Numerical Analysis of Fractional-Order Parabolic Equations via Elzaki Transform

Deanship of Joint First Year Umm Al-Qura University Makkah, Saudi Arabia Department of Mathematics, College of Science, King Khalid University, 9004 Abha, Saudi Arabia Department of Mathematics, Faculty of Science, Al-Azhar University, 71524 Assiut, Egypt Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan


Introduction
The present research work is dedicated to studying the analytical solution of fractional-order parabolic equations. The literature is well recognized that a broad range of physics, engineering, nuclear physics, and mathematics problems can be defined as unique boundary and initial value problems. Homogeneous beam's transverse vibrations are controlled by fractional single fourth-order parabolic partial differential equations (PDEs). Such problem types occur in viscoelastic and inelastic flow mathematical modeling, layer deflection theories, and beam deformation [1][2][3][4][5][6][7][8][9][10][11][12]. Analyses of these problems have taken several physicist's and mathematician's attention [13][14][15].
The time fractional parabolic PDEs with variable coefficient: where κðϕ, φ, ψÞ, μðϕ, φ, ψÞ, and ρðϕ, φ, ψÞ are positive. With initial conditions, with boundary conditions For which, h ℓ , g ℓ , k ℓ , h ℓ , g ℓ , and k ℓ are continuous variables, and ℓ differs between 0 and 1 and beam's flexural stiffness ratio [1] in its volume per unit mass, like, and its mentions [1,3,4,6,8,10,11]. Many researchers [10,16,17] have attempted to study the analytical solutions of the parabolic equation of the fourth-order. Different techniques have been suggested recently, such as the B-spline method [18], decomposition method [19], the implicit scheme [20], the explicit scheme [11], and the spline method [21] to analyze the solution of the partial differential fourth-order parabolic equation. Biazar and Ghazvini [22] have used He's iterative technique for the solution of parabolic PDE's. The modified version of this method was introduced in [23] to solve singular fourth-order parabolic PDEs. The fourth-order parabolic PDE analytical solution was examined in [24]. The modified Laplace discussed variational iteration technique [25] to solve singular fourth-order parabolic PDEs.
G. Adomian is an American scientist who has developed the Adomian decomposition method. It focuses on the search for a set of solutions and the decomposition of the nonlinear operator into a sequence in which Adomian polynomials [26] are recurrently computed to use the terms. This method is improved with the aid of Elzaki transformation such that the improved method is known as the Elzaki decomposition method (EDM). Elzaki Transform (ET) is a modern integral transform introduced by Tarig Elzaki in 2010. ET is a modified transform of Sumudu and Laplace transforms. It is important to note that there are many differential equations with variable coefficients that Sumudu and Laplace cannot accomplish transforms but can be conveniently done by using ET [27][28][29][30]. Many mathematicians have been solving differential equations with the aid of ET, such as Navier-Stokes equations [30], heat-like equations [31] and Burgers-Huxley equation [32].

Idea of NDM
The general fractional-order PDE is given as In Equation (9), we represent the linear part of the equation with L and the nonlinear part with N, and D β = ∂ β /∂τ β denotes the Caputo fractional derivatives.
With initial condition, We have applied the Elzaki transformation to Equation and using Elzaki Transform's differentiation property, we 2 Journal of Function Spaces Now, μðϕ, 0Þ = kðϕÞ.
The following infinite series represent the EDM solution μðϕ, τÞ.
Applying the Elzaki transformation's linearity, We can generally write Equation (18) and Equation (20) implement the inverse Elzaki transformation 4. Numerical Implementation 4.1. Problem. Consider fractional-order one-dimensional parabolic equation: with initial conditions with boundary conditions The Elzaki transform of Equation (22): Simplify and replace Equation (23) condition.
Use of inverse Elzaki transformation Equation (28) correction function is provided by the first term 3 Journal of Function Spaces then we got for j = 0, The following terms are The series form of problem (1) such as: when β = 2, then integer EDM solution is The exact result is given as In Figure 1, the exact and the EDM solutions of problem 1 at β = 1 are shown by subgraphs, respectively. From the given figure, it can be seen that both the EDM and exact results are in close contact with each other. In Figure 2, the EDM solutions of problem 1 are investigated at different fractional order β = 0:8 and 0:6. It is analyzed that timefractional problem results are convergent to an integerorder effect as time-fractional analysis to integer order.

Problem. Consider fractional-order two-dimensional parabolic equation:
with initial conditions with boundary conditions In the Elzaki transformation of Equation (37), we get Simplify and replace Equation (38) condition.

Equation (43) correction function is provided by
the first term then we get for j = 0, The following terms are In the series form of problem (2), we get

Journal of Function Spaces
Then, β = 2, the integer EDM result as The exact solution is In Figure 3, the exact and the EDM solutions of problem 2 at β = 1 are shown by subgraphs, respectively. From the given figure, it can be seen that both the EDM and exact results are in close contact with each other. In Figure 4, the EDM solutions of problem 2 are investigated at different fractional order β = 0:8 and 0:6. It is analyzed that timefractional problem results are convergent to an integerorder effect as time-fractional analysis to integer order.

Problem. Consider fractional-order three-dimensional parabolic equation:
with initial conditions with boundary conditions In the Elzaki transformation of Equation (52), we get Simplify and replace Equation (54) condition.
using the inverse Elzaki transform " " " " Equation (59) correction function is provided by the first term Journal of Function Spaces then we get for j = 0, The following terms are In the integer-order solution of EDM of Equation (52) at β = 2, we get The exact solution is given as In Figure 5, the exact and the EDM solutions of problem 3 at β = 1 are shown by subgraphs, respectively. From the given figure, it can be seen that both the EDM and exact results are in close contact with each other. In Figure 6, the EDM solutions of problem 3 are investigated at different fractional order β = 0:8 and 0:6. It is analyzed that timefractional problem results are convergent to an integerorder effect as time-fractional analysis to integer order.

Conclusion
In the present article, an efficient analytical technique is used to solve fractional-order parabolic equations. The present method is the combination of two well-known methods, namely, Elzaki transform and Adomian decomposition method. The Elzaki transform is applied to the given problem, which makes it easier. After this, we implemented Adomian decomposition method and then inverse Elzaki transform to get closed form analytical solutions for the given problems. The proposed method required small number of calculation to attain closed form solutions and is therefore considered to be one of the best analytical method to solve fractional-order partial differential equations.

Data Availability
The numerical data used to support the findings of this study are included within the article.