On the Fuzzy Solution of Linear-Nonlinear Partial Differential Equations

: In this article, we present the fuzzy Adomian decomposition method (ADM) and fuzzy modiﬁed Laplace decomposition method (MLDM) to obtain the solutions of fuzzy fractional Navier– Stokes equations in a tube under fuzzy fractional derivatives. We have looked at the turbulent ﬂow of a viscous ﬂuid in a tube, where the velocity ﬁeld is a function of only one spatial coordinate, in addition to time being one of the dependent variables. Furthermore, we investigate the fuzzy Elzaki transform, and the fuzzy Elzaki decomposition method (EDM) applied to solving fuzzy linear-nonlinear Schrodinger differential equations. The proposed method worked perfectly without any need for linearization or discretization. Finally, we compared the fuzzy reduced differential transform method (RDTM) and fuzzy homotopy perturbation method (HPM) to solving fuzzy heat-like and wave-like equations with variable coefﬁcients. The RDTM and HPM solutions are simpler than other already existing methods. Several examples are provided to illustrate the methods that have been offered. The results obtained using the scheme presented here agree well with the analytical solutions and the numerical results presented elsewhere. These studies are important in the context of the development of the theory of fuzzy partial differential

Fractional calculus and fractional differential equations naturally arise in a number of fields, such as diffusion processes, viscoelasticity, electrochemistry, rheology, etc. Fractional calculus is usually used to replace the time derivative in a given evolution equation with a fractional derivative.For a general overview and applications of fractional differential equations in signal processing, as well as in the complex dynamic in biological tissues, the readers are referred to [20][21][22][23][24][25][26][27][28] and the references therein.
The Laplace ADM is an efficient analytical techniques for solving linear-nonlinear equations [42][43][44][45].This method is devoid of any small or large parameters and has advantages over other approximation techniques such as perturbation [46][47][48].
Elzaki invented the Elzaki transform (ET) from the classical Fourier integral in [49,50], and it is used to facilitate the solution of ordinary and partial differential equations (PDEs) in the time domain.Elzaki Transform is a mathematical method for solving differential equations, similar to Fourier Transform, Laplace Transform, and Sumudu Transform.Lately, this method has been considered by various researchers; see [51][52][53][54].
In electric circuit analysis, Zhou [55] developed DTM and solved engineering models principally.It is an iterative procedure that generates an analytic way out in the form of a polynomial using Taylor series expansions.This approach is addressed thoroughly in [56][57][58].
Keskin et al. devised the reduced DTM [59,60] and fractional reduced DTM (FRDTM) [61] to overcome the complex calculation flaw of DTM.These approaches have proven to be trustworthy semi-analytical methods, and have been used to estimate numerical solutions to PDEs and fractional order PDEs.RDTM and FRDTM have significant applications [40,57,62].
The HPM was developed by He [63] and used the homotopy in topology for nonlinear problems [64].Various other authors have discussed the HPM.Altaie et al. [65] used the HPM to develop an approximate analytical solution for the fuzzy PDEs.Ates et al. [66] studied the application of the HPM to two-point boundary-value problems with a fractionalorder derivative of the Caputo type.Sakar et al. [67] discussed the HPM applied to solve fractional PDEs with proportional delays.Jameel et al. [68] presented the application of HPM to solving one-dimensional heat-like and wave-like equations in a fuzzy environment.Osman et al. [40] investigated the comparison of fuzzy HPM and other methods to get the solutions of a fuzzy (1 + n)-dimensional Burgers' equation.
In this paper, we establish the comparison of fuzzy ADM and fuzzy modified LDM to get the solutions of fuzzy time-fractional Navier-Stokes equations in a tube.The nonlinear fuzzy time-fractional Navier-Stokes equations have no general solutions.Relatively, few circumstances can solve the problem exactly, assuming/given a simple fluid condition and flow pattern.We consider the unsteady flow of a viscous fluid in a tube where the velocity field is a function of just one space coordinate.Moreover, we studied the fuzzy EDM to give the exact solution for the fuzzy linear and nonlinear Schrodinger differential equations.The Schrodinger equations are often used in several more areas of physics and engineering science, such as optics, plasma physics, quantum mechanics, and others.Finally, we study the fuzzy RDTM and fuzzy HPM to solve fuzzy heat-like and wavelike equations with variable coefficients.These techniques are flexible and can solve the underlined problems without having to calculate complicated Adomian polynomials or make unrealistic assumptions about nonlinear behavior.
This work is organized as follows: In Section 2, we review some fundamental definitions, and the theorems that will be used are presented.In Section 3, we propose fuzzy fractional Navier-Stokes equations utilizing fuzzy ADM and modified LDM.In Section 4, we investigate the fuzzy EDM for solving the fuzzy linear-nonlinear Schrodinger differential equation.In Section 5, we apply the fuzzy RDTM and HPM to deduce the solutions of fuzzy heat-like and wave-like equations.Finally, conclusions are given in Section 6.

Basic Concepts
In this section, the most fundamental notations utilized in this article are presented as follows: The set of all real numbers is denoted by the letter R, and the set of all fuzzy numbers that are contained within R is denoted by the letter E 1 .A fuzzy number is a mapping ω : R → [0, 1] that possesses the following qualities [18]: w is normal, i.e., there exists ψ 0 ∈ R with w(ψ 0 ) = 1; 2. w stands for a convex fuzzy set (i.e., w(αψ + (1 − α)φ) ≥ min{ w(ψ), w(φ)}, for all α ∈ [0, 1], ψ, φ ∈ R); 3.
w is semicontinuous on R; 4.
supp w = {ψ ∈ R| w(ψ) > 0} is the support of the w; in addition, its closure cl(supp w) is compact.
D( 1 w, 2 w) = | 1 − 2 |D( w, 0), ∀ w ∈ E 1 , and 1 , 2 ∈ R, with 1 • 2 ≥ 0. Now, we state the definition of the Hukuhara difference from [20].Let w and τ ∈ E 1 .A Hukuhara difference is defined by the set h such that w H τ = h ⇔ w = τ ⊕ h.The H-difference is unique, but it does not always exist (a necessary condition for w H τ to exist is that w contains a translate {c} ⊕ τ of τ).

Fuzzy Fractional Calculus
We denote C F [a, b] as a space of all fuzzy-valued functions which are continuous on [a, b].In addition, we denote the space of all Lebesgue integrable fuzzy-valued functions on the bounded interval [a, b] ⊂ R by L F [a, b], refs.[71].
The fuzzy Riemann-Liouville integral of fuzzy-valued function f is defined as: Suppose that the σ-level representation of fuzzy-valued function f as f (ψ; σ) = [ f (ψ; σ), f (ψ; σ)], for 0, σ 1, then we can indicate the fuzzy Riemann-Liouville integral of fuzzy-valued function f based on the lower and upper functions as follows: and the fuzzy Riemann-Liouville integral of fuzzy-valued function f is defined as: where 0 ≤ σ ≤ 1 and Definition 8.The Mittag-Leffler function E (t) with > 0 is expressed as follows: Then, f is said to be Caputo's gH-differentiable at ψ when

Analytical Solution of Fuzzy Fractional Navier-Stokes Equation
In this section, we analyze fuzzy time-fractional Navier-Stokes equations via fuzzy ADM and fuzzy modified LDM as follows: with the initial condition w(℘, 0) = g(℘), (8) where P = − ∂p ρ ∂z , is a parameter describing the order of the time fractional derivative.

Fuzzy Adomian Decomposition Method
We consider the following parametric of fuzzy fractional Navier-Stokes Equation ( 7) of the form: ∂w(℘, t; σ) ∂t The time derivative term ( 9) and ( 10) takes the fractional derivative form Employing the process of decomposition, we define ( 11) and ( 12) in an operator from where D t , L ℘ , and L ℘℘ symbolize ∂ ∂t , ∂ ∂℘ and ∂ 2 ∂℘ 2 , respectively.The procedure is determined by using the operator J , which is the inverse of D t , for both sides of ( 13) and ( 14), we get Presupposing the existence of a series solution for w(℘, t; σ) = [w(℘, t; σ), w(℘, t; σ)], expressed as where wn (℘, t; σ) = [w n (℘, t; σ), w n (℘, t; σ)] is obtained recursively.( 18) into (15) gives and we use the recursive relations as and The components wn (℘, t; σ), n ≥ 1 can be entirely identified such that each term is obtained by the prior term.As w0 (℘, t; σ) is given and The series solution is defined by
Theorem 3. Let T w(℘, t; σ) = −R w(℘, t; σ) − N w(℘, t; σ) be a semicontinuous (i.e., the restriction of (−R − N ) to the segments of H is continuous, in H weak) and satisfies the hypotheses H 1 , H 2 as: For any g(℘, t; σ) ∈ H , the fuzzy nonlinear functional Equation (29) admits a unique solution w(℘, t; σ) ∈ H.Moreover, if the solution w(℘, t; σ), it can be assimilated as: Consequently, the fuzzy ADM diagram corresponding to the functional equation under study converges strongly to w(℘, t; σ) ∈ H, which is the unique solution to the functional equation.
The proof of this theorem is similar to the proof of Theorem 3.3 in [40].

Fuzzy Modified Laplace Decomposition Method
We consider the fuzzy nonlinear fractional PDEs as follows: where denotes the linear and nonlinear terms in ℘, respectively, and g(℘, t; σ) = [g(℘, t; σ), g(℘, t; σ)] denotes continuous fuzzy-valued functions.Therefore, through firstly using the Laplace transform for (31), we obtain In order to compute the fractional derivative, the differential property of the Laplace transform must be used To simplify, From (36) and (37), we obtain in which G(℘, t; σ) = [G(℘, t; σ), G(℘, t; σ)] denotes the term resulting from the source term and the specified initial conditions.The fuzzy LDM allows for the existence of a solution of the form N w(℘, t; σ) = [N w(℘, t; σ), N w(℘, t; σ)] is a nonlinear term that can be decomposed as where Ãµ (σ) = A µ (σ), A µ (σ) denotes Adomian polynomials with coefficients w0 , w1 , w2 , • • • wn , and it can be determined using the following formula: Taking ( 43) into (38), we obtain We obtain the following relationship by equating the terms ( 46) and ( 47) and According to the modified LDM, the fuzzy-valued function G(℘, t; σ) stated above should be divided into two pieces, G0 (℘, t; σ) and G1 (℘, t; σ) However, we consider the variations below . . . and . . .

Convergence Analysis
The series expressed from fuzzy ADM in ( 40) and ( 41) rapidly and uniformly converge to the exact solution of the system.The series of solutions represented by using fuzzy ADM (40) and ( 41) is as follows: where n = 1, 2, 3, . . . .State the convergence conditions of { wn (℘, t; σ)} in the following theorem.
Theorem 4. Let Ψ be a Banach space and T : Ψ → Ψ a contraction map and h ∈ (0, 1) be a contractive; then, T has a unique point w(℘, t; σ) such that T( w(℘, t;

Examples
In this part, we applied the methods for solving fuzzy fractional Navier-Stokes equation.In the ending, two examples are proposed.
Example 1.Consider the fuzzy time-fractional Navier-Stokes equation below: with the initial condition where

Case [A]. Fuzzy Adomian decomposition method
Using the FADM, we simply substitute the initial condition ( 56) into (48), we obtain and The first few terms of the decomposition series are defined by and where µ = 0, 1, 2, 3, • • • .Thus, we can obtain the solution as follows:

Case [B]. Fuzzy modified Laplace decomposition method
Using the fuzzy Laplace transform in (55) yields L[w(℘, t; σ)] = (5 Taking the fuzzy inverse Laplace transform ( 63) and ( 64), it follows that Allowing an infinite series solution of the type ( 40) and ( 41), and using the technique above, we obtain Using the fuzzy fractional LDM, we obtain and we can obtain the solution as follows: Example 2. We consider the following fuzzy time-fractional Navier-Stokes equation with the initial condition where (n = 1, 2, 3, • • • ).

Case [A]. Fuzzy Adomian decomposition method
The decomposition series' first few terms are provided as , and , Hence, the solution as

Case [B]. Fuzzy modified Laplace decomposition method
Using the same approach as in the preceding example, we obtain the decomposition series' starting terms are obtained from , and , Thus, the solution of the standard fuzzy Navier-Stokes equation, when = 1 as:

Fuzzy Linear and Nonlinear Schrodinger Equations
In this section, we propose some theorems of fuzzy Elzaki transform and fuzzy EDM for solving linear-nonlinear Schrodinger differential equations.

•
Consider the following fuzzy linear Schrodinger equation: with the initial condition • Consider the fuzzy nonlinear Schrodinger equation as: with the initial condition and with the initial condition where ζ is a constant and w(ψ; t) is a complex fuzzy-valued function.Equation (79) discusses the time evolution of a free particle.

Elzaki Transform
We present the fuzzy Elzaki transform of the fuzzy-valued functions belonging to a class A, where A = w(t) : ∃M, 1 , 2 > 0 so that |u(t and defined by the setting: Here are some Elzaki transform properties:

Fuzzy Elzaki Adomian Decomposition Method
The nonlinear Schrodinger differential equation is represented by the fuzzy complex valued function w(ψ, t; σ) = [w(ψ, t; σ), w(ψ, t; σ)] of the form with the initial condition and boundary condition When ζ = 0 in (86), we obtain The more general Schrodinger Equation (86), we can define as Hence, for both sides of ( 91) and (92), we apply the fuzzy Elzaki transform given in (85): We obtain by combining the differentiation property of the fuzzy Elzaki transform with the initial condition and The following step is to substitute an infinite series for the arbitrary fuzzy-valued function w(ψ, t; σ).

Convergence Analysis
We consider the following fuzzy convergence analysis of fuzzy EADM for the general fuzzy nonlinear partial differential equations given by The 2nd order operator is shown by L = ∂ 2 ∂t , the linear operator of order less than L is denoted by R, the nonlinear operator is N , and the source term is denoted by g(ψ, t).Theorem 5. Assume N : H → H a nonlinear operator.H denotes Hilbert space and suppose w(ψ, t, σ), an exact solution to (112).∑ ∞ i=0 wi (ψ, t; σ), which is obtained by Equation (111) converges to w(ψ, t; σ), if ∃γ, 0 ≤ γ < 1, such that w +1 (ψ, t; σ) ≤ γ w (ψ, t; σ) , for all ∈ N ∪ {0}.

Examples
In this part, we investigate the fuzzy EDM for solving the fuzzy linear-nonlinear Schrodinger differential equation.
According to (128) with the initial condition (129), we obtain and The first few components of w(ψ, t; σ) = [w(ψ, t; σ), w(ψ, t; σ)] are and Thus, when using the above iterations, we can obtain the exact solution as:

The Fuzzy Nonlinear Schrodinger Equation
In this part, we show two examples of the fuzzy nonlinear Schrodinger differential equation.

(141)
The few components are and Thus, we can obtain the exact solution as: Example 6.Consider the fuzzy nonlinear Schrodinger differential equation with ζ = 2 and ℘ = 1 with the initial condition where (n = 1, 2, 3, . ..).
According to (144) and the initial condition, (145) yields and where A n (σ) and A n (σ) are the Adomian polynomials to be determined from the nonlinear term given in Equations ( 140) and ( 141).Consequently, we express the few components as: and Thus, we can obtain the exact solution as follows: In the preceding instances, Figure 1 demonstrates that the left-hand functions of the σ-level set of w (w lower) are always increasing functions of σ and the right-hand functions of the σ-level set of w (w upper) are always decreasing functions of σ.

Fuzzy Heat-like and Wave-like Equations with Variable Coefficients
In this section, we apply the fuzzy RDTM and HPM to obtain the fuzzy solutions of heat-like and wave-like equations with variable coefficients as follows: • Consider the fuzzy heat-like equation of the form with the initial condition • Consider the fuzzy wave-like equation of the form with the initial condition w(ψ, φ, η, 0) = θ4 (ψ, φ, η), wt (ψ, φ, η, 0) = θ5 (ψ, φ, η). (153)
Definition 10.If a fuzzy-valued function w(ψ, t; σ) is analytic and differentiated continuously with respect to time t and space ψ in the domain of interest, then let where the t-dimensional spectrum fuzzy-valued function wj (ψ; σ) is the transformed function.
Definition 11.The fuzzy differential inverse transform of wj (ψ; σ) is defined as: Subsequently, combining Equations (159) into (156), we obtain Next, using the aforementioned definitions, we can find that the concept of fuzzy RDTM is derived from the expansion of the power series.To illustrate the basic concepts of fuzzy RDTM, we consider the following fuzzy nonlinear PDE written in the operator form with the initial condition w(ψ, 0) = f (ψ), where L = ∂ ∂t , R is a linear operator which has fuzzy partial derivatives, N w(ψ, t; σ) = [N w(ψ, t; σ), N w(ψ, t; σ)] is a fuzzy nonlinear operator and g(ψ, t; σ) = [g(ψ, t; σ), g(ψ, t; σ)] is an fuzzy inhomogeneous term.Applying the fuzzy RDTM, we obtain where wj (ψ; σ), R wj (ψ; σ), N wj (ψ; σ), and Gj (ψ; σ) are the fuzzy transformations of the fuzzy-valued functions L w(ψ, t), R w(ψ, t), N w(ψ, t) and g(ψ, t), respectively.According to the initial condition (163), we obtain From ( 167) into (164) and by straightforward iterative calculation, we obtain the following W j (ψ; σ), W j (ψ; σ) values.Moreover, the fuzzy inverse transformation of the set of values wj (ψ; σ) n j=0 gives the n-terms approximation solution as: Thus, the exact solution of the problem is given by Next, the basic mathematical operations performed by RDTM proposed in [60,61] as follows:

The Fuzzy Homotopy Perturbation Method
Consider the fuzzy nonlinear differential equation as: where 1. Ã( w) a fuzzy differential operator, which means A( w) and A( w) are differential operator, 2.
Ã( w)(σ) = f (℘, σ) and Ã( w under the boundary condition where B stand for boundary operator and ∂Ψ stand for boundary of the domain Ψ.The fuzzy operator Ã can be divided into two parts L and N , where L is a linear operator while N is a nonlinear operator.Moreover, Equation (172) can be rewritten as follows: By the fuzzy homotopy technique, we construct a homotopy: and where ℘ ∈ Ψ, and p ∈ [0, 1] is an embedding parameter, w0 (σ) = [w 0 (σ), w 0 (σ)] is the initial approximation to (172), which satisfies the boundary conditions.According to (176) and (177), we have and and the changing process of p from zero to unity is just that τ(℘, p; In topology, this is called deformation, L( τ)(σ) and Aτ(σ) − f (℘, σ), Aτ(σ) − f (℘, σ), are called Homotopy.The Homotopy parameter p is used as an expanding parameter by the fuzzy HPM to obtain Finally, on setting p = 1, this results in the formal solution

Examples
In this part, we present the exact solutions to the fuzzy heat-like and wave-like equations with variable coefficients discussed by Osman et al. [39] to assess the efficiency of the fuzzy RDTM and HPM.
Therefore, the exact solution of (207) is obtained as: Example 8. Consider the two-dimensional fuzzy heat-like equation with variable coefficients in the form [39] wt (ψ, with the initial condition where n = 1, 2, 3, . . .
Taking the initial condition (230), we have Substituting ( 234) into (231), we obtain According to the fuzzy inverse transformation of the set of values, wj (ψ, φ, η; σ) n j=0 gives n-terms approximation solutions as: The exact solution can be obtained as:
Substituting (246) into (243), we obtain Applying the fuzzy inverse transformation of the set of values wj (ψ; σ) n j=0 gives n-terms approximation solutions as: The exact solution is given as:

Discussion
The comparison between fuzzy RDTM and HPM with the fuzzy ADM, and VIM in [39] shows that, although the results of these methods when applied to the fuzzy heat-like and wave-like equations are the same, fuzzy RDTM, like fuzzy HPM, does not require specific algorithms and complex calculations such as fuzzy ADM or construction of correction functionals using general Lagrange multipliers in the fuzzy variational iteration method.Therefore, the fuzzy RDTM and HPM are more readily implemented and promising approaches to solving fuzzy partial differential equations with variable coefficients.

Conclusions
In this article, we successfully compared the fuzzy Adomian decomposition method (ADM) and fuzzy modified Laplace decomposition method (LDM) to obtain fuzzy fractional Navier-Stokes equations in a tube under fuzzy fractional derivative.The analytical results were expressed as a power series with easily calculated terms.Furthermore, we investigated the fuzzy Elzaki decomposition method (EDM) applied to solving fuzzy linear-nonlinear Schrodinger differential equations.For all four numerical problems investigated, the technique works wonderfully since the solutions found yield outstanding exact solutions.Finally, we proposed the comparison of the fuzzy reduced differential transform method (RDTM) and fuzzy homotopy perturbation method (HPM) to solving fuzzy heat-like and wave-like equations with variable coefficients.The results demonstrate that the methods are efficient and reliable, and a comparison of the approaches to other analytical methods accessible in the literature reveals that, while the results are similar, RDTM and HPM are more convenient and efficient.All these results demonstrate that the methods are powerful mathematical tools for solving fuzzy linear and nonlinear partial differential equations.

4. 4 . 1 .Example 3 .
The Fuzzy Linear Schrodinger Equation Here are two examples of the fuzzy linear Schrodinger differential equation.We take into account the fuzzy linear Schrodinger differential equation with ζ = 0