Approximate and Closed-Form Solutions of Newell-Whitehead-Segel Equations via Modified Conformable Shehu Transform Decomposition Method

In this study, we introduced a novel scheme to attain approximate and closed-form solutions of conformable Newell-Whitehead-Segel (NWS) equations, which belong to the most consequential amplitude equations in physics. The conformable Shehu transform (CST) and the Adomian decomposition method (ADM) are combined in the proposed method. We call it the conformable Shehu decomposition method (CSDM). To assess the eﬃciency and consistency of the recommended method, we demonstrate 2D and 3D graphs as well as numerical simulations of the derived solutions. As a result, CSDM demonstrates that it is a useful and simple mathematical tool for getting approximate and exact analytical solutions to linear-nonlinear fractional partial diﬀerential equations (PDEs) of the given kind. The convergence and absolute error analysis of the series solutions is also oﬀered.


Introduction
While the most well-known fractional derivatives, particularly Caputo and Riemann-Liouville (R-L), are used by many mathematicians [1][2][3][4], there is some evidence that these operators have limitations, as shown in the following [5,6]. Khalil et al. in 2014 recommended a novel fractional derivative operator, so-called conformable fractional derivative (CFD) [7], that contains variety of the features in an operator that are impossible to be satisfied with the prevailing operators. is operator is relatively similar to the characterization of derivatives in the traditional limit method, and it is relatively effortless to manage. As a result, it was quickly recognized and became the subject of various scientific studies [8][9][10][11][12].
In the context of science and engineering, we derive logical and physical processes that, when viewed through the lens of mathematical representations, occur as differential equations (DEs). For example, the equation of simple harmonic motion, the equation of motion, and the deflection of the beam, and as a consequence, all are categorized through DEs. Subsequently, the explanations of DEs are necessitated. e various DEs emitted by applications become so complicated as a result that it is frequently unreasonable to participate in closed-form results. Numerical methods afford an appreciable alternate means for explaining the DEs under specified early circumstances.
Transforms are interesting in their sense, and they make it easier for researchers to solve mathematical problems such as ordinary, partial, and fractional-order DEs. e Laplace transform and the Fourier transform are two well-known transforms that were used to solve ordinary and partial DEs in the beginning. Later, in the domain of fractional calculus, these transformations were applied to fractional-order DEs [13][14][15]. Many other transformations have been proposed by researchers in recent years, addressing a wide range of mathematical problems. To handle fractional-order DEs, these transformations are combined with additional analytical, numerical, or homotopy-based approaches. e Laplace transform [16], fractional complex transform [17], travelling wave transform [18], Elzaki transform [19], Sumudu transforms [20], and ZZ transforms [21], among others, are still used to solve fractional-order DEs. Recently, Shehu and Zhao [22] introduced a Laplace-type integral transform called the Shehu transform, which is used to solve DEs in the time domain. e proposed integral transform is successfully derived from the classical Fourier integral transform and is applied to both ordinary and partial DEs. Numerous mathematicians have lately become interested in the Shehu transform, which has been employed by many researchers for fractional-order DEs [23][24][25][26].
e main advantages of the recommended transform can be summarized as follows: (i) e natural, the Sumudu, and the Elzaki integral transforms are all more difficult to comprehend than the suggested transform. (ii) When variable v � 1 is used, the recommended transform becomes a Laplace transform, and when the variable u � 1 is used, it becomes a Yang integral transform. (iii) It is a generalization of the Sumudu and the Laplace transforms. (iv) It can be used to solve exactly and numerically fractional-order DEs easily and efficiently.
During the past years, abundant procedures have been put forward to explain fractional-order DEs, including the Elzaki residual power series method [27], natural decomposition method [28], Legendre wavelet method [29], Laplace variational iteration scheme [30], differential transform method [31], homotopy perturbation with the Sumudu transformation approach [32], operational matrix method [33], Aboodh decomposition method [34], residual power series method [35], and the rational symmetric contraction mappings approach [36]. e variations in the sand, the appearances of the seashells, and various further striped shapes happen in many spatial structures that are possibly demonstrated through the amplitude model. e NWS model is one of the most important amplitude models in the practical sciences, and it explains how stripe patterns appear in two-dimensional structures [37,38].
In the current effort, the CSDM is utilized to acquire the approximate and closed-form results of the linear and nonlinear fractional NWS equations of the procedure specified as follows [39]: where T α τ is CFD, n is a positive integer, and q, k, and g ∈ R q > 0, τ ≥ 0, 0 < α ≤ 1, and x ∈ R. Here, Φ(x, τ) possibly engaged employing the velocity of a fluid or temperature distribution in a thin and infinitely long pipe. e solution of the time-fractional Newell-Whitehead-Segel (NWS) in the sense of Caputo and conformable derivative has attracted attention in recent years. Prakash et al. [40] have used the fractional variation iteration method to obtain approximate analytical solutions for the fractional NWS equations in the sense of the Caputo derivative. e variational iteration approach was used by Nadeem et al. [41] to develop approximate and precise solutions to the Caputo time-fractional NWS problem. Using the fractional power series methodology, Ali et al. [42] solved the Caputo timefractional NWS equation. To solve NWS equations, Benattia and Belghaba [39] employed the conformable Sumudu transform and the Adomian decomposition approach. e Sumudu decomposition approach was utilized by Ahmed and Elbadri [43] to determine approximate and definite solutions to the Caputo time-fractional NWS equations. Saadeh et al. [44] obtained the approximate analytical solutions to the fractional-order NWS equations in the sense of Caputo by using the residual power series method. In the work of Prakash and Verma [45], they used the Adomian decomposition method to solve time-fractional NWS equations in the sense of Caputo.
For the first time in research, we established a new procedure by using the conformable Shehu transform (CST) and an Adomian decomposition method (ADM) for solving fractional-order NWS equations in the sense of conformable fractional derivative (CFD). e results obtained by the recommended method are in very good agreement with the results already existing in the literature. e advantage of the CSDM is that it significantly reduces the amount of numerical computation required to find approximate and exact solutions to these types of equations compared to the current methods, especially when compared to the Caputo base methods.
is paper is schematized as it goes along with Section 2, where we recall considerable elementary explanations and significant proceedings utilizing the CST and CFD. e essential suggestion beyond the CSDM and convergence consideration for the conformable time-fractional NWS model is established in Section 3. In Section 4, we verified numerical illustrations of NWS models to illustrate the capability, potential, and uncomplicated nature of the adapted method. To end, in Section 5, significance is congregated in the conclusions.

Fundamental Concepts
Fractional calculus (FC) is a branch of mathematical analysis that studies the integral and derivative of any real or complex order. FC is correspondingly recognized utilizing non-Newtonian calculus and generalized calculus. e FC has developed influential contrivances in numerous domains of physics, engineering, biology, chemistry, image processing, solid-state, stochastic-based finance, control theory, economics, signal, viscoelasticity, and fiber optics by interpreting these problems into mathematical representations [46,47].
In this part, we will elucidate substantial expressive compatible explanations and theorems in the Shehu 2 Mathematical Problems in Engineering transform (ST) and conformable fractional derivative (CFD) senses, which we will utilize in this paper.

Definition 3.
e Laplace transform of a function Φ(τ) τ > 0 is defined as where F(u) is the Laplace transform of Φ(τ).

Definition 4. 32 e Sumudu transform of a function Φ(τ)
is defined as where P(v) is the Sumudu transform of Φ(τ).
Theorem 2 (see 48). Let Φ: [0, ∞) ⟶ R be a real value function and 0 < α ≤ 1, then we have Theorem 3 (see 48). Supposing a and c ∈ R and 0 < α ≤ 1, we have the following: In the succeeding part, we generate the foremost suggestion of the CSDM to obtain the results for linear and nonlinear NWS equations and deliberate on the convergence of the series solution and maximum error analysis.

The Algorithm of the CSDM with Convergence and Error Analysis
We discuss the following conformable PDEs in common operator systems to demonstrate the fundamental conceptualization of CSDM: Subject to the initial conditions, where T α τ represents the CFD of order α in τ, D n x is the uppermost order linear classical derivative in x, L represents further linear expression through lesser derivatives, N demonstrates nonlinear expression, and M(x, τ) is the nonhomogenous part. Now, applying the CST to (10), we have Mathematical Problems in Engineering By using the linear property of CST, (12) becomes as follows: We get the following by utilizing eorem 2 in equation (13): When inverse CST is implemented, (14) becomes as follows: So, according to the Shehu decomposition method (SDM), we can acquire the solution Φ(x, τ) to (10) as follows: e nonlinear operator N(Φ(x, τ)) is decomposed as where Now, by replacing (16) and (17) with (15), we attain as follows: Equating both sides of (19), we come to be Also, where i � 0, 1, 2, . . .. e following theorem clarifies and governs the condition for convergence of the expansion solution, (16).
Proof. Deliberate the subsequent succession It is essential to validate that successions of ith partial sums H i (x, τ) are a Cauchy series in Banach space. Intended for this, we contemplate the following: For every i, j ∈ Ν, i ≤ j, we get as follows: (24) becomes as follows when the triangle inequality is used: Inequality (25) can be expressed in the following way: By simple calculation, inequality (26) can be written as As a result, inequality (27) is as follows: So, 0 < z < 1, and 1 − z i− j ≤ 1. erefore, from inequality (28), we get as As a result, Φ 0 (x, τ) is bounded, and we have erefore, H i is a Cauchy series in Banach space, so the expansion solution (16) converges.
In the subsequent theorem, we offer an absolute error investigation of the proposed procedure. □ Theorem 5. Let Φ(x, τ) be the approximate solution of the truncated finite series κ i�0 Φ i (x, τ). Assume it is attainable to acquire a real number z ∈ (0, 1) in order that ‖Φ i+1 (x, τ)‖ ≤ z‖Φ i (x, τ)‖ ∀i ∈ Ν. Furthermore, the utmost absolute error is Proof. Let the series κ i�0 Φ i (x, τ) be finite, then is proof is complete. In the next section, we determine the appropriateness of the recommended method.

Numerical Examples and Concluding Remarks
In this section, three problems with conformable NWS equations are recognized to illustrate the performance and appropriateness of the suggested method.
Example 1. Consider the linear conformable fractional NWSE as follows: Subject to the initial condition, Using the conformable Shehu transform on (30), we get Using eorem 2, equation (32) is transformed as follows: By using the inverse conformable Shehu transform, (33) becomes as By using the procedure of CSDM, as explained in Section 3, the expansion solution of (34) can be represented by the expansion form as follows: We get as by substituting (35) into (34), Using the approach outlined in Section 3, we can get the following from: We obtain as a result of applying the 2 nd part of eorem 3, We can also extract the following from (36) using the methodology discussed in Section 3:

Mathematical Problems in Engineering
By repeating the iteration process in (39), we obtain the following results: As a result, the series solution can be found as When α � 1, we get a closed-form solution of (30) in the following form: Figure 1(a) shows the behavior of the 5 th term approximate and exact solutions of (30) at several values of α, the approximate result corresponds with the precise result at α � 1, and this admits the effectiveness and precision of the suggested method. Figures 1(b) and 1(c) demonstrate 3D and 2D graphs of absolute errors in the intervals τ, x ∈ [0, 1] and τ ∈ [0, 1] over the 5 th approximate and accurate solutions of (30) at α � 1, respectively. As of the figures, the approximate solution is in a preeminent compact with the precise solution.
Error functions are available to distinguish the precision and capability of the scheme. To indicate the accuracy and capability of CSDM, we selected residual, recurrence, and absolute errors functions. Table 1 displays absolute and relative errors at reasonable nominated grid points in the interval τ ∈ [0, 1] amongst the 5 th approximate and precise solutions of (30) at α � 1, when x � 1, attained using CSDM. From Table 1, it can be perceived that the approximate solutions are in eminent contract with the precise solution, and this sanctions the efficiency of the recommended method. e convergence of the CSDM approximate solution to the exact solution for (30) has been shown numerically, as in Table 2. From the obtained results, it is evident that the present technique is an effective and convenient algorithm to solve certain classes of fractional order DEs with fewer calculations and iteration steps.
Example 2. Consider the conformable nonlinear fractional NWSE as follows: With the initial condition, Using the CST on (43), By using eorem 2 and making some simple calculations, we get the following from equation (45): (49) Using inverse CST on (46) yields the following: Suppose that (47) has the following expansion solution by using the procedure discussed in Section 3: (51) Using (48), in (47), we get as follows: e nonlinear term is represented by the following Adomian polynomials: A few Adomian polynomial components obtained from (50) are as follows:

(54)
Utilizing these Adomian polynomials in (49) and employing the iterative process of CSDM, we get as follows: As a result, the series solution is as follows: (53) can be written as follows: (57) e (54) can be written as follows: For, α � 1, we get the closed-form solution of (43) in the following form: 8 Mathematical Problems in Engineering (59) Figure 2(a) illustrates the performance of the 5 th approximate and exact CSDM solutions of (43) for diverse values of α when ϕ � 0.10 in the interval τ ∈ [0, 1]. Indisputably, results in instances of fractional values of α converge to the results in case of α � 1. As well, the imprecise results correspond with the precise solution at α � 1 and this supports the efficiency and exactness of the recommended method. Figures 2(b) and 2(c) describe the 3D and 2D graphs of absolute errors in the intervals τ, x ∈ [0, 1] and τ ∈ [0, 1] when ϕ � 0.10, respectively, over the 5 th approximate and accurate solutions of (43) at α � 1, attained utilizing CSDM. As of the figure, it can be predicted that the approximate solution is very close to the exact solutions which show the efficiency of the CSDM. Table 3 displays absolute and relative errors at suitable chosen grid points in the interval τ ∈ [0, 1] between the 5 th approximate and precise solutions of (43) at α � 1 when ϕ � 0.10 accomplished employing CSDM. Moreover, numerical comparisons are made to validate the accuracy of our method by calculating the recurrence errors for the approximate solution of (43) obtained for various values of when, as shown in Table 4. From Tables 3 and 4, it can be perceived that the approximate solution is in distinguished contract with the precise solution, and this is in agreement with the efficacy of the CSDM.
Example 3. Consider the conformable nonlinear fractional NWSE as follows: Subject to the initial conditions, By utilizing the CST on (57), we get as Using eorem 2 and some calculations based on equation (59), we get to the following: By applying inverse CST to (60), we get as Assume that (61), using the approach outlined in Section 3, yields the following expansion solution: By using (62), in (61), e Adomian polynomials for nonlinear terms are given as e following are a few components of Adomian polynomials calculated from (64): τ). .
Φ(x, τ) � βe 2τ α /α + 3 2 β 2 e 2τ α /α 1 − e 2τ α /α + (71) (68) can be represented by the closed-form as follows: For, α � 1, we get a closed-form solution in the following form of (57): (73) Figure 3(a) exemplifies the performance of the 5 th approximate CSDM solutions of (57), correspondingly for assorted values of α and exact solutions at α � 1. Figures 3(b) and 3(c) represent the 3D and 2D graphs of absolute errors in the interval τ ∈ [0, 1] with β � 0.05, over the 5 th imprecise and accurate results of (57) at α � 1. One can observe the corresponding findings portrayed for Examples 1 and 2. Table 5 displays absolute and relative errors at appropriate preferred grid points in the interval [0, 1] among the 5 th terms in exact and closed-form results of (57) at α � 1 when β � 0.05. Moreover, numerical comparisons are made to validate the accuracy of our method by calculating the recurrence errors for the approximate solution of (57) obtained for various values of α when β � 0.05, as shown in Table 6. From Tables 5 and 6, one can notice the identical conclusions described for Examples 1 and 2.

Conclusion
is paper elaborates an effective novel coupling method of the CST and ADM for the time-fractional NWS equations in the sense of CFD. With the assistance of the CST, the calculation of this method is very effortless and unpretentious. In the examples, it is realized that the five-step imprecise results of the linear-nonlinear complications accord us an exceptionally precise solution. Consequently, this confirms that CSDM is an efficient and uncomplicated mathematical means for gaining the approximate analytical results of the linear and nonlinear conformable PDEs of the presumed kind.
Furthermore, to evaluate the effectiveness and consistency of the proposed method for conformable PDEs, absolute, recurrence, and relative errors of three complications are considered graphically and numerically. e implications demonstrate that the CSDM is more effective and accurate with fewer calculations than existing schemes. Subsequently, we further demonstrate that this procedure can be applied to solve more linear and nonlinear DEs.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.