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

Authors: Muhammad Imran Liaqat; Adnan Khan; Md. Ashraful Alam; M. K. Pandit; Sina Etemad; Shahram Rezapour
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

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.


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## Body

## 1. Introduction

While the most well-known fractional derivatives, particularly Caputo and Riemann–Liouville (R-L), are used by many mathematicians [1–4], 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. This 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 [8–12].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. The 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. The 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 [13–15]. 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. The 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. The 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 [23–26].The main advantages of the recommended transform can be summarized as follows:(i)
The natural, the Sumudu, and the Elzaki integral transforms are all more difficult to comprehend than the suggested transform.(ii)
When variablev=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].The 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. The 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]:(1)TταΦx,τ=qDx2Φx,τ+kΦx,τ−gΦnx,τ,where Tτα is CFD, n is a positive integer, and q,k,andg∈Rq>0,τ≥0,0<α≤1, and x∈R. Here, Φx,τ possibly engaged employing the velocity of a fluid or temperature distribution in a thin and infinitely long pipe.The 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. The 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 time-fractional NWS equation. To solve NWS equations, Benattia and Belghaba [39] employed the conformable Sumudu transform and the Adomian decomposition approach. The 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). The results obtained by the recommended method are in very good agreement with the results already existing in the literature. The 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.This paper is schematized as it goes along with Section2, where we recall considerable elementary explanations and significant proceedings utilizing the CST and CFD. The 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.


## 2. 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. The 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 transform (ST) and conformable fractional derivative (CFD) senses, which we will utilize in this paper.Definition 1 (see8).
A functionΦ:[0,∞)⟶R was specified as well the CFD of Φ order α is demarcated employing(2)TταΦτ=lim∈⟶0Φτ+∈τ1−α−Φτ∈,
Forτ>0, and α∈(0,1], if Φ is α-differentiable in some 0,θθ>0, and limτ⟶0+TαΦτ occurs subsequently, it is demarcated as(3)TαΦ0=limτ⟶0+TαΦτ.Definition 2 (see22).
Shehu transform (ST) is a novel integral transformation that is expounded for the function of exponential order. We employ a function belonging to the set'ϒ′ explicated through ϒ=Φτ:∃,ℓ1,ℓ2>0,Φτ<ηeτ/ℓi,ifτ∈[0,∞).
The ST which is characterized throughS. for a function Φτ is demarcated by way of(4)SΦτ=Yu,v=∫0∞Φτe−uτ/vdτ,τ>0.
The S-T of a functionΦτ is Yu,v, then Φτ is termed the inverse of Yu,v that is stated as S−1Yu,v=Φτ,forτ≥0,S−1 which is called inverse S-T.Definition 3.
The Laplace transform of a functionΦττ>0 is defined as(5)LΦτ=Fu=∫0∞e−uτΦτdτ,where Fu is the Laplace transform of Φτ.Definition 4.
32 The Sumudu transform of a function Φτ is defined as(6)SΦτ=Pv=∫0∞e−τΦvτdτ,where Pv is the Sumudu transform of Φτ.Definition 5.
48 Let 0<α≤1, and Φ:[0,∞)⟶R be a real value function. Subsequently, the CST of order α is defined by(7)SαΦ=Yu,v=∫0∞e−u/vτα/αΦτα−1dτ.Definition 6.
19 The Mittag–Leffler function is defined as follows:(8)Eατ=∑i=0∞τiΓαi+1,α∈C,Reα>0.Theorem 1 (see [8]).
Let0<α≤1,Φ1, andΦ2beα−differentiable at a pointτ>0 subsequently.(i)
Tτατc=cτc−α,∀c∈R.(ii)
Tταc=0,wherec∈R.(iii)
TταΦ1Φ2=Φ1TταΦ2+Φ2TταΦ1.(iv)
TταΦ1/Φ2=Φ2TταΦ1−Φ1TταΦ2/Φ22.(v)
Tταc1Φ1+c2Φ2=c1TταΦ1+c2TταΦ2,∀c1,c2∈R.Theorem 2 (see48).
LetΦ:[0,∞)⟶R be a real value function and 0<α≤1, then we have(9)SαTταΦx,τ=uvSαΦx,τ−Φx,0.Theorem 3 (see48).
Supposingaandc∈Rand0<α≤1, we have the following:(i)
Sαc=cv/u.(ii)
Sαexpaτα/α=Yu,v=v/u−av,u/v>0.(iii)
Sαsinaτα/α=Yu,v=av2/u2+a2v2,u/v>0.(iv)
Sαcosaτα/α=Yu,v=uv/u2+a2v2,u/v>0.(v)
Sαsinhaτα/α=Yu,v=av2/u2−a2v2,u/v>a.(vi)
Sαcoshaτα/α=Yu,v=uv/u2−a2v2,u/v>a.(vii)
Sατc=Yu,v=αc/αv/uc/α+1Γ1+c/α.
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.


## 3. 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:(10)TταΦx,τ+DxnΦx,τ+LΦx,τ+NΦx,τ=Mx,τ,τ>0,x>0,0<α≤1,Subject to the initial conditions,(11)Φx,0=fx,where Tτα represents the CFD of order α in τ,Dxn is the uppermost order linear classical derivative in x, L represents further linear expression through lesser derivatives, N demonstrates nonlinear expression, and Mx,τ is the nonhomogenous part.Now, applying the CST to (10), we have(12)SαTταΦx,τ+DxnΦx,τ+LΦx,τ+NΦx,τ=SαMx,τ.By using the linear property of CST, (12) becomes as follows:(13)SαTταΦx,τ+SαDxnΦx,τ+SαLΦx,τ+NΦx,τ=SαMx,τ,SαTταΦx,τ+SαDxnΦx,τ+SαLΦx,τ+NΦx,τ=SαMx,τ.We get the following by utilizing Theorem2 in equation (13):(14)SαΦx,τ=vuSαMx,τ+vuΦx,0−vuSαDxnΦx,τ−vuSαLΦx,τ+NΦx,τ.When inverse CST is implemented, (14) becomes as follows:(15)Φx,τ=Sα−1vuSαMx,τ+vuΦx,0−Sα−1vuSαDαnΦx,τ−Sα−1vuSαLΦx,τ+NΦx,τ.So, according to the Shehu decomposition method (SDM), we can acquire the solutionΦx,τ to (10) as follows:(16)Φx,τ=∑i=0∞Φix,τ.The nonlinear operatorNΦx,τ is decomposed as(17)NΦx,τ=∑i=0∞Ai,where(18)Ai=1i!didƛiN∑κ=0iƛκΦκx,τƛ=0,i=0,1,2,….Now, by replacing (16) and (17) with (15), we attain as follows:(19)∑i=0∞Φix,τ=Sα−1vuSαMx,τ+vuΦx,0−Sα−1vuSαDxn∑i=0∞Φix,τ−Sα−1vuSαL∑i=0∞Φix,τ+N∑i=0∞Ai.Equating both sides of (19), we come to be(20)Φ0x,τ=Sα−1vuSαMx,τ+vuΦx,0.Also,(21)Φi+1x,τ=−Sα−1vuSαDxn∑i=0∞Φix,τ−Sα−1vuSαL∑i=0∞Φix,τ+N∑i=0∞Ai,where i=0,1,2,….The following theorem clarifies and governs the condition for convergence of the expansion solution, (16).Theorem 4.
LetB represent a Banach space; subsequently, the expansion result of Φx,τ converges uncertainty; there occurs z,0<z<1, so that Φx,τi≤zΦi−1x,τ, for all i∈Ν.Proof.
Deliberate the subsequent succession(22)Hix,τ=Φ0x,τ+Φ1x,τ+Φ2x,τ+⋯+Φix,τ.
It is essential to validate that successions ofith partial sums Hix,τ are a Cauchy series in Banach space. Intended for this, we contemplate the following:(23)Hi+1x,τ−Hix,τ≤Φi+1x,τ≤zΦix,τ≤z2Φi−1x,τ…≤zi+1Φ0x,τ.
For everyi,j∈Ν, i≤j, we get as follows:(24)Hix,τ−Hjx,τ=Hj+1x,τ−Hjx,τ+Hj+2x,τ−Hj+1x,τ+⋯+Hix,τ−Hi+1x,τ.
(24) becomes as follows when the triangle inequality is used:(25)Hix,τ−Hjx,τ≤Hj+1x,τ−Hjx,τ+Hj+2x,τ−Hj+1x,τ+⋯+Hix,τ−Hi+1x,τ.
Inequality (25) can be expressed in the following way:(26)Hix,τ−Hjx,τ≤zj+1Φ0x,τ+zj+2Φ0x,τ+⋯+ziΦ0x,τ.
By simple calculation, inequality (26) can be written as(27)Hix,τ−Hjx,τ≤zj+11+z+z2+⋯+zi−j−1Φ0x,τ,as 1−zi−j/1−z=1+z+z2+⋯+zi−j−1.
As a result, inequality (27) is as follows:(28)Hix,τ−Hjx,τ≤zj+11−zi−j1−zΦ0x,τ.
So,0<z<1, and 1−zi−j≤1.
Therefore, from inequality (28), we get as (29)Hix,τ−Hjx,τ≤zi+11−zΦ0x,τ.
As a result,Φ0x,τ is bounded, and we have(30)limi,j⟶∞Hix,τ−Hjx,τ=0.
Therefore,Hi 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.
LetΦx,τ be the approximate solution of the truncated finite series ∑i=0κΦix,τ. Assume it is attainable to acquire a real number z∈0,1 in order that Φi+1x,τ≤zΦix,τ∀i∈Ν. Furthermore, the utmost absolute error is(31)Φx,τ−∑i=0κΦix,τ≤zκ+11−κΦ0x,τ.Proof.
Let the series∑i=0κΦix,τ be finite, then(32)Φx,τ−∑i=0κΦix,τ=∑κ+1∞Φix,τ≤∑κ+1∞Φix,τ≤∑κ+1∞zκΦ0x,τ≤zκ+11+z+z2+z3+⋯Φ0x,τ≤zκ+11−zΦ0x,τ.
This proof is complete.
In the next section, we determine the appropriateness of the recommended method.


## 4. 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:(33)TταΦx,τ=Dx2Φx,τ−2Φx,τ,0<α≤1.
Subject to the initial condition,(34)Φx,0=fx=ex.
Using the conformable Shehu transform on (30), we get(35)SαTταΦx,τ=SαDx2Φx,τ−2Φx,τ.
Using Theorem2, equation (32) is transformed as follows:(36)SαΦx,τ=vu+2vex+vu+2vSαDx2Φx,τ.
By using the inverse conformable Shehu transform, (33) becomes as(37)Φx,τ=Sα−1vu+2vex+Sα−1vu+2vSαDx2Φx,τ.
By using the procedure of CSDM, as explained in Section3, the expansion solution of (34) can be represented by the expansion form as follows:(38)Φx,τ=∑i=0∞Φix,τ.
We get as by substituting (35) into (34),(39)∑i=0∞Φix,τ=Sα−1v2v+uex+Sα−1v2v+uSαDα2∑i=0∞Φix,τ.
Using the approach outlined in Section3, we can get the following from:(40)Φ0x,τ=Sα−1vu+2vex.
We obtain as a result of applying the 2nd part of Theorem 3,(41)Φ0x,τ=e−2τα/αex,Φ0x,τ=ex−2τα/α.
We can also extract the following from (36) using the methodology discussed in Section 3:(42)Φi+1x,τ=Sα−1v2v+uSαDα2Φix,τ,i=0,1,2,….
By repeating the iteration process in (39), we obtain the following results:(43)Φ1x,τ=τααex−2τα/α,Φ2x,τ=τ2α2α2ex−2τα/α,Φ3x,τ=τ3α6α3ex−2τα/α,Φ4x,τ=τ4α24α4ex−2τα/α,Φ5x,τ=τ5α120α5ex−2τα/α.
As a result, the series solution can be found as(44)Φx,τ=Φ0x,τ+Φ1x,τ+Φ2x,τ+⋯.Φx,τ=ex−2τα/α+τααex−2τα/α+τ2α2α2ex−2τα/α+τ3α6α3ex−2τα/α+τ4α24α4ex−2τα/α+τ5α120α5ex−2τα/α+⋯.Φx,τ=ex−2τα/α1+ταα+τ2α2α2+τ3α6α3+τ4α24α4+τ5α120α5+⋯.Φx,τ=ex−2τα/αeτα/α.Φx,τ=ex−τα/α.
Whenα=1, we get a closed-form solution of (30) in the following form:(45)Φx,τ=ex−τ.
Figure1(a) shows the behavior of the 5th 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 5th 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.
Table1 displays absolute and relative errors at reasonable nominated grid points in the interval τ∈0,1 amongst the 5th 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. The 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.Figure 1
(a) The behavior of the 5th approximate and exact results. (b) The absolute errors of Example 1. (c) The absolute errors of Example 1.
(a)(b)(c)Table 1
The absolute and relative errors of Example1.
τΦx,τΦ5x,τRel.error=Φ−Φ5/ΦAbs.error=Φ−Φ50.12.4596031111569500122.4596031080212050110.00000000127489877120.00000000313574499130.22.2255409284924680232.2255407617804353320.00000007490854492310.00000016671203262350.32.0137527074704766132.0137511297524755330.00000078347157288140.00000157771800113180.41.8221188003905089311.8221114341414937230.00000404268317391110.00000736624901521290.51.6487212707001282131.6486979166666667420.00001416493732238130.00002335403346154210.61.4918246976412703311.4917667311842309130.00003885607815123310.00005796645703948230.71.3498588075760032131.3497372847160471340.00009002634888478560.00012152285995603530.81.2214027581601699321.2211776016963252430.00018434252120394760.00022515646384468340.91.1051709180756475231.1047912974871723320.00034349491310918430.00037962058847518211.01.0000000000000000230.9994058153511914110.00059418464880857440.0005941846488085602Table 2
The recurrence errorsΦ5x,τ−Φ4x,τ of the fifth approximate solution with different values of α, when x=1 for Example 1.
τα=0.7α=0.8α=0.9α=1.00.20.0001909996045321340.0000554854093328720.00001628587102969450.000004858983467708020.40.0012118933507937540.0005324288162665870.00023447008487459710.000104226368696334550.60.0030573659449498730.0017011921718397160.00094661555294280130.000530537527994532200.80.0053584300245729840.0034977782538910210.00228194118048423500.001498621640960755501.00.0077407253582818230.0056744934119778870.00415720881965411900.00306566201027972830Example 2.
Consider the conformable nonlinear fractional NWSE as follows:(46)TταΦx,τ=5Dx2Φx,τ+2Φx,τ+Φ2x,τ,0<α≤1.
With the initial condition,(47)Φx,0=ϕ.
Using the CST on (43),(48)SαTταΦx,τ=Sα5Dx2Φx,τ+2Φx,τ+Φ2x,τ.
By using Theorem2 and making some simple calculations, we get the following from equation (45):(49)SαΦx,τ=vu−2vϕ+vu−2vSα5Dx2Φx,τ+vu−2vSαΦ2x,τ.
Using inverse CST on (46) yields the following:(50)Φx,τ=Sα−1vu−2vϕ+vu−2vSα5Dx2Φx,τ+Sα−1vu−2vSαΦ2x,τ.
Suppose that (47) has the following expansion solution by using the procedure discussed in Section 3:(51)Φx,τ=∑i=0∞Φix,τ.
Using (48), in (47), we get as follows:(52)∑i=0∞Φix,τ=Sα−1vu−2vϕ+Sα−1vu−2vSα5Dx2∑i=0∞Φix,τ+Sα−1vu−2vSα∑i=0∞Ai.
The nonlinear term is represented by the following Adomian polynomials:(53)Ai=1i!didƛiN∑κ=0iƛκΦκx,τƛ=0,i=0,1,2,….
A few Adomian polynomial components obtained from (50) are as follows:(54)A0=Φ02x,τ,A1=2Φ0x,τΦ1x,τ,A2=2Φ0x,τΦ2x,τ+Φ12x,τ,A3=2Φ0x,τΦ3x,τ+2Φ1x,τΦ2x,τ.
Utilizing these Adomian polynomials in (49) and employing the iterative process of CSDM, we get as follows:(55)Φ0x,τ=ϕe2τα/α,Φ1x,τ=12ϕ2e2τα/αe2τα/α−1,Φ2x,τ=122ϕ3e2τα/αe2τα/α−12,Φ3x,τ=123ϕ4e2τα/αe2τα/α−13,Φ4x,τ=124ϕ5e2τα/αe2τα/α−14,Φ5x,τ=125ϕ6e2τα/αe2τα/α−15.
As a result, the series solution is as follows:(56)Φx,τ=Φ0x,τ+Φ1x,τ+Φ2x,τ+⋯,Φx,τ=ϕe2τα/α+12ϕ2e2τα/αe2τα/α−1+122ϕ3e2τα/αe2τα/α−12+123ϕ4e2τα/αe2τα/α−13+124ϕ5e2τα/αe2τα/α−14+125ϕ6e2τα/αe2τα/α−15+⋯.
(53) can be written as follows:(57)Φx,τ=ϕe2τα/α(1+12ϕe2τα/α−1+12ϕe2τα/α−12+12ϕe2τα/α−13+12ϕe2τα/α−14+12ϕe2τα/α−15+⋯).
The (54) can be written as follows:(58)Φx,τ=ϕe2τα/α11−1/2ϕe2τα/α−1,Φx,τ=2ϕe2τα/α2+ϕ1−e2τα/α.
For,α=1, we get the closed-form solution of (43) in the following form:(59)Φx,τ=2ϕe2τ2+ϕ1−e2τ.
Figure2(a) illustrates the performance of the 5th 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 5th 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.
Table3 displays absolute and relative errors at suitable chosen grid points in the interval τ∈0,1 between the 5th 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.Figure 2
(a) The behavior of the 5th approximate and exact results. (b) The absolute errors of Example 2. (c) The absolute errors of Example 2.
(a)(b)(c)Table 3
The absolute and relative errors of Example2.
τΦx,τΦ5x,τRel.error=Φ−Φ5/ΦAbs.error=Φ−Φ50.10.12350752110735164010.1235075211071243300.00000000000184040840.000000000000227304200.20.15294354028701950010.1529435402531964200.0000000002.211475100.00000000003.382308300.30.19002295211085280020.1900229511941394000.00000000482422459730.000000000916713399000.40.23708176305009931030.2370817504989038100.00000005294036680800.00000001.255119549900.50.29737706302579140010.2973769434360635000.00000040214845987600.000000119589727898000.60.37558132123693270030.3755804058952446610.00000243713314880080.0000009.15341688056020.70.47863650900689303040.0000060822419489220.00001270743421044820.000006082241948923700.80.61731693555919860020.0000368053097749020.00005962141592885480.000036805309774901000.90.80929830249991450040.8090886524298937010.00025905166163480280.000209650070020828141.01.08575219362268150051.0845982862445835010.00106277232030990950.00115390737809795140Table 4
The recurrence errorsΦ5x,τ−Φ4x,τ of the fifth approximate solution with different values of α, when ϕ=0.10 for Example 2.
τα=0.7α=0.8α=0.9α=1.00.20.000000649904853801130.00000006.02803790605760.000000007.97190011780.000000001.34158898100.40.000074056336466595060.0000070393785388145850.000001009361079765500.000000192275834724190.60.002428311904951922700.0002326373793077226300.000034570982701091520.000006975137270680700.80.046349402882127376000.0045490798533370290000.000703274716853774600.000149407732889476801.00.647828892995555600000.0665701580303999200000.010873548354564798000.00245822987732897180Example 3.
Consider the conformable nonlinear fractional NWSE as follows:(60)TταΦx,τ=Dx2Φx,τ+2Φx,τ−3Φ2x,τ,0<α≤1.
Subject to the initial conditions,(61)Φx,0=β.
By utilizing the CST on (57), we get as(62)SαTταΦx,τ=SαDx2Φx,τ+2Φx,τ−3Φ2x,τ.
Using Theorem2 and some calculations based on equation (59), we get to the following:(63)SαΦx,τ=vu−2vβ+vu−2vSαDx2Φx,τ−3vu−2vSαΦ2x,τ.
By applying inverse CST to (60), we get as(64)Φx,τ=Sα−1vu−2vβ+Sα−1vu−2vSαDx2Φx,τ−Sα−13vu−2vSαΦ2x,τ.
Assume that (61), using the approach outlined in Section 3, yields the following expansion solution:(65)Φx,τ=∑i=0∞Φix,τ.
By using (62), in (61),(66)∑i=0∞Φix,τ=Sα−1vu−2vβ+Sα−1vu−2vSαDx2∑i=0∞Φix,τ−Sα−13vu−2vSα∑i=0∞Ai.
The Adomian polynomials for nonlinear terms are given as(67)Ai=1i!didƛiN∑κ=0iƛκΦκx,τƛ=0,i=0,1,2,….
The following are a few components of Adomian polynomials calculated from (64):(68)A0=Φ02x,τ.A1=2Φ0x,τΦ1x,τ.A2=2Φ0x,τΦ2x,τ+Φ12x,τ.A3=2Φ0x,τΦ3x,τ+2Φ1x,τΦ2x,τ.
If we put the Adomian polynomials into (63), and use the iterative scheme of CSDM, we get as follows:(69)Φ0x,τ=βe2τα/α,Φ1x,τ=32β2e2τα/α1−e2τα/α,Φ2x,τ=322β3e2τα/α1−e2τα/α2,Φ3x,τ=323β4e2τα/α1−e2τα/α3,Φ4x,τ=324β5e2τα/α1−e2τα/α4,Φ5x,τ=325β6e2τα/α1−e2τα/α5.
Hence, the solution to (57) is given by(70)Φx,τ=Φ0x,τ+Φ1x,τ+Φ2x,τ+⋯.Φx,τ=βe2τα/α+32β2e2τα/α1−e2τα/α+322β3e2τα/α1−e2τα/α2+323β4e2τα/α1−e2ταα3+324β5e2τα/α1−e2τα/α4+325β6e2τα/α1−e2τα/α5+⋯,(71)Φx,τ=βe2τα/α(1+32β1−e2τα/α+32β1−e2τα/α2+32β1−e2τα/α3+32β1−e2τα/α4+32β1−e2τα/α5+⋯).
(68) can be represented by the closed-form as follows:(72)Φx,τ=βe2τα/α11−3/2β1−e2τα/α,Φx,τ=2βe2τα/α2−3β1−e2τα/α.
For,α=1, we get a closed-form solution in the following form of (57):(73)Φx,τ=2βe2τ2−3β1−e2τ.
Figure3(a) exemplifies the performance of the 5th 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 5th imprecise and accurate results of (57) at α=1. One can observe the corresponding findings portrayed for Examples 1 and 2.
Table5 displays absolute and relative errors at appropriate preferred grid points in the interval 0,1 among the 5th 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.Figure 3
(a) The behavior of the 5th approximate and exact solutions. (b) The absolute errors of Example 3. (c) The absolute errors of Example 3.
(a)(b)(c)Table 5
The absolute and relative errors of Example3.
τΦx,τΦ5x,τRel.error=Φ−Φ5/ΦAbs.error=Φ−Φ50.10.062101343564302670.0621013435630008110.00000000002.096338620.000000000001.301853210.20.077448049652299650.0774480494572073720.000000002.51900824510.0000000001.95092275620.30.097092561535772800.0970925562004460700.00000005.495093174620.000000005.335326722020.40.122540418824418450.1225403449296217400.0000006.0302386282830.00000007.3894796712100.50.156020650005630530.1560199353183599210.000004580722299186020.0000007.14687270614310.60.200977734324247650.2009721550879290430.000027760469772277630.0000055792363186091200.70.263030897553600040.2629928249838322510.000144745617803437820.0000380725697677886110.80.352016598222026100.3517775348825888010.000679125190814189710.0002390633394372954000.90.486872434979752000.4854357911112320040.002950760333309475730.0014366438685200489011.00.709366465797620000.7007791300528785020.012105640961031000010.008587335744741464005Table 6
The recurrence errorsΦ5x,τ−Φ4x,τ of the fifth approximate solution with different values of α, when β=0.05 for Example 3.
τα=0.7α=0.8α=0.9α=1.00.20.00000246760749177610.0000002.28877064245620.000000030268308259910.000000005.093845662600.400.0002811826525216030.000026727640389561620.000003832417849734650.0000007.30047309968400.60.00921999676411433200.000883295049559009300.000131261699943206860.0000264837243246158060.80.17598288906807738000.017272287568139030000.002670246190554175600.0005672824858147323001.00.17598289990680773800.017279928756813903000.002680246190554175600.009333591565483440000


## 5. Conclusion

This 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. The 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.



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