New Configurations of the Fuzzy Fractional Differential Boussinesq Model with Application in Ocean Engineering and Their Analysis in Statistical Theory

1  Introduction

Numerous multidimensional algorithms have previously been efficaciously constructed to study physical phenomena. The Korteweg-de Vries (KdV) equation, for example, is primarily utilized to study the transmission of waves that propagate in one region of space, whilst the Boussinesq equation (BSQe) characterizes physical phenomena in multiple directions. The BSQe is furthermore applied to investigate various mechanisms that occur in magnetic fields in electrode materials [1], magnetosound vibrations in fluid [2] and ferroelectric vibrations in superparamagnetic [3]. Since the 1970s, these formulas and numerous others have piqued the interest of a huge proportion of theoretical physicists and cosmologists due to their usage in discovering the intrinsic structures of some intricate physical processes. As a result, various researchers have innovated and revealed a diversity of innovative strategies for creating numerical solutions for the majority of nonlinear PDEs. The finite difference method [4], the iterative process [5], the sine-cosine principle [6], the Weierstrass elliptic function approach [7], the tanh-sech technique [8], the F-expansion technique [9] and Hirota’s bilinear principle [10] are certain formulated instruction in ways and fundamentals. Other methodologies’ documentation can all be found in [11–13]. In this research, we will apply the fuzzified Aboodh transform in connection with the Adomian decomposition method to procure analytical findings of fourth-order time-FBSQe. In 1978, Bear [14] defined the conventional BSQe shown as:

∂∂u(QuΥ∂Υ∂u)+∂∂ξ(QωΥ∂Υ∂ξ)+W=U∂Υ∂y,(1)

where Qu signifies swamped flowability in the u manner (W/T); The concentrated permeability in the ξ way is denoted by Qξ, (W/T); Υ denotes the hydraulic gradient (S); U denotes the accurate production (dimensionless); and P indicates the recharge/discharge rate (W/T).

When calculating the FBSQe, the following hypothesis are considered:

  i.   Assume that the Dupuit-Forchhimer assumptions, along with Darcy’s law, are valid.

 ii.   The liquid (water) is non-compressible in the constant volume.

iii.   A power-law component governs the flux perturbations in the monitoring volume.

The (1) estimation was created as conducting an investigation into acoustic waves in tropical waters. This was later improved to address purification filtration issues in highly permeable groundwater structures. In addition, (1) is commonly used in oceanic and seaside renovation to rectify seawater desalination challenges in nanostructured subsurface structures. BSQe also has the potential to develop a procedure for an assortment of configurations that communicate with shallow aquifers’ liquidity and irrigation channels [15–17]. To calculate the FBSQe, the researchers [18] considered the index-law variability of fluid flow in detection capacity and fractional Taylor encapsulates. Studying BSQe [19] intermittent, nonlinear and concussive flow remedies were studied using fractional variational methods. Zhuang et al. [20] implemented two unique supercomputing methodologies for FBSQe: finite volume and finite element strategies relying on bifurcation-facilitating capabilities. Abassy et al. [21] used an re-configured VIM to generate an FBSQe. Wu et al. [22] optimised transfer function BSQe strategies for a concentration of underground water with interactively changeable floods. Mainstream calculus, however, is incompetent at addressing a situation like this. As a consequence, naturalistic explanations are needed to identify this issue. One of the most influential methodologies for attempting to articulate this situation is fuzzy set theory [23].

The researchers presented an innovative notion of fractional integral and derivative, known as the Atangana-Baleanu-Caputo fractional derivative (ABCFD) and integral, in [24], which extrapolates the Riemann-Liouville and Caputo integral and derivative into a standard pattern. The researchers suggested a discrete variant of the ABCFD and achieved a quantitative methodology for solving a linear FDE with ABCFD in [25,26]. The authors [27] looked into the chaotic behaviour and stabilization effects of FDEs within the ABCFD operator. In recent times, fuzzy interpretation and fuzzified DEs have been envisioned to tackle the unpredictability resulting from inadequate data in several quantitative or mathematical simulations of such predetermined real-world manifestations. So far, this hypothesis has also been established and a wide range of implementations have been considered in [28–30] and the descriptions herein. The notion of fuzzified sort Riemann-Liouville differentiability predicated on Hukuhara differentiability (HD) was introduced in [31], and the researchers developed the presence of certain fuzzified iterative methods utilizing adequate compressibility type prerequisites utilizing the Hausdorff estimate of non-compactness. Numerous techniques and strategies premised on HD or generalized HD (see [32]) were then recognized in a variety of studies in the literature [33,34], and we are currently analyzing several of these findings momentarily. Allahviranloo et al. [35] helped in providing an expressive approximate solutions to the fuzzified conic equations. The suggested system’s authenticity and perseverance were evaluated in attempt to show that it was intrinsically rigorous. Arqub et al. [36] developed the fuzzified FDE by incorporating the non-singular component into the differential implementation of the AB operator. Zhao et al. [37] considered a fuzzied-based method for dealing with the new therapeutic Corona-virus epidemic.

The Adomian decomposition method (ADM) is an operational approach for interacting with nonlinear functions that emerge in scientific domains; Adomian [38] became the best to introduce it. The outcome is interpreted as the cumulative of an infinite series that inevitably culminates in the intended correctness. Because this technique is precise and convenient, it does not require the use of an irreducible matrix, composite multivariable calculus, or infinite series viewpoints. There are no negative instances associated with this methodology. This approach has been utilized by many researchers [39,40].

Because of the foregoing tendency, figuring out the exact-approximate findings of fuzzified fractional PDEs is a complicated process. The objective of this scientific work was to establish a reliable technique for obtaining predicted values for a fuzzified FBSQe, such that a generalized BSQe with propagation components that are imprecise in initial conditions (ICs) due to the AADM, which designs the progression under consideration, so the AADM and the Aboodh transform are closely correlated and the ADM is referred to as the fuzzified AADM. The FBSQe has been investigated employing novel methodology. Furthermore, the truly innovative fractional derivative notion makes it easier to evaluate fuzzy FDEs rather than examining specific challenges with CFD and ABCFD operators. In addition, introducing the computational underpinnings for the fuzzy CFD and ABC fractional HD research findings necessitates a fuzzy derivative of fractional-order. In summary, we presented the comparison techniques for the aforementioned fractional derivatives using the analysis of variance technique. Illustrative findings suggest that both the techniques are reliable when fuzzy set theory is involved.

The organization of this paper is to clarify the influence of FBSQe as follows: Section 2 demonstrates the preliminary concepts of fractional calculus and fuzzy set theory. Section 3 elaborates on the roadmap of the semi-analytical technique in the convolution of fuzzy set theory, the Adomian decomposition method, and the Aboodh transform. Section 4 presents the test examples of the fourth-order FBSQe in R~,R~n and 2ndth-order in R~ with their graphical illustrations and physical interpretations. Statistical analysis also shows the efficacy of the proposed technique. Section 5 presents the concluding remarks based on experimental and numerical results.

2  Preliminaries

This section will attempt to make some basic observations about the Aboodh transform, in addition to certain key features pertaining to fuzzy set and fractional calculus theory. We refer to [24,26] for further documentation.

Definition 2.1. ([41,42]) A fuzzy set Ξ: R~↦[0,1] is said to be the fuzzy number (Fn), if the following suppositions are true:

1. Ξ is normal (for some ϰ0∈R~;Ξ(u0)=1),

2. Ξ is upper semi continuous,

3. Ξ(u1Υ+(1−Υ)u2)≥(Ξ(u1)∧Ξ(u2))∀Υ∈[0,1],u1,u2∈R~, i.e., Ξ is convex;

4. cl{u∈R~,Ξ(u)>0} is compact.

Definition 2.2. ([41]) A Fn Ξ is said to be the σ-level set presented as follows:

[Ξ]σ={Υ∈R~: Ξ(Υ)≥1},(2)

where σ∈[0,1] and Υ∈R~.

Definition 2.3. ([41]) A parametric representation of an Fn is denoted by [Ξ_(σ),Ξ¯(σ)] such that σ∈[0,1] if the following suppositions are true:

1. Ξ_(σ) is non-decreasing, left continuous, bounded over (0,1] and left continuous at 0.

2. Ξ¯(σ) is non-increasing, right continuous, bounded over (0,1] and right continuous at 0.

3. Ξ_(σ)≤Ξ¯(σ).

Also, σ is defined to be crisp number if Ξ_(σ)=Ξ¯(σ)=σ.

Definition 2.4. ([43]) For σ∈[0,1] and a scalar Υ. Assume that for two Fns ϑ1~=(ϑ1_,ϑ1¯),ϑ2~=(ϑ2_,ϑ2¯), then the algebraic properties are presented as

1. ϑ1~⊕ϑ2~=(ϑ1_(σ)+ϑ2_(σ),ϑ1¯(σ)⊕ϑ2¯(σ)),

2. ϑ1~⊖ϑ2~=(ϑ1_(σ)−ϑ2_(σ),ϑ1¯(σ)−ϑ2¯(σ)),

3. Υ⊙ϑ1~={(Υϑ1_,Υϑ1¯)Υ≥0,(Υϑ1¯,Υϑ1_)Υ<0.

Definition 2.5. ([44]) Suppose a fuzzy function Υ: E~×E~↦R~ containing two fuzzy numbers ϑ1~=(ϑ1_,ϑ1¯), ϑ2~=(ϑ2_,ϑ2¯), then Υ-distance between ϑ1~ and ϑ2~ is stated as

Υ(ϑ1~,ϑ2~)=supσ∈[0,1][max{|ϑ1_(σ)−ϑ2_(σ)|,|ϑ1¯(σ)−ϑ2¯(σ)|}].(3)

Definition 2.6. ([44]) Assume that a fuzzy function Υ: R~↦E~, if for any ε>0 ∃ δ>0 and the fixed value of u0∈[a1,a2], we have

Υ(w(u),w(u0))<ε;whenever |u−u0|<δ,(4)

then Υ is said to be continuous.

Definition 2.7. ([45]) Let δ1,δ2∈E~, if δ3∈E~ and δ1=δ2+δ3. The H-difference δ3 of δ1 and δ2 is presented as δ1⊖Hδ2. Clearly, δ1⊖Hδ2≠δ1+(−1)δ2.

Definition 2.8. ([45]) Consider Υ:(b1,b2)↦E~ and ε0∈(b1,b2). Then, Υ is known as strongly generalized differentiable at ε0 if Υ′(ε0)∈E~ exists such that

(i) Υ′(ε0)=limσ↦0Υ(ε0+σ)⊖gHΥ(ε0)σ=limσ↦0Υ(ε0)⊖gHΥ(ε0−σ)σ,

(ii) Υ′(ε0)=limσ↦0Υ(ε0)⊖gHΥ(ε0+σ)−σ=limσ↦0Υ(ε0−σ)⊖gHΥ(ε0)−σ.

All through this inquiry, we will employ the representation Υ is (1)-differentiable and (2)-differentiable, respectively, if it is differentiable under the supposition (i) and (ii) stated in the aforesaid notion.

Theorem 2.9. ([43]) Surmise that a fuzzy valued mapping Υ: R~↦E~ such that Υ(ε0;σ)=[Υ_(ε0;σ),Υ¯(ε0;σ)] and σ∈[0,1]. Then

I. Υ_(ε0;σ) and Υ¯(ε0;σ) are differentiable, if Υ is a (1)-differentiable, and

[Υ′(ε0)]σ=[Υ_′(ε0;σ),Υ¯′(ε0;σ)].(5)

II. Υ_(ε0;σ) and Υ¯(ε0;σ) are differentiable, if Υ is a (2)-differentiable, and

[Υ′(ε0)]σ=[Υ¯′(ε0;σ),Υ_′(ε0;σ)].(6)

Definition 2.10. ([44]) Surmise that a fuzzy mapping ΥgH(r)=Υ(r)∈CF[0,p]⋂LF[0,p]. Then, fuzzy gH-fractional Caputo differentiability of fuzzy-valued mapping Υ is stated as

(gHcDβΥ)(ζ)=𝒥a1r−β⊙(Υ(r))(ε)=1Γ(r−β)⊙∫a1ζ(ζ−u)r−β−1⊙Υ(r)(u)du,β∈(r−1,r],r∈N,ζ>a1.(7)

Thus, the parametric formulations of Υ=[Υ_σ(ζ),Υ¯σ(ζ)], σ∈[0,1] and ζ0∈(0,p), then fuzzified CFD is defined as

[D(i)−gHβΥ(ζ0)]σ=[D(i)−gHβΥ_(ζ0),D(i)−gHβΥ¯(ζ0)], σ∈[0,1],(8)

where r=[σ].

[D(i)−gHβΥ_(ζ0)]=1Γ(r−β)[∫0ζ(ζ−u)r−β−1drdurΥ_(i)−gH(u)du]ζ=ζ0,[D(i)−gHβΥ¯(ζ0)]=1Γ(r−β)[∫0ζ(ζ−u)r−β−1drdurΥ¯(i)−gH(u)du]ζ=ζ0.(9)

Definition 2.11. Consider a fuzzy function Υ~(ζ)∈H~1(0,T) and β∈[0,1], then fuzzy gH-fractional Atangana-Baleanu differentiabilty of fuzzy-valued mapping is defined as

(gHDβΥ)(ζ)=ABC(β)1−β⊙[∫0ζΥ_′(u)⊙Eβ[−β(ζ−u)β1−β]du].(10)

Therefore, the parametric version of Υ=[Υ_σ(ζ),Υ¯σ(ζ)], σ∈[0,1] and ζ0∈(0,p), then the fuzzy Atangana-Baleanu derivative in the Caputo context is described as

[ABCD(i)−gHβΥ~(ζ0;σ)]=[ABCD(i)−gHβΥ_(ζ0;σ),ABCD(i)−gHβΥ¯(ζ0;σ)],σ∈[0,1],(11)

where

ABCD(i)−gHβΥ_(ζ0;σ)=ABC(β)1−β[∫0ζΥ_(i)−gH′(u)Eβ[−β(ζ−u)β1−β]du]ζ=ζ0,ABCD(i)−gHβΥ¯(ζ0;σ)=ABC(β)1−β[∫0ζΥ¯(i)−gH′(u)Eβ[−β(ζ−u)β1−β]du]ζ=ζ0,(12)

where the normalized function is denoted by ABC(β) and ABC(0)=ABC(1)=1. Furthermore, type (i)−gH exists. Consequently, there is no supposition to let (ii)−gH differentiability.

Initially, authors [46] described the novel transform. This notion is extended to fuzzy set analysis.

Definition 2.12. Aboodh transform for mapping Υ(ζ) of exponential order over the set of mappings is stated as

𝒜={Υ: |Υ(ζ)|<Mexp⁡(κ)ι|ζ|, if ζ∈(−1)ℓ×[0,∞),ℓ=1,2; (M,κ1,κ1>0)},(13)

as

A[Υ~(ζ),ρ]=H(ρ)=1ρ∫0+∞exp⁡(−ρζ)⊙Υ~(ζ)dζ,ζ≤0,ρ∈[κ1,κ2].(14)

In (14), Υ~ satisfied the supposition of the nonincreasing Υ_ and nondecreasing diameter Υ¯, respectively, of a fuzzy function Υ.

In view of Salahshour et al. [47], we have

1ρ∫0+∞exp⁡(−ρζ)⊙Υ~(ζ)dζ=(1ρ∫0+∞exp⁡(−ρζ)Υ_(ζ)dζ,1ρ∫0+∞exp⁡(−ρζ)Υ¯(ζ)dζ).(15)

Furthermore, taking into account the classical Aboodh transform [46], we find

A[Υ_(ζ;σ)]=1ρ∫0+∞exp⁡(−ρζ)Υ_(ζ;σ)dζ(16)

and

A[Υ¯(ζ;σ)]=1ρ∫0+∞exp⁡(−ρζ)Υ¯(ζ;σ)dζ.(17)

The aforementioned representations can then be documented as

A[Υ~(ζ)]=(A[Υ_(ζ;σ)],A[Υ¯(ζ;σ)])=(𝒜_(ρ),𝒜¯(ρ)).(18)

In the context of the Aboodh transform, Awuya et al. [48] suggested the ABCFD formulation. Besides that, we apply the concept of fuzzified ABCFD in the context of a fuzzified Aboodh transform as shown below:

Definition 2.13. Suppose that Y(ρ) is the aboodh transform of Υ(ζ)∈C and Υ(ω) is te Laplace transform of Υ(ζ)∈C, then the Aboodh transform of ABCFD is obtained as follows:

A[gℋABC𝒟ζβΥ~(ζ)]=φβABC(β)β+(1−β)φβ(ρ)⊙(Υ~(ρ)⊖φ−2⊙Υ~(0)).(19)

Also, implying the idea of Salahshour et al. [47], we have

φβABC(β)β+(1−β)φβ(ρ)⊙(Υ~(ρ)⊖φ−2⊙Υ~(0))=(φβABC(β)β+(1−β)φβ(ρ)(Υ_(ρ)−φ−2Υ_(0)),φβABC(β)β+(1−β)φβ(ρ)(Υ¯(ρ)−φ−2Υ¯(0))).(20)

3  Configuration of Semi-Analytical Scheme in Fuzzy the Sense

The following is an investigation of an iterative mechanism for accumulating numerical solutions to the one-dimensional FBSQe employing the CFD and ABCFD formulations in the fuzzified Aboodh transform:

Dζ(β)Υ~(u,ζ;σ)=χ⊙Du(4)Υ~(u,ζ;σ)⊕ϑ⊙Du(2)Υ~(u,ζ;σ)⊕Φ⊙Du(4)Υ~2(u,ζ;σ)⊖4Φ⊙Υ~2(u,ζ;σ),u∈R~,ζ>0,(21)

supplemented with ICs

Υ~(u,0)=Υ(σ)⊙g(u;σ),(22)

where σ∈[0,1] denotes the fuzzy valued interval and Υ~(σ)=[Υ_(σ),Υ¯(σ)]=[σ−1,1−σ].

The parametric extension of (21) is shown as

{Dζ(β)Υ_(u,ζ;σ)=χDu(4)Υ_(u,ζ;σ)+ϑDu(2)Υ_(u,ζ;σ)+ΦDu(4)Υ_2(u,ζ;σ)−4ΦΥ~2(u,ζ;σ),Υ_(u,0;σ)=(1−σ)g_(u;σ),Dζ(β)Υ¯(u,ζ;σ)=χDu(4)Υ¯(u,ζ;σ)+ϑDu(2)Υ¯(u,ζ;σ)+ΦDu(4)Υ¯2(u,ζ;σ)−4ΦΥ¯2(u,ζ;σ),Υ¯(u,0;σ)=(σ−1)g¯(u;σ).(23)

Considering the Aboodh transform of the first foregoing scenario of (23), we get

A[Dζ(β)Υ_(u,ζ;σ)]=A[χDu(4)Υ_(u,ζ;σ)+ϑDu(2)Υ_(u,ζ;σ)+ΦDu(4)Υ_2(u,ζ;σ)−4ΦΥ~2(u,ζ;σ)].

Considering (22), then we have

φβ𝒰_(u,ρ;σ)−∑ℓ=0q−1φβ−2−ℓΥ_(ℓ)(u,0;σ)=A[χDu(4)Υ_(u,ζ;σ)+ϑDu(2)Υ_(u,ζ;σ)+ΦDu(4)Υ_2(u,ζ;σ)−4ΦΥ~2(u,ζ;σ)].

Furthermore, the reconstructed mapping in the fuzzified ABCFD context

φβABC(β)β+(1−β)φβ[𝒰_(u,ρ;σ)−φ−2Υ_(u,0;σ)]=A[χDu(4)Υ_(u,ζ;σ)+ϑDu(2)Υ_(u,ζ;σ)+ΦDu(4)Υ_2(u,ζ;σ)−4ΦΥ~2(u,ζ;σ)],

or likewise, we have

A[𝒰_(u,ρ;σ)]=(σ−1)φ−2g_(u;σ)+1φβA[χDu(4)Υ_(u,ζ;σ)+ϑDu(2)Υ_(u,ζ;σ)+ΦDu(4)Υ_2(u,ζ;σ)−4ΦΥ~2(u,ζ;σ)](24)

and

A[𝒰_(u,ρ;σ)]=(σ−1)φ−2g_(u;σ)+(β+(1−β)φβφβABC(β))A[χDu(4)Υ_(u,ζ;σ)+ϑDu(2)Υ_(u,ζ;σ)+ΦDu(4)Υ_2(u,ζ;σ)−4ΦΥ~2(u,ζ;σ)].(25)

The solution to the unidentified sequence is written as

Υ_(u,ζ;σ)=∑q=0+∞Υ_(u,ζ;σ),(26)

and the nonlinear component are treated by A^

𝒩_(u,ζ;σ)=∑q=0+∞A_q(u,ζ;σ),(27)

where A_q is widely recognized that the Adomian polynomial is given as

A_q=1q!dqdλq[𝒩_(∑q=0+∞λqΥ_q(u,λ;σ))]λ=0.(28)

Merging (24), (26) and (27) with (25), yields the following expressions:

A[∑q=0+∞Υ_(u,ζ;σ)]=(σ−1)φ−2g_(u;σ)+1φβA[χ(∑q=0+∞Υ_q(u,ζ;σ))uuuu+ϑ(∑q=0+∞Υ_q(u,ζ;σ))uu+Φ(∑q=0+∞A_q(Υ_))uuuu−4Φ∑q=0+∞B_q(Υ_)](29)

and

A[∑q=0+∞Υ_(u,ζ;σ)]=(σ−1)φ−2g_(u;σ)+(β+(1−β)φβφβABC(β))A[χ(∑q=0+∞Υ_q(u,ζ;σ))uuuu+ϑ(∑q=0+∞Υ_q(u,ζ;σ))uu+Φ(∑q=0+∞A_q(Υ_))uuuu−4Φ∑q=0+∞B_q(Υ_)].(30)

We determine the aforementioned recursive expressions by fuzzified CFD and ABCFD operators using the inverse Aboodh transform and considering both sides analogous exponents of (29) and (30), respectively:

Υ_0(u,ζ;σ)=A−1[(σ−1)φ−2g_(u;σ)],Υ_1(u,ζ;σ)=A−1[1φβA[χ(Υ_0(u,ζ;σ))uuuu+ϑ(Υ_0(u,ζ;σ))uu+Φ(A_0(Υ_))uuuu−4ΦB_0(Υ_)]],Υ_2(u,ζ;σ)=A−1[1φβA[χ(Υ_1(u,ζ;σ))uuuu+ϑ(Υ_1(u,ζ;σ))uu+Φ(A_1(Υ_))uuuu−4ΦB_1(Υ_)]],⋮Υ_q+1(u,ζ;σ)=A−1[1φβA[χ(Υ_q(u,ζ;σ))uuuu+ϑ(Υ_q(u,ζ;σ))uu+Φ(A_q(Υ_))uuuu−4ΦB_q(Υ_)]].

As an outcome, we get

Υ_0(u,ζ;σ)=A−1[(σ−1)φ−2g_(u;σ)],Υ_1(u,ζ;σ)=A−1[β+(1−β)φβABC(β)φβA[χ(Υ_0(u,ζ;σ))uuuu+ϑ(Υ_0(u,ζ;σ))uu+Φ(A_0(Υ_))uuuu−4ΦB_0(Υ_)]],Υ_2(u,ζ;σ)=A−1[β+(1−β)φβABC(β)φβA[χ(Υ_1(u,ζ;σ))uuuu+ϑ(Υ_1(u,ζ;σ))uu+Φ(A_1(Υ_))uuuu−4ΦB_1(Υ_)]],⋮Υ_q+1(u,ζ;σ)=A−1[β+(1−β)φβABC(β)φβA[χ(Υ_q(u,ζ;σ))uuuu+ϑ(Υ_q(u,ζ;σ))uu+Φ(A_q(Υ_))uuuu−4ΦB_q(Υ_)]].

As a consequence, the intended approximation is composed as

Υ_(u,ζ;σ)=Υ_0(u,ζ;σ)+Υ_1(u,ζ;σ)+⋯.

Practise the same procedure for the other part of (23). As a result, we display the solution as we try to follow:

{Υ_(u,ζ;σ)=Υ_0(u,ζ;σ)+Υ_1(u,ζ;σ)+⋯,Υ¯(u,ζ;σ)=Υ¯0(u,ζ;σ)+Υ¯1(u,ζ;σ)+⋯.

4  Application to the Fuzzy FBSQe

In what follows, the sets of solutions to the FBSQe will be established here using an Aboodh transform associated with the ADM, which is articulated by the fuzzified CFD and ABCFD techniques.

4.1 Fourth-Order Fuzzy FBSQe in ℝ˜

Example 1. Surmise that the general 1D fuzzified FBSQe is denoted by

Dζ(β)Υ~(u,ζ;σ)=χ⊙Du(4)Υ~(u,ζ;σ)⊕ϑ⊙Du(2)Υ~(u,ζ;σ)⊕Φ⊙Du(4)Υ~2(u,ζ;σ)⊖4Φ⊙Υ~2(u,ζ;σ),u∈R~,ζ>0,(31)

supplemented with fuzzified ICs

Υ~(u,0)=Υ(σ)⊙exp⁡(u),(32)

where σ∈[0,1] denotes the fuzzy valued interval and Υ~(σ)=[Υ_(σ),Υ¯(σ)]=[σ−1,1−σ].

The parametric extension of (31) is shown as

{Dζ(β)Υ_(u,ζ;σ)=χDu(4)Υ_(u,ζ;σ)+ϑDu(2)Υ_(u,ζ;σ)+ΦDu(4)Υ_2(u,ζ;σ)−4ΦΥ~2(u,ζ;σ),Υ_(u,0;σ)=(1−σ)exp⁡(u)Dζ(β)Υ¯(u,ζ;σ)=χDu(4)Υ¯(u,ζ;σ)+ϑDu(2)Υ¯(u,ζ;σ)+ΦDu(4)Υ¯2(u,ζ;σ)−4ΦΥ¯2(u,ζ;σ),Υ¯(u,0;σ)=(σ−1)exp⁡(u).(33)

Case I. To begin, we apply the Aboodh transform to the first part of (33), as well as gH–differentiability via the CFD formula.

Considering the Aboodh transform of the first foregoing scenario of (33), we have

A[Dζ(β)Υ_(u,ζ;σ)]=A[χDu(4)Υ_(u,ζ;σ)+ϑDu(2)Υ_(u,ζ;σ)+ΦDu(4)Υ_2(u,ζ;σ)−4ΦΥ~2(u,ζ;σ)].

Considering (33), we have

ζβ𝒰_(u,ρ;σ)−∑ℓ=0q−1ζβ−2−ℓΥ_(ℓ)(u,0;σ)=A[χDu(4)Υ_(u,ζ;σ)+ϑDu(2)Υ_(u,ζ;σ)+ΦDu(4)Υ_2(u,ζ;σ)−4ΦΥ~2(u,ζ;σ)].

or likewise, we have

A[𝒰_(u,ρ;σ)]=(σ−1)φ−2exp⁡(u)+1φβA[χDu(4)Υ_(u,ζ;σ)+ϑDu(2)Υ_(u,ζ;σ)+ΦDu(4)Υ_2(u,ζ;σ)−4ΦΥ~2(u,ζ;σ)].(34)