Fuzzy Analysis for Thin-Film Flow of a Third-Grade Fluid Down an Inclined Plane

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Introduction
e fuzzy set theory (FST) concept was first proposed by Zadeh [1]. e FST is a useful technique for defining situations when information is ambiguous, hazy, or unsure. e membership function, or belongingness, of FST defines it. e membership function (MF) in FST assigns a number form of the [0, 1] interval to each element of the discourse universe. A fuzzy number (FN) is a function with a range between zero and one. Every numerical value in the range is allocated an MF grade, with "0" indicating the lowest grade and "1" signifying the highest grade. Numerous authors have created arithmetic operations on FNs, for example, [1,2]. Triangular, trapezoidal, and Gaussian fuzzy numbers are all examples of FNs. In this article, we will employ TFN to keep things simple.
When the partial or ordinary DEs are converted through dynamic systems, information is sometimes fragmentary, ambiguous, or uncertain. e fuzzy differential equations (FDEs) are a valuable tool for modeling dynamical systems with ambiguity or uncertainty.
is impreciseness or vagueness can be mathematically defined using FNs or TFNs. FDEs have been the subject of some investigations in recent years. e fuzzy differentiability notion was first developed by Seikala [3]. In [4], Kaleva addressed fuzzy differentiation and integration. FDEs were first reported by Kandel and Byatt [5], while Buckley et al. [6] used two ways to solve them using the extension principle and FNs. Nieto [7] investigated the Cauchy problem using FDEs. In [8], Lakshmikantham and Mohapatra examined the initial value problems with help of FDEs. For the existence and uniqueness solution of FDE, Park and Han [9] employed successive approximation techniques. Hashemi et al. [10] employed the homotopy analysis method (HAM) to determine a system of fuzzy differential equations (SFDEs). Mosleh [11] used universal approximation and fuzzy neural network methods to solve the SFDEs. Gasilov et al. [12,13] established the symmetrical method to solve SFDEs. Khastan and Nieto [14] used a generalized differentiability concept to solve the second-order FDE. Salahsour et al. [15] applied FDE and TFNs to evaluate the fuzzy logistic equation and alley impact. Nadeem et al. [16] numerically examined the effect of thermal radiation and natural convective flow on third-grade fuzzy hybrid nanofluid between two upright plates. Recently, Nadeem et al. [17] explored Magnetohydrodynamic (MHD) and ohmic heating on a third-grade fluid in an inclined channel in a fuzzy atmosphere, using the triangle MF to address the uncertainty. Siddique et al. [18] studied the Couette flow and heat transfer on third-grade fuzzy hybrid (SWCNT + MWCNT/Water) nanofluid over the inclined plane under a fuzzy environment. Many scientists and engineers have used FST to attain well-known achievements in science and technology [19][20][21][22][23][24][25][26][27][28][29][30]. e above literature review motivates us to initiate the application of FDE in fluid mechanics.
In science and engineering, fluid flow is extremely important.
ere are increases in a wide variety of problems such as magnetic effect, chemical diffusion, and heat transfer. Physical problems are transformed into linear or nonlinear DEs and may contain some ambiguous information. Physical problems such as parameters, geometry, initial, and boundary conditions have a significant impact on the solution of DEs. e parameters, initial, and boundary conditions are not crisp due to mechanical imperfections, experimental inaccuracies, and measurement errors. In this situation, FDEs play an important role in reducing uncertainty and providing an appropriate manner to explain physical problems that originate from unknown parameters, initial, and boundary conditions. "A fluid is a substance that deforms continuously when shear stress or an external force is applied. Newtonian and non-Newtonian fluids are the two main types of fluid. Newtonian fluids, such as air, mineral oil, water, thin motor oil, gasoline, glycerol, and alcohol, follow Newton's law of viscosity, whereas non-Newtonian fluids are the polar opposite of Newtonian fluids. e importance of non-Newtonian fluids with developments in industries and technology like polymer, petroleum, pulp, etc, is as follows. Various industrial ingredients fall into this cluster, such as cosmetics, soap, paints, tars, shampoos, mayonnaise, blood, yogurt, syrups biological solutions, and glues. It is difficult to build a unique model that can represent the features of all non-Newtonian fluids because of the fluid's complexity. A third-grade fluid [31] is a non-Newtonian fluid that exhibits non-Newtonian phenomena including shearthickening, shear-thinning, and normal stresses. So, the third-grade fluid has received superior attention from researchers. In this paper, considered fluids are a thirdgrade (differential type), which have been successfully investigated in a variety of flow scenarios [32,33] and references therein. Siddiqui et al., [34] used the perturbation method (PM) [35,36] and homotopy perturbation method (HPM) [37] to find out the solution of nonlinear DE formulated for fluids of third grade. He proved that PM provides more reliable and accurate results than HPM. Later on, Hayat et al. [38] calculated the exact solution to the same problem under certain norms. Different authors like Sajid and Hayat [39] used the HAM. Shah et al. [40] used HAM, Siddiqui et al. [41] used He's variational iteration method (VIM) and Adomian decomposition method (ADM), and Iqbal and Abualnaja [42] used Galerkin's finite element method. Variation of parameter method (VPM) was utilized by Zaidi et al. [43] to describe the thin-film flow of third-grade fluid down an inclined plane. Khan et al. [44] studied the impact of thermal radiation and MHD on Non-Newtonian fluid over a curved surface. Koriko et al. [45] considered the impact of viscosity dissipation on Non-Newtonian Carreau nanofluids and dust fluids. ere are some further studies about the thinfilm flow given in [46][47][48][49][50]. Linear regression is a statistical data-driven prediction tool. e goal of regression is to use a sequence of explanatory or independent variables to explain the uncertainty and variability in a dependent variable, resulting in a prediction equation. Fuzzy linear regression is an effort to expand linear regression to fuzzy number applications. It gives an alternate strategy in circumstances where crisp linear regression is not achievable, such as when stringent assumptions are not followed or when the underlying data or process has visible fuzziness. Animasaun et al. [51] investigated heat transfer analysis through linear regression via data points. Wakif et al. [52] studied the meta-analysis of nanosize particles in various fluids. Shah et al. [53] measured the linear regression analysis of Grashof number in different fluids with convective boundary conditions.
In the review of literature, third-grade fluid problems were studied for only crisp or classical cases. So, the abovementioned works motivated us to extend the work of Siddiqui et al. [34] for the fuzzy analysis of thin-film flow of a third-grade fluid down an inclined plane under the fuzzy environment.
is article discussed the uncertain flow mechanism through FDEs and the generalization of Siddiqui et al. [34]. Also, it discusses the fuzzy regression analysis via data points of the fuzzy velocity profile. e goal of this article is to affect the fuzzy velocity profile on various parameters, using a statistical technique for quantifying the rate of increase or decrease and scrutinizing the consistent effects.
e article is systematized as follows. Section 2 contains some essential preliminaries connected to the current research. Section 3 develops the governing equations of the proposed study and also changes governing equations in the fuzzy form to solve by a regular PM. Results and discussion in graphical and tabular form are presented in Section 5. Section 6 gives some conclusions.

Preliminaries
is section discussed some basic notations and definitions that are used in the present work.
Definition 1 (Zadeh [1]). "Fuzzy set is defined as the set of ordered pairs such that where X is the universal set, and μ 0U (x) is membership function of U and mapping defined as Definition 2 (Gasilov et al. [12]). "α-cut or α-level of a fuzzy set U is a crisp set U α and defined by e TFN with peak (or center) a 2 , left width a 2 − a 1 > 0, right width a 3 − a 2 > 0, and these TFNs being transformed into interval numbers through α-cut approach is written as where α ∈ [0, 1] as shown in Figure 1. An arbitrary TFN satisfies the following conditions: are bounded on left continuous and right continuous at [0, 1] respectively." Definition 4 (Seikala [3]): "Let I be a real interval. A mapping u: I ⟶ F is called a fuzzy process, defined as

Formulation of a Crisp Model into a Fuzzy Model.
e thin-film flow of an incompressible third-grade fluid down an inclined plane of inclination θ ≠ 0 with the assumptions that surface tension is negligible, the ambient air is stationary, and in the absence of a pressure, gradient is governed by the following boundary value problem (see Figure 2) [35,36].
where w is the velocity along the inclined plane, ρ is the fluid density, β 3 and β 2 are material constants of third-grade fluid, g is the acceleration due to gravity, μ is the dynamic viscosity, and δ is the thickness of the thin layer. We introduced the following nondimensionless variables in (2) and (3): After dropping the sign of asterisks, equation (2) and the boundary conditions (3) become where λ � (β 2 + β 3 )] 2 /μδ 4 is the third-grade fluid parameter and c � gδ 3 sinθ/] 2 is an inclined parameter.
To deal with this problem, we used TFNs and discretization in the form of (a 1 , a 2 , a 3 ) and (d, e, f) for the fuzzy parameter α − cut. Due to fuzziness at boundaries, this discretization is used at the boundary of the inclined plate in Figures 3 for Figure 4 has certain flow behaviors. Using α-cut approach, the fuzzy boundary conditions can be decomposed into an interval form. Hence, governing equation (5) with boundary conditions (6) is converted into coupled FDE and fuzzy boundary conditions as given as follows: And also, it can be written as for 0 ≤ α ≤ 1, subject to fuzzy boundary conditions After simplification of (7) and (9), fuzzy boundary conditions are

Solution of the Problem in a Fuzzy
Environment. e method of the PM [35,36] for solving FDEs: fuzzy and the crisp velocities u(x) are in the form where u 0 , v 0 , u 1 , v 1 , u 2 , and v 2 are zero-, first-, and secondorder solutions, respectively.
Zeroth-order fuzzy problem is e zeroth-order fuzzy boundary conditions for the above equation are e first-order fuzzy problem is g δ x · y · θ ≠ 0  Mathematical Problems in Engineering e first-order fuzzy boundary condition for the above equation is e second-order fuzzy problem is e second-order fuzzy boundary condition for the above equation is e zeroth-order fuzzy solution is e first-order fuzzy solution is e second-order fuzzy solution is Combining equations (18)- (20), which give the approximate fuzzy solution for a lower and upper velocity, e solution of crisp velocity is

Discussion of Observed Results.
We extend the work of Siddiqui et al. [34] under the fuzzy environment. e TFNs are used to fuzzify the boundary conditions and the governing equations, which are then solved by a modified FPM. e effect of numerous fluid and fuzzy parameters on fluid velocity is analyzed in graphical and tabular forms. e comparison of HPM, VPM, PM, and numerical solutions is presented in Table 1. It can be examined that PM has good agreement with HPM, VPM, and numerical results at λ � 0.3 and c � 0.5.
In Figures 3 and 4, membership functions of the fuzzy velocity profiles are plotted with the influence of λ, c , and α − cut at x � 2. e horizontal axis represents the fuzzy velocity while the vertical axis shows the variation of the α − cut. We observed that v(x; α) increases and u(x; α) decreases correspond to values of λ and c with increasing α − cut, so the solution is strong. e crisp solution is always between the fuzzy solutions; when α − cut increases, the  Mathematical Problems in Engineering width between u(x; α) and v(x; α) of fuzzy velocity profiles decreases and at α − cut � 1 the coherent is with one another. It is proved that uncertainties in physical parameters and boundary conditions have a nonnegligible impact on the fuzzy velocity profile. Also, the width between u(x; α) and v(x; α) fuzzy velocity is less than uncertainty. Achieved (x; α) and v(x; α) bounds of velocity profiles are plotted in Figures  5-13 for different values of α-cut (α � 0, 0.3, 0.7, 1). It may be observed that as α − cut increases from 0 to 1, the fuzzy velocity profile has a narrow width, and the uncertainty decreases significantly, which finally provides crisp results (see Figures 8, 9, 13 and 14) for

Mathematical Problems in Engineering
growing the values of λ due to a rise in the boundary layer thickness. Also in Figure 9, it can be observed that if λ � 0, the solution reduces to the Newtonian fluid.  Figure 13, we can see that at α-cut � 1, fuzzy boundary conditions convert into crisp boundary conditions. It is exciting that for equal responses, fuzzy solutions of u(x; α) and v(x; α) bounds of velocity profiles are the same at α-cut � 1. However, further evidence provided by the fuzzy velocity profiles at different levels of possibility (i.e., different α-cut) may help decision-makers. Figure 14 shows the crisp velocity behavior for different values of c. It is seen that the crisp velocity increases as the c increases. e reason is that when c is increased, the fluid velocity upsurges due to the effect of inclined geometry with an increase in the boundary layer thickness. It is encouraging to note that the u(x; α) and

Fuzzy Regression
Analysis. e method of slope linear regression via data points on Microsoft Excel is applied in this section.
To explain the approach, the effect of the third-grade fluid parameter (λ) on the fuzzy velocity profile is examined as shown in Table 2. e formula in Excel for α − cut and u(x, α) � Slope (A1 : A2, B1 : B2). Similarly, we use the formula in Excel for α − cut, v(x, α) and mid values. Using the slope linear regression through the fuzzy velocity data points suggested by [51][52][53], it is worth deducing from Table 2 Figure 15, we conclude that u(x, α) increases with increasing the value of λ while v(x; α) decreases with increasing the value of λ sand α − cut at x � 0.7. From Figure 16, u(x; α) upsurges with increasing the value of c while v(x; α) declines with growing the value of c and α − cut for x � 0.7. Also we can see that in both figures when α − cut � 1, they give the same behavior. e impact of the inclined parameter (c) on the fuzzy velocity profile is examined as shown in Table 3. It is seen that u(x, α) increases with α − cut at the rate of 0.368764 for c � 0.50. When c � 0.55, as α − cut increases, u(x, α) now increased at the larger rate of 0.394109. is is because the membership functions are associated with fuzzy numbers or TFNs including imperative and valuable information that is not included in crisp regression. Also, the fuzzy velocity profile shows the maximum rate of flow as compared to mid values (crisp velocity).

Conclusion and Recommendation
In this work, we analyzed the thin-film flow problem of a third-grade fluid on an inclined plane under a fuzzy environment.
e governing equations as well as the boundary conditions which are fuzzified using the TFNs developed by α-cut are solved by the fuzzy perturbation technique. As α-cut increases from 0 to 1, the uncertainty of fuzzy velocity profile decreases gradually, and u(x; α) and v(x; α) bounds of velocity profile give the crisp behavior at α-cut � 1. So, from the above observations, we can conclude that the upper and lower bounds of a TFN coincide with the crisp value of the original problem. Furthermore, the current findings are in good accord with previous findings in the literature when conducted in a crisp environment. Using fuzzy slope regression analysis, the fuzzy velocity profile also displays the highest rate of flow when compared to the crisp velocity.

Data Availability
No data were used to perform this work.