Novel Numerical Investigations of Fuzzy Cauchy Reaction–Diffusion Models via Generalized Fuzzy Fractional Derivative Operators

: The present research correlates with a fuzzy hybrid approach merged with a homotopy perturbation transform method known as the fuzzy Shehu homotopy perturbation transform method (SHPTM). With the aid of Caputo and Atangana–Baleanu under generalized Hukuhara differentiability, we illustrate the reliability of this scheme by obtaining fuzzy fractional Cauchy reaction–diffusion equations (CRDEs) with fuzzy initial conditions (ICs). Fractional CRDEs play a vital role in diffusion and instabilities may develop spatial phenomena such as pattern formation. By considering the fuzzy set theory, the proposed method enables the solution of the fuzzy linear CRDEs to be evaluated as a series of expressions in which the components can be efﬁciently identiﬁed and generating a pair of approximate solutions with the uncertainty parameter λ ∈ [ 0,1 ] . To demonstrate the usefulness and capabilities of the suggested methodology, several numerical examples are examined to validate convergence outcomes for the supplied problem. The simulation results reveal that the fuzzy SHPTM is a viable strategy for precisely and accurately analyzing the behavior of a proposed model.

An estimated analytical approach has the advantages of being able to solve complex problems without ascribing motives to numerical solutions to the precise solution to assess its validity. It also has quick estimation accuracy. In [43], a Chinese mathematician, J. -H. He, developed the homotopy perturbation method (HPM) on the premise of homotopy in topology [44]. In HPM, the approximate result is represented as a series that rapidly converges to the exact solution. The versatility of HPM allows it to yield approximate and exact solutions to both linear and nonlinear problems without the necessity for discretization and linearization, as with analytical methods [45]. Various studies have extensively used the HPM to analyze linear and nonlinear PDEs [46][47][48].
In this research, we employed a hybrid approach of the Shehu transform connected with the homotopy perturbation method for finding the applicability of the fuzzy fractional CRDEs of the type based on prior work. The main objective of this study was to expand the implementation of the SHPTM to develop numerical solutions to fractional CRDEs via fuzziness. The findings of the fractional-order with the uncertainty factor were examined by advanced techniques and methods. The strength of SHPTM is that its value comes from its ability to combine two powerful strategies for obtaining numerical findings for complex equations. Some comparison plots illustrate the supremacy of the Hukuhara generalized fractional derivative of CFD and ABC operators. It is worth noting that the proposed algorithm is capable of reducing the amount of computing costs as compared to conventional systems while maintaining good numerical accuracy maintained by the uncertain term λ ∈ [0, 1]. Several physical phenomena can be addressed by the projected method.

Basic Notions of Fractional and Fuzzy Calculus
This section clearly exhibits some major features connected to the stream of fuzzy set theory and FC, as well as certain key findings about the Shehu transform. For more details, we refer the reader to [49].
Remark 1. In (14),Ψ fulfills the assumption of the decreasing diameter Ψ and increasing diameter Ψ of a fuzzy mapping Ψ. If ν = 1, then fuzzy the Shehu transform is reduced to fuzzy Laplace transform.
Using the fact of Salahshour et al. [30], we have: Furthermore, considering the classical Shehu transform [54], we obtain: and: Then, the aforesaid expressions can be written as Then, we will define the fuzzy Shehu transform of the Caputo generalized Hukuhara derivative c gH D ϑ t 1 Ψ(t 1 ), as can be seen in [53].
Definition 12 ([53]). Suppose there is an integrable fuzzy-valued mapping c gH D ϑ t 1Ψ (t 1 ), and Ψ(t 1 ) is the primitive of c gH D ϑ t 1Ψ (t 1 ) on [0, +∞), then the CFD of order ϑ is presented as Again, using the fact of Salahshour et al. [30], we have: Bokhari et al. [55] defined the ABC fractional derivative operator in the Shehu sense. Furthermore, we extend the idea of fuzzy ABC fractional derivative in a fuzzy Shehu transform sense as follows: 1]; then, the Shehu transform of the fuzzy ABC of order ϑ ∈ [0, 1] is described as follows: Furthermore, using the fact of Salahshour et al. [30], we have:

Description of the Fuzzy SHPTM
In this unit, we exhibit the fundamental technique of the fuzzy SHPTM to establish the general solution for the one-dimensional fuzzy fractional Cauchy reaction-diffusion equation.
The parameterized formulation of (22) is exhibited as where * 0 D ϑ t 1 represents the CFD or AB fractional derivative in the Caputo sense and the linear term is denoted by L . and the nonlinear factor is denoted by N . . Taking into consideration the fuzzy Shehu transform elaborated in Definition 12 and Definition 13, respectively, we characterize the iterative process for the solution of (22). For this, we proceed with the first case of (24) and transformed mappings for the fuzzy CFD operator, then for fuzzy AB fractional derivative in the Caputo sense as Furthermore, the transformed function in the fuzzy ABC derivative sense: It follows that: and: S θ( , t 1 ; λ) , respectively. By employing the perturbation method, we acquire the solution of the first case of (24) as The nonlinear term in (24) can be calculated from: and the components of: F κ ( , t 1 ; λ) are the He's polynomials [56] as Substituting (27) and (28) into (25), we attain the iterative terms which yield the solution for the fuzzy fractional CFD operator: and again, plugging (27) and (29) into (26), we attain the iterative terms which yield the solution for the fuzzy AB fractional derivative operator in the Caputo sense: Then, by equating powers of η in (30), we compute the following CFD homotopies: Moreover, by equating powers of η in (31), we compute the following ABC operator homotopies: After applying the inverse Shehu transform, the components of Ψ κ ( , t 1 ; λ) are easily computed to the convergence series form, when η → 1; thus, we acquire the approximate solution of (22): Repeating the same procedure for the upper case of (24). Therefore, we mention the solution in a parameterized version as follows:

Convergence Analysis of Fuzzy SHPTM
Now, we describe the convergence analysis of the fuzzy SHPTM for the generalized fuzzy operator equation by employing the approach proposed by [57]:
For the sake of simplicity, the proof is followed by Osman et al. [58].

Functioning of the SHPTM and Mathematical Findings
Here, we elaborate the approximate-analytical solution of the fuzzy fractional Cauchy reaction-diffusion models via the Shehu homotopy perturbation transform method involving the CFD and ABC fractional derivative operators, respectively. Throughout this investigation, the MATLAB 2021 software package was considered for the graphical representation processes. Problem 1. Assume the fuzzy time-fractional Cauchy-reaction diffusion model: subject to fuzzy ICs:Ψ where: The parameterized formulation of (35) is presented as

Case 1.
Firstly, take into consideration the CFD coupled with the Shehu homotopy perturbation transform method in the first case of (37).
Considering the process stated in Section 3, we have: Considering the fuzzy IC, making use of the inverse Shehu transform implies that: Now implementing the HPM, we have: Equating the coefficients of the same powers of η, we have: The series form solution is presented as follows: which implies that: Finally, we have: Case 2. Now, we employ the fuzzy ABC derivative operator in the first case of (37) as follows.
Considering the process stated in Section 3, we have: Considering the fuzzy IC, making use of the inverse Shehu transform implies that: Now, by implementing the HPM, we have: By equating the coefficients of the same powers of η, we have: The series form solution is presented as follows: which implies that: Finally, we have: Figure 1a,b reveal how the effectiveness of multiple (lower and upper bound accuracy) surface graphs for Problem 2 interacting with the fuzzy CFD and Shehu transform is being exhibited in this investigation. The pattern specifies the fluctuation in the mapping Ψ( , t 1 ; λ) on the space co-ordinate with the consideration of t 1 and the uncertainty parameter λ ∈ [0, 1]. The figure illustrates that, as time passes, the mappingΨ( , t 1 ; λ) will become more intricate. Figure 2a highlights the impact of the suggested technique on the mappingΨ( , t 1 ; λ) for the fuzzy CFD to have an uncertainty parameter λ ∈ [0, 1]. With the slight increase inΨ( , t 1 ; λ), it then clearly demonstrates a large decrease in the mapping Ψ( , t 1 ; λ). Figure 2b highlights the effect of the recommended approach on the mappingΨ( , t 1 ; λ) for the fuzzy CFD to have a fixed fractional order with the varying uncertainty parameter λ ∈ [0, 1]. With a small increase in the uncertainty parameter, the mappingsΨ( , t 1 ; λ) and Ψ( , t 1 ; λ) are stable. Figure 3a highlights the impact of the suggested technique on the mappingΨ( , t 1 ; λ) for the fuzzy ABC to have an uncertainty parameter λ ∈ [0, 1]. With the slight increase inΨ( , t 1 ; λ), it then clearly demonstrates a large decrease in the mapping Ψ( , t 1 ; λ). Figure 3b highlights the effect of the recommended approach on the mappingΨ( , t 1 ; λ) for the fuzzy ABC to have a fixed fractional order with the varying uncertainty parameter λ ∈ [0, 1]. With a small increase in the uncertainty parameter, the mappingsΨ( , t 1 ; λ) and Ψ( , t 1 ; λ) are stable. Figure 4a,b illustrate the comparison between the lower and upper bound accuracies for fuzzy CFD and fuzzy ABC fractional derivative operators for Problem 1 established by the SHPTM for standard motion, i.e., at ϑ = 1.
The graphs in Figures 1-4 assist in recognizing how time and space variation statistically interact. In addition, the proposed method will facilitate scientists' work on pattern formation, diffusion, instability theory, and monitoring competence by employing inferential statistical testing.

Problem 2.
Assume that the fuzzy time-fractional Cauchy-reaction diffusion model: is subject to the fuzzy initial condition: is the fuzzy number.
The parameterized formulation of (35) is presented as Case 1. Firstly, we take into consideration the fuzzy CFD coupled with the Shehu homotopy perturbation transform method in the first case of (40).
Considering the process stated in Section 3, we have: Considering the fuzzy IC, making use of the inverse Shehu transform implies that: Now, by implementing the HPM, we have: Equating the coefficients of the same powers of η, we have: The series form solution is presented as follows: which implies that: Finally, we have:

Case 2.
Now, we employ the fuzzy ABC derivative operator in the first case of (37) as follows.
Finally, we have: Figure 5a,b reveal how the effectiveness of multiple (lower and upper bound accuracy) surface graphs for Problem 2 interacting with the fuzzy CFD and Shehu transform is being exhibited in this investigation. The pattern specifies the fluctuation in the mapping Ψ( , t 1 ; λ) on the space co-ordinate with the consideration of t 1 and the uncertainty parameter λ ∈ [0, 1]. The figure illustrates that, as time passes, the mappingΨ( , t 1 ; λ) will become more intricate. Figure 6a highlights the impact of the suggested technique on the mappingΨ( , t 1 ; λ) for the fuzzy CFD to have an uncertainty parameter λ ∈ [0, 1]. With the slight increase inΨ( , t 1 ; λ), it then clearly demonstrates a large decrease in the mapping Ψ( , t 1 ; λ). Figure 6b highlights the effect of the recommended approach on the mappingΨ( , t 1 ; λ) for the fuzzy CFD to have a fixed fractional order with the varying uncertainty parameter λ ∈ [0, 1]. AWith a small increase in the uncertainty parameter, the mappingsΨ( , t 1 ; λ) and Ψ( , t 1 ; λ) are stable. Figure 7a highlights the impact of the suggested technique on the mappingΨ( , t 1 ; λ) for the fuzzy ABC to have an uncertainty parameter λ ∈ [0, 1]. With the slight increase inΨ( , t 1 ; λ), it then clearly demonstrates a large decrease in the mapping Ψ( , t 1 ; λ). Figure 7b highlights the effect of the recommended approach on the mappingΨ( , t 1 ; λ) for the fuzzy ABC to have a fixed fractional order with the varying uncertainty parameter λ ∈ [0, 1]. With a small increase in the uncertainty parameter, the mappingsΨ( , t 1 ; λ) and Ψ( , t 1 ; λ) are stable. Figure 8a,b illustrate the comparison between the lower and upper bound accuracies for fuzzy CFD and fuzzy ABC fractional derivative operators for Problem 2 established by the SHPTM for standard motion, i.e., at ϑ = 1.
The graphs in Figures 5-8 assist in recognizing how time and space variation statistically interact. In addition, the proposed method will facilitate scientists' work on pattern formation, diffusion, instability theory, and monitoring competence by employing inferential statistical testing.
subject to the fuzzy initial condition: 1] is the fuzzy number.
The parameterized formulation of (44) is presented as Case 1. Firstly, we take into consideration the CFD coupled with the Shehu homotopy perturbation transform method in the first case of (46). Considering the process stated in Section, we have: Considering the fuzzy initial condition, making use of the inverse Shehu transform implies that: Now, by implementing the HPM, we have: Equating the coefficients of the same powers of η, we have: . . . .
Considering the process stated in Section 3, we have: Considering the fuzzy initial condition, making use of the inverse Shehu transform implies that: Now, by implementing the HPM, we have: Equating the coefficients of the same powers of η, we have: . . . .
The series form solution is presented as follows: Ψ( , t 1 ; λ) =Ψ 0 ( , t 1 ; λ) +Ψ 1 ( , t 1 ; λ) +Ψ 2 ( , t 1 ; λ) +Ψ 3 ( , t 1 ; λ) + ... , which implies that: Finally, we have: Figure 9a,b reveal how the effectiveness of multiple (lower and upper bound accuracy) surface graphs for Problem 2 interacting with the fuzzy CFD and Shehu transform is being exhibited in this investigation. The pattern specifies the fluctuation in the mapping Ψ( , t 1 ; λ) on the space co-ordinate with the consideration of t 1 and the uncertainty parameter λ ∈ [0, 1]. The figure illustrates that, as time passes, the mappingΨ( , t 1 ; λ) will become more intricate. Figure 10a highlights the impact of the suggested technique on the mappingΨ( , t 1 ; λ) for the fuzzy CFD to have an uncertainty parameter λ ∈ [0, 1]. With the slight increase inΨ( , t 1 ; λ), it then clearly demonstrates a large decrease in the mapping Ψ( , t 1 ; λ). Figure 10b highlights the effect of the recommended approach on the mappingΨ( , t 1 ; λ) for the fuzzy CFD to have a fixed fractional order with the varying uncertainty parameter λ ∈ [0, 1]. With a small increase in the uncertainty parameter, the mappingsΨ( , t 1 ; λ) and Ψ( , t 1 ; λ) are stable. Figure 11a highlights the impact of the suggested technique on the mappingΨ( , t 1 ; λ) for the fuzzy ABC to have an uncertainty parameter λ ∈ [0, 1]. With the slight increase inΨ( , t 1 ; λ), it then clearly demonstrates a large decrease in the mapping Ψ( , t 1 ; λ). Figure 11b highlights the effect of the recommended approach on the mappingΨ( , t 1 ; λ) for the fuzzy ABC to have a fixed fractional order with the varying uncertainty parameter λ ∈ [0, 1]. With a small increase in the uncertainty parameter, the mappingsΨ( , t 1 ; λ) and Ψ( , t 1 ; λ) are stable. Figure 12a,b illustrate the comparison between the lower and upper bound accuracies for fuzzy CFD and fuzzy ABC fractional derivative operators for Problem 3 established by the SHPTM for standard motion, i.e., at ϑ = 1.
The graphs in Figures 9-12 assist in recognizing how time and space variation statistically interact. In addition, the proposed method will facilitate scientists' work on pattern formation, diffusion, instability theory, and monitoring competence by employing inferential statistical testing.
Case 2. Now, we employ the ABC derivative operator in the first case of (37) as follows.
Considering the process stated in Section, we have: Considering the fuzzy initial condition, making use of the inverse Shehu transform implies that: Now, by implementing the HPM, we have: By equating the coefficients of same powers of η, we have: The series form solution is presented as follows: Ψ( , t 1 ; λ) =Ψ 0 ( , t 1 ; λ) +Ψ 1 ( , t 1 ; λ) +Ψ 2 ( , t 1 ; λ) +Ψ 3 ( , t 1 ; λ) + ... , which implies that: Finally, we have: Figure 13a,b reveal how the effectiveness of multiple (lower and upper bound accuracy) surface graphs for Problem 2 interacting with the fuzzy CFD and Shehu transform is being exhibited in this investigation. The pattern specifies the fluctuation in the mapping Ψ( , t 1 ; λ) on the space co-ordinate with the consideration of t 1 and the uncertainty parameter λ ∈ [0, 1]. The figure illustrates that, as time passes, the mappingΨ( , t 1 ; λ) will become more intricate. Figure 14a highlights the impact of the suggested technique on the mappingΨ( , t 1 ; λ) for the fuzzy CFD to have an uncertainty parameter λ ∈ [0, 1]. With the slight increase inΨ( , t 1 ; λ), it then clearly demonstrates a large decrease in the mapping Ψ( , t 1 ; λ). Figure 14b highlights the effect of the recommended approach on the mappingΨ( , t 1 ; λ) for the fuzzy CFD to have a fixed fractional order with the varying uncertainty parameter λ ∈ [0, 1]. With a small increase in the uncertainty parameter, the mappingsΨ( , t 1 ; λ) and Ψ( , t 1 ; λ) are stable. Figure 15a highlights the impact of the suggested technique on the mappingΨ( , t 1 ; λ) for the fuzzy ABC to have an uncertainty parameter λ ∈ [0, 1]. With the slight increase inΨ( , t 1 ; λ), it then clearly demonstrates a large decrease in the mapping Ψ( , t 1 ; λ). Figure 15b highlights the effect of the recommended approach on the mappingΨ( , t 1 ; λ) for the fuzzy ABC to have a fixed fractional order with the varying uncertainty parameter λ ∈ [0, 1]. With a small increase in the uncertainty parameter, the mappingsΨ( , t 1 ; λ) and Ψ( , t 1 ; λ) are stable. Figure 16a,b illustrate the comparison between the lower and upper bound accuracies for fuzzy CFD and fuzzy ABC fractional derivative operators for Problem 4 established by the SHPTM for standard motion, i.e., at ϑ = 1.

Conclusions
The principal aim of this investigation was to provide an approximate-analytical solution to the fuzzy fractional Cauchy reaction-diffusion equation by taking into consideration the generalized Hukuhara derivative of Caputo and AB fractional derivatives. A stability analysis of the proposed study was presented. The findings will be pertinent in the evaluation of nonlinear complex processes that arise in both scientific disciplines. By employing the fuzzy set theory, this analysis shows that the SHPTM is simple, powerful, adequate, and applicable to a wide range of nonlinear equations. It is remarkable that SHPTM yields solutions in pairs, which often becomes an advantage in selecting the best feasible solution for the governing model. Furthermore, it is clear from graphical views that the approximate findings of ABC operators are in close contact with the CFD operators. When it concerns reducing the size of the computational cost, the SHPTM technique is effective. Because the SHPTM investigates fractional equations without using Adomian's polynomials, it delivers a wide analytical outcome capacity. This is one of the advantages of the SHPTM method over the decomposition method. Finally, the SHPTM is well suited to all analytical methods and has a wide range of applicability in science and technology.