Analytical investigation of fractional-order Newell-Whitehead-Segel equations via a novel transform

: In this paper, we ﬁnd the solution of the time-fractional Newell-Whitehead-Segel equation with the help of two di ﬀ erent methods. The newell-Whitehead-Segel equation plays an e ﬃ cient role in nonlinear systems, describing the stripe patterns’ appearance in two-dimensional systems. Four case study problems of Newell-Whitehead-Segel are solved by the proposed methods with the aid of the Antagana-Baleanu fractional derivative operator and the Laplace transform. The numerical results obtained by suggested techniques are compared with an exact solution. To show the e ﬀ ectiveness of the proposed methods, we show exact and analytical results compared with the help of graphs and tables, which are in strong agreement with each other. Also, the results obtained by implementing the suggested methods at various fractional orders are compared, which conﬁrms that the solution gets closer to the exact solution as the value tends from fractional-order towards integer order. Moreover, proposed methods are interesting, easy and highly accurate in solving various nonlinear fractional-order partial di ﬀ erential equations.


Introduction
Fractional calculus (FC) is a subject that dates back to 1695 and is regarded to be as old as ordinary calculus. Ordinary calculus made it impossible to model nonlinear real-world phenomenon in nature, hence fractional calculus became popular among researchers. The fractional derivative is the derivative of arbitrary order in applied mathematics and mathematical analysis. In the nineteenth century, Riemann Liouville [1] implemented the fractional derivative when simulating real-world problems. Some of the Nobel contributions of mathematicians are listed here, Caputo [2], Kemple and Beyer [3], Abbasbandy [4], Jafari and Seifi [5,6], Miller and Ross [7], Podlubny [8], Kilbas and Trujillo [9], Diethelm et al. [10], Hayat et al. [11], Debanth [12], Momani and Shawagfeh [13], etc. Fractional calculus has become one of the fascinating science and engineering research areas in recent years. Viscoelasticity and damping, diffusion and wave propagation, electromagnetism and heat transfer, biology, signal processing, robotics system classification, physics, mechanics, chemistry, and control theory are the most important scientific fields that use fractional calculus at the moment.
Researchers are interested in FC because of its wide applications in physics, engineering, and reallife sciences. Fractional differential equations accurately represent these physical facts. Fractional differential equations are significantly greater generalizations of integer-order differential equations. Fractional differential equations (FDEs) have generated much interest in recent years. As a result of their frequent appearance in diverse applications, such as quantum mechanics [14], chaotic dynamics [15], plasma physics [16,17], theory of long-range interaction [18], mechanics of non-Hamiltonian systems [19], physical kinetics [20], anomalous diffusion and transport theory [21], mechanics of fractional media [22], astrophysics [23], and so on. FDEs have been the subject of numerous investigations. Many works have been dedicated to developing efficient methods for solving FDEs, but it is important to remember that finding an analytical or approximate solution is difficult, therefore, accurate methods for obtaining FDE solutions are still being researched. In the literature, there are several analytical and numerical approaches for solving FDEs. For example, the generalized differential transform method (GDTM) [24], adomian decomposition method (ADM) [25], homotopy analysis method (HAM) [26], variational iteration method (VIM) [27], homotopy perturbation method (HPM) [28], Elzaki transform decomposition method (ETDM) [29], iterative Laplace transform method (ILTM) [30], fractional wavelet method (FWM) [31,32], residual power series method (RPSM) [33,34].
In this paper, we used two powerful techniques with the aid of the Antagana-Baleanu fractional derivative operator and Laplace transform for solving time fractional NWSEs. The two well-known methods that we implement are LTDM and VITM. The suggested techniques give series form solutions having quick convergence towards the exact solutions. Four non-linear NWSEs case study issues are resolved using the given methodology. Newell and Whitehead [35] developed the non-linear NWSE. The diffusion term's influence interacts with the reaction term's nonlinear effect in the Newell-Whitehead-Segel equation model. The fractional NWSE is written in the following way: where r is a positive integer and g, h are real numbers with k > 0. The first term D τ µ(ξ, τ) on the left hand side in (1.1) shows the deviations of µ(ξ, τ) with time at a fixed location, while the right hand side first term D 2 ξ µ(ξ, τ) shows the deviations with spatial variable ξ of µ(ξ, τ) at a specific time and the right hand side remaining terms gµ − hµ r , is the source terms. µ(ξ, τ) is a function of the spatial variable ξ and the temporal variable τ in (1.1), with ξ ∈ R and τ ≥ 0. The function µ(ξ, τ) might be considered the (nonlinear) temperature distribution in an infinitely thin and long rod or as the fluid flow velocity in an infinitely long pipe with a small diameter. Many researchers find the analytical solution of NWSEs [36,37] due to their wide range of applications in mechanical and chemical engineering, ecology, biology, and bioengineering.

Preleminaries
Some basic definitions related to fractional calculus are expressed here in this section.
Definition 2.1. The Caputo fractional-order derivative is given as where n < ≤ n + 1.
Definition 2.3. The Atangana-Baleanu derivative in Caputo manner is given as and E γ represent the Mittag-Leffler function.
Definition 2.4. The Atangana-Baleanu derivative in Riemann-Liouville manner is given as Definition 2.5. The Laplace transform connected with the Atangana-Baleanu operator is define as (2.5) Definition 2.6. Consider 0 < < 1, and g is a function of , then the fractional-order integral operator of is given as

The idea of LTDM
The solution by LTDM for partial differential equations having fractional-order is described in this section.
Employing the Laplace transform to (3.1), and we acquire By the virtue of Laplace differentiation property, we have where where Θ(ξ, ω) demonstrates the terms occurring from source factor. LTDM determines the solution of the infinite sequence of µ(ξ, τ) and decomposing the nonlinear operator N 1 as where A m are Adomian polynomials given as Putting (3.5) and (3.7) into (3.4) , gives The following terms are described: Thus all components for m ≥ 1 are calculated as (3.11)

VITM formulation
The VITM solution for FPDEs is defined in this section.
with initial source µ(ξ, 0) = g 1 (ξ), (4.2) where D τ = ∂ ∂τ is stand for fractional-order AB operator, M is a linear operator and N is nonlinear term and P indicates the source term.
The Laplace transform is applied to Eq (4.1), we have By the property of LT differentiation, we get The iteration technique for (4.4) as where (s) is Lagrange multiplier and with the application of inverse Laplace transform, (4.5) series form solution is given by

Convergence analysis
Here we discuss uniqueness and convergence analysis.
where I is contraction as From Banach fixed point theorem, the result of (3.1) is unique.

Applications
Four cases of nonlinear NWSEs are presented to demonstrate the suggested technique's capability and reliability.
Applying Laplace transform to (6.1), we get On taking Laplace inverse transform, we get Assume that the solution, µ(ξ, τ) in the form of infinite series given by where µ 2 = ∞ m=0 A m are the so-called Adomian polynomials that represent the nonlinear terms, thus (6.3) having certain terms are rewritten as According to (3.7), the decomposition of nonlinear terms by Adomian polynomials is defined as Thus, on comparing both sides of (6.5) µ 0 (ξ, τ) = Υ.
For m = 0, we have For m = 1, we have The approximate series solution is expressed as Therefore, we have Particularly, putting = 1, we get the exact solution The analytical results by VITM: The iteration formulas for (6.1), we have For m = 0, 1, 2, . . ., we have . . .
Therefore, we obtain Particularly, putting = 1, we get the exact solution, see Figure 1 and Table 1. .
(6.23)   Example 3. The NWSE has given in (1.1) for g = 1, h = 1, k = 1 and r = 4 becomes Applying Laplace transform to (6.24), we get Assume that the solution, µ(ξ, τ) in the form of infinite series given by where µ 4 = ∞ m=0 A m are the so-called Adomian polynomials that represent the nonlinear terms, thus (6.26) having certain terms are rewritten as According to (3.7), the decomposition of nonlinear terms by Adomian polynomials is defined as Thus, on comparing both sides of (6.28) For m = 0, we have For m = 1, we have The approximate series solution is expressed as Therefore, we obtain µ(ξ, τ) = 1 1 + exp 3ξ 2 τ 2 Γ(2 + 1) Particularly, putting = 1, we get the exact solution The analytical results by VITM: The iteration formulas for (6.24), we have .
Therefore, we obtain  Table 3.    Example 4. The NWSE has given in (1.1) for g = 3, h = 4, k = 1 and r = 3 becomes with initial source Applying Laplace transform to (6.36), we get (6.38) Assume that the solution, µ(ξ, τ) in the form of infinite series given by where µ 3 = ∞ m=0 A m are the so-called Adomian polynomials that represent the nonlinear terms, thus (6.38) having certain terms are rewritten as According to (3.7), the decomposition of nonlinear terms by Adomian polynomials is defined as, Thus, on comparing both sides of (6.40) For m = 0, we have For m = 1, we have The approximate series solution is expressed as Therefore, we obtain Particularly, putting = 1, we get the exact solution The analytical results by VITM: The iteration formulas for (6.36), we have For m = 0, 1, 2, . . ., we have Particularly, putting = 1, we get the exact solution, see Figures 6,7 and Table 4.     Figure 3, the two and three dimensional different fractional order graphs of Example 2. The figures show that our solution approaches the exact solution as the fractional order goes towards the integer-order. Figure 4, demonstrate the layout of the exact and analytical solution while Figure 5 shows the error comparison of the exact and analytical results of Example 3. The error confirms the efficiency of the suggested techniques. The graphical view of Example 4 for exact and our solution can be seen in Figure 6, however, Figure 7 shows the error comparison of both results. Furthermore, the behavior of the exact and proposed method solution with the aid of absolute error at different orders of is shown in Tables 1-4. Finally, it is clear from the figures and tables that the proposed methods have a sufficient degree of accuracy and quick convergence towards the exact solution.

Conclusions
The LTDM and VITM were used for solving time fractional Newell-Whitehead-Segel equation. The solution we obtained is a series that quickly converges to exact solutions. Four cases are studied, which shows that the proposed methods solutions strongly agree with the exact solution. It is found that the suggested techniques are easy to implement and need a small number of calculations. This shows that LTDM and VITM are very efficient, effective, and powerful mathematical tools easily applied in finding approximate analytic solutions for a wide range of real-world problems arising in science and engineering.