73264sinh41838ψcosh641838ψ12476099cosh9418/38ψ1B2ςς2℘2ςΓ2ς+1+2ς1−ς℘ςΓς+1+1−ς2.
The series solution is obtained using equation (24); therefore, we write
(34)U~ψ,℘=U~0ψ,℘+U~1ψ,℘+U~2ψ,℘+U~3ψ,℘+U~4ψ,℘+⋯.
The upper and bottom portion forms can be written as(35)U¯ψ,℘=U¯0ψ,℘+U¯1ψ,℘+U¯2ψ,℘+U¯3ψ,℘+U¯4ψ,℘+⋯,U¯ψ,℘=U¯0ψ,℘+U¯1ψ,℘+U¯2ψ,℘+U¯3ψ,℘+U¯4ψ,℘+⋯.U¯ψ,℘=k¯r−41811−270361418tanh41838ψ+330361418tanh341838ψ+k¯r−90418−152cosh341838ψ+209cosh41838ψ+88sinh41838ψcosh241838ψ−242sinh41838ψ16859cosh5418/38ψ1Bςς℘ςΓς+1+1−ς+k¯r180418288574sinh41838ψcosh441838ψ−2495625sinh41838ψcosh241838ψ+2635380sinh41838ψ+444752cosh541838ψ−436810cosh341838ψ−63536cosh741838ψ+73264sinh41838ψcosh641838ψ12476099cosh9418/38ψ1B2ςς2℘2ςΓ2ς+1+2ς1−ς℘ςΓς+1+1−ς2+⋯,U¯ψ,℘=k¯r−41811−270361418tanh41838ψ+330361418tanh341838ψ+k¯r−90418−152cosh341838ψ+209cosh41838ψ+88sinh41838ψcosh241838ψ−242sinh41838ψ16859cosh5418/38ψ1Bςς℘ςΓς+1+1−ς+k¯r180418288574sinh41838ψcosh441838ψ−2495625sinh41838ψcosh241838ψ+2635380sinh41838ψ+444752cosh541838ψ−436810cosh341838ψ−63536cosh741838ψ+73264sinh41838ψcosh641838ψ12476099cosh9418/38ψ1B2ςς2℘2ςΓ2ς+1+2ς1−ς℘ςΓς+1+1−ς2+⋯.
The exact result of equation (31) is given as
(36)U~ψ,℘=k~−1ρ+6019ρ−38θρ2+℘tanhμ+120θρ3tanh3μ.
Thenμ=ρψ+℘ and ρ=0.522/19, on the interval [−1, 1].Figure 1
(a) 3D upper and lower fuzzy branch plots of the approximate series result and (b) the different fractional order ofς.Figure 2
(a) 2D upper and lower fuzzy branch plots of the approximate series result and (b) the different fractional order ofς.Figure3 represents the two-dimensional upper and lower fuzzy branch plots of the approximate series result and different fractional order of ς. Figure 4 represents the two-dimensional upper and lower fuzzy branch plots with respect to time.Figure 3
(a) 2D upper and lower fuzzy branch plots of the approximate series result and (b) the different fractional order ofς.Figure 4
(a) 2D upper and lower fuzzy branch plots with respect to time and (b) different fractional-order ofς.


## 5. Conclusion

Finally, we suggest that an efficient method for computing semianalytical or approximate solutions to two-dimensional fuzzy fractional-order Kuramoto-Sivashinsky equations as external terms has been implemented. The techniques of the novel iterative transform approach were used to analyze a semianalytical result in the series form solution. These techniques are applicable to a variety of fractional-order issues. Additionally, we validated our generic solution strategies using three specific examples. All two models were studied using the aforementioned procedures, with the fuzzy number multiplied at two different fractional orders to account for the uncertainty in the beginning values. The reported findings for both scenarios were simulated using two distinct graph formats: two dimensions and three dimensions. The observed solutions were also achieved in the uncertainty form, using fuzzy solutions with upper and lower branches. Additionally, we can take alternative fractional-order values and compare them to the integer order. We can converge to the values of integer order 1 by raising the order ofq. In the future, this approach may be applied to a variety of partial differential equations with series-type solutions in both integer and noninteger orders. Additionally, the adopted methodologies can be employed as a robust tool for studying nonlinear and linear dynamical systems of fractional order.



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