# Effective Analytical Computational Technique for Conformable Time-Fractional Nonlinear Gardner Equation and Cahn-Hilliard Equations of Fourth and Sixth Order Emerging in Dispersive Media

Authors: Mohammed Al-Smadi; Shrideh Al-Omari; Yeliz Karaca; Shaher Momani
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The aim of this paper is to investigate the approximate solutions of nonlinear temporal fractional models of Gardner and Cahn-Hilliard equations. The fractional models of the Gardner and Cahn-Hilliard equations play an important role in pulse propagation in dispersive media. The time-fractional derivative is observed in the conformable framework. In this orientation, a reliable computationally algorithm is designed and developed by following a residual error and multivariable power series expansion. Basically, the approximate solutions of pulse wave function of the fractional higher-order Gardner and Cahn-Hilliard equations are obtained in the form of a conformable convergent fractional series. Relevant consequences are theoretically and numerically investigated under the conformable sense. Besides, the analysis of the error and convergence of the developed technique are discussed. Some of the unidirectional homogeneous physical applications of the posed models in a finite compact regime are tested to confirm the theoretical aspects, demonstrate different evolutionary dynamics, and highlight the superiority of the novel developed algorithm compared to other existing analytical methods. For this purpose, associated graphs are displayed in two and three dimensions. Growing and decaying modes of the fractional parameters are analyzed for severalα values. From a numerical viewpoint, the simulations and results declare that the proposed iterative algorithm is indeed straightforward and appropriate with efficiency for long-wavelength solutions of nonlinear partial differential equations.


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## Body

## 1. Introduction

Nonlinear parabolic partial differential equations are superable mathematical tools for describing different evolutionary dynamics, long-wave propagation, growing and decaying modes, and phase separation of many nonlinear physical system [1–4]. Many applications of nonlinear temporal evolution and phase field models may be obtained from various engineering topics, for instance, cosmology, gravity waves, aerodynamics, blood flow, thermodynamics, incompressible, and inviscid fluid [5–9]. On the other aspect as well, the fractional partial systems play out a significance role in modeling several fascinating nonlinear physical complex systems and realizing the interactions of particles, basic physics, phase transition, and the process of dynamic that rule such systems. Indeed, they, in the recent past, have been witnessed by scientists owing to its excellent applications in different fields of sciences, including magneto-acoustic propagation in plasma, electromagnetic, chemical kinetics, control theory, quantum mechanics, dissipative systems, gas-solid flows, granular fluids, and hydrodynamics [10–18]. Nevertheless, different types of fractional operators have been moderated by Riemann-Liouville, Atangana-Baleanu, Erdelyi-Kober, Riesz-Caputo, Hadamard, Grünwald-Letnikov, and local-fractional derivatives and conformable. Although the concept of the nonlocal fractional is more acceptable because of the physical long-term features, a deficiency is there as chain, quotient, and Leibniz rules. In this point, the local fractional operators are based on natural generalization of fractional derivatives to avoid the violation of nonnormal rules, keep the local nature of the derivatives, and explore features of certain convergence [19–23].So long, many effective techniques have been successfully developed and implemented to deal with various categories of temporal fractional nonlinear evolution equations, such as variational iteration method, differential transform method, homotopy perturbation method, reproducing kernel method, operational matrix method, and Galerkin finite element method, in addition to many traveling wave techniques, including tan(ϕξ/2)-expansion method, generalized Kudryashov method, tanh-coth method, and exp−ϕϵ method [24–31]. Finding exact traveling wave, approximate and soliton solutions of higher order temporal fractional partial differential equations in nonlinear wave situations are an issue for knowing the dynamics system of dispersive waves in the phase fields. In this framework, we plan to build approximate and accurate analytical solution for a class of nonlinear homogeneous time-fractional parabolic partial differential equations of higher order equipped with appropriate initial conditions in terms of conformable sense using a novel analytical-computational algorithm. The main contribution lies in designing a superb iteration algorithm to obtain accurate approximate solutions of the posed models in the form of a rapid convergence series at a lower cost of calculations. This algorithm is free of linearization, perturbation, and any restrictive assumptions for handling dispersive and nonlinear terms. To begin with, we consider the following well-known model for the nonlinear third-order time fractional Gardner equation [13, 14]
(1)∂tαu+6u−λ2u2ux+uxxx=0,0<α≤1,where λ is nontrivial constant parameter, α signifies the order of time-dependent derivatives of fractional order, and u=ux,t is wave-profile function scaling spatiotemporal durations of x∈a,b and t≥0. Typically, ∂tα represents the variance of u with time and fixed location, and the nonlinear terms uux and u2ux refer to wave steepening while the linear dispersive term uxxx refers to wave effects. Therefore, it has a vital role regarding interactions of dispersion and nonlinearity in soliton theory. Hereinafter, ∂tα stands to the temporal conformable derivative. The aforementioned phase-field model is widely used in several practical applications, such as phase incompressible and inviscid fluids, quantum field theory, curvature flows, quantum mechanics, and gravitational field. Further, it describes a variety of nonlinear wave propagation phenomena in plasma and solid states [13].In this investigation as well, we focus on the fourth and sixth order time-fractional Cahn-Hilliard equations [24, 25]:
(2)∂tαu=μux+−uxx−u+u3xx,0<α≤1,and
(3)∂tαu=μuux+uxx+u−u3xxxx,0<α≤1,where μ is constant parameter with μ≠0. Herein, the nonlinear terms denote the chemical potential of the model, while uxxxx and uxxxxxx denote the dispersive wave effects of the fourth and sixth order system, respectively. This model is profitably used in multiphase incompressible fluid flows, phase ordering dynamics, tumor growth simulation, surface reconstruction, phase separation, image inpainting, spinodal decomposition, and microstructures with elastic inhomogeneity, see [24, 32] for a detailed discussion. Furthermore, the posed models (1)–(3) are involved with initial condition
(4)ux,0=f0x,x∈a,b,where f0x is a smooth analytic function of x.Several types of nonlinear temporal fractional evolution equations have been established in the literature, but several do not assume soliton solutions [33]. Anyhow, the nonlinear time-fractional Gardner and Cahn-Hilliard models are profitably used to describe many nonlinear dispersive wave phenomena arising in nonlinear optics, capillary waves, and plasma physics [34–41]. In [34], nonlocal fourth-order fractional Cahn-Hilliard equation with advection and reaction terms has been considered in the sense of Caputo to investigate the approximate solutions using the homotopy analysis method. Using the new iterative method and q-homotopy analysis method [24], Akinyemi et al. successfully obtained analytical-approximate solutions of the nonlinear fourth and sixth order time-fractional Cahn-Hilliard equations. In [25], homotopy perturbation method has been applied to solve fourth-order Cahn-Hilliard equation with Caputo fractional derivative. Akagi et al. discussed the existence and uniqueness of weak solutions to space-fractional Cahn-Hilliard equation in a bounded domain [35]. Ran and Zhou constructed an implicit difference scheme for the fourth-order time-fractional Cahn-Hilliard equations [36]. In [37], Fourier spectral method has been implemented for time-fractional nonlinear Allen-Cahn and Cahn-Hilliard phase-field models. Prakasha et al. [13] proposed two computational methods for time-fractional Gardner equation and fourth-order Cahn-Hilliard equation in light of Caputo's concept. Arafa and Elmahdy [38] designed residual power series algorithm to solve fractional nonlinear Gardner and Cahn-Hilliard equations under Caputo sense. Hosseini et al. [39] proposed a new technique based on expansion method for finding exact solutions of time-fractional Conformable Cahn-Hilliard equation. Moreover, Jafari et al. [40] developed the fractional subequation method to construct exact-analytical solutions for fractional Cahn-Hilliard model.By and large, no conventional approach can be found to produce analytical solutions, soliton solutions, or traveling wave solutions of closed-form for such nonlinear fractional-types dispersive PDEs. So, there have been found demand for sophisticated reliable methods to find analytical and approximate solutions to such problems. This paper formulates an iterative computational algorithm for creating analytical-approximate solutions of a class of nonlinear higher-order time-fractional parabolic partial differential equations by utilizing a novel fractional parameter, the conformable derivative. Estimation of errors for the said algorithm is derived as well. Indeed, several numerical examples have been checked in one-dimensional space to verify the great flexibility and efficiency of the novel developed algorithm, among which the third-order homogeneous time-fractional Gardner equation and fourth and sixth-order homogeneous time-fractional Cahn-Hilliard equations. For this purpose, a comparison study is performed between the presented method and other existing methods. Hereinafter, some notations and auxiliary results are retrieved. In Section3, an analytical algorithm is expanded to solve nonlinear time-fractional parabolic partial differential equations. In Section 4, certain applications are stated to back up the theoretical concept. Further, several numerical techniques and discussions are reported. In Section 5, some concluding remarks are given.


## 2. Preliminaries and Principal Results

Fractional calculus has been introduced and developed an interesting tool to explain the memory and characteristics of many processes in a variety of fields of pure and applied science. In recent literature, it has been used to formulate many nonlinear partial differential equation systems and exploited to provide a comprehensive and clear explanation of dynamics, dispersion, wave propagation, and evolutionary models in view of spacetime change. In this direction, different fractional derivatives have been suggested to handle such partial equations like Feller, Riemann-Liouville, Caputo-Fabrizio, Riesz, Grünwald, Mittag-Leffler, and conformable concepts [41–44]. Consequently, the conformable operator has been modified as a natural generalization of the standard notation of derivatives [45]. In this portion, the primary concept of conformable fractional derivative and some interesting properties is highlighted. It also briefly illustrates the concept and characteristics of the residual series expansion under the conformable operator to complete the theoretical aspect of this work.Definition 1 (see [45]).
Given a real-valued functionut on 0,∞, the conformable derivative of ut at α∈0,1 is given by
(5)∂tαut=limε⟶0ut+εt1−α−utε,t>0,where ∂tαu0 is understood to mean ∂tαu0=limt⟶0+∂tαut.Definition 2 (see [46]).
Given a real-valued functionut on s,∞, if ut is α-differentiable; then, the α-fractional integral is given by
(6)Isαut=∫stuξξ1−αdξ,t>s≥0,α∈0,1,provided that the integral is Riemann improper.The following results are some of the basic characteristics gained in terms of∂tαut. For additional properties, we refer to [47–50] and the references therein.Lemma 3.
Letα∈0,1 and the functions vt, ut be α-differentiable at a point t∈0,∞. Then, for all real constants a1,a2,a3,a4, the following properties hold:
(i)
∂tαℯ1vt+ℯ2ut=ℯ1∂tαvt+ℯ2∂tαut(ii)
∂tαvtut=vt∂tαut+ut∂tαvt(iii)
∂tαvt/ut=ut∂tαvt−vt∂tαut/u2t,ut≠0(iv)
∂tαta3=a3ta3−α(v)
∂tαa4=0(vi)
Ifut is differentiable, then it also holds that ∂tαut=t1−αdut/dtLemma 4 (see [46]).
Given the real-valued functionsut and vt on 0,∞, let α∈0,1, ut be first order differentiable and α-differentiable and let vt be first order differentiable on the range of ut. So, the use of the known chain rule yield
(7)∂tαu∘vt=t1−αv′tu′vt.Definition 5 (see [47]).
Given a real-valued functionux,t on a,b×s,∞, the αth order conformable partial derivative at a point t∈0,∞ is defined as
(8)∂tαux,t=limε⟶0ux,t+εt−s1−α−ux,tε,α∈0,1.Definition 6 (see [47]).
Given a real-valued functionux,t on a,b×s,∞, the αth order conformable integral is defined as
(9)Isαux,t=∫stux,ξξ−s1−αdξ,α∈0,1.Definition 7 (see [47]).
The fractional series expansion att0 can be defined as follows
(10)∑i=0∞Cixt−t0iα=C0x+C1xt−t0α+C2xt−t02α+⋯,t0>0,where α∈0,1,Cix is the ith unknown coefficient, t∈t0,t0+γ1/α,γ>0, and γ1/α is the radius of convergence.Theorem 8 (see [47]).
Given a real-valued functionux,t on a,b×t0,t0+r1/α, let ux,t has many conformable partial derivatives at any point t∈0,∞ with the following fractional series expansion at t0:
(11)ux,t=∑i=0∞Cixt−t0iα,α>0,t0>0.
Then,Cix,i=0,1,2,⋯, can be calculated by
(12)Cix=∂t0iαux,t0i!αi,in which ∂t0iαux,t0 is the ith conformable partial derivative of ux,t about t0 so that ∂t0iαux,t0=∂t0α.∂t0α⋯∂t0αux,t0 (i-times).


## 3. Fundamentals of Conformable Fractional Residual Series Approach

Conformable fractional residual series (CFRS) technique is a semianalytic computational algorithm specifically developed to deal with emerging partial differential equations in various nonlinear dynamical phenomena. This technique is based on extending the generalized arbitrary order Taylor series and minimizing the residual errors to detect the unknown compounds. It possesses many attractive and stimulating features and remarkable ability to deal with nonlinear terms profitably without putting any constraints or transformation of the governing models. Consequently, it has gained wide popularity and has recently become an exciting focus of research and a hot tool used in various applied and computational sciences [41–44]. In this portion, a new algorithm is developed to obtain accurate approximate solutions of the nonlinear homogeneous higher-order time-fractional parabolic partial differential equation involving initial conditions in a limited space time domain. In this context, let us see the nonlinear generalized time fractional sixth order partial differential equation as follows
(13)∂tαux,t+Nu,u2,u3,ux,uxx,u3x,u4x,ux2,uxx2+u6xx,t=0,0<α≤1,along with the condition
(14)vx,0=f0x,x∈a,b,t≥0,α is the order of conformable time-fractional index, uix=∂iux,t/∂xi,i=3,4,5,6, f0x is a given analytical function, and ux,t is an unknown sufficiently differentiable wave-profile function. Herein, N indicates the nonlinear operator from a Banach space B to itself in terms of uux,uuxx2,ux2uxx,uuxu3x, and u2u4x over a one-dimensional spatiotemporal domain.Based on the proposed algorithm, the solutionux,t of (13) has following form of fractional series expansion at t0=0:
(15)ux,t=∑i=0∞Cixtiαi!αi,t≥t0,provided that vx,0=C0x=f0x. So, the n-term truncated series solution unx,t of ux,t in view of the initial condition (14) can be described as
(16)unx,t=C0x+∑i=1nCixtiαi!αi.Basically, the residual errorRsx,t of model (13) is defined as
(17)Rsx,t=∂tαux,t+Nu,u2,u3,ux,uxx,u3x,u4x,ux2,uxx2+u6xx,t,and thus the n-term truncated residual of Rsx,t is expressed by
(18)Rsnx,t=∂tαunx,t+Nun,un2,⋯,unxx2+un6xx,t,where unkx=∂kunx,t/∂xk, Rsx,t=0=∂tn−1αRsx,t,n=1,2,3,⋯,x∈a,b, 0≤t<T,T≡t0+r1/α, and ∂tn−1αRsnx,tt=0≡0 for each n=1,2,3,⋯.To demonstrate the main steps of the residual series algorithm in finding out the values of unknown parametersCix of the n-term truncated solution (16), set n=1 and equate Rs1x,t to zero at t=0, so that C1x can be acquired. Thereafter, by applying the operator ∂tα on both sides of the resulting relevant equation for n=2, and solving ∂tαRs2x,0=0, the coefficient C2x can be acquired as well. Continuing likewise, the rest of coefficients Cix for each i≥3 of the fractional series expansion (16) can be acquired. To complete the presentation and clarification, the underlying algorithm is dedicated.Algorithm 1
Consider the nonlinear generalized time fractional sixth order partial differential equation (13) along with the initial condition (14). Let ux,t be a solution of model (13) and (14) that has nth order partial derivatives in conformable sense at any point t∈t0,T. Then, to obtain the nth approximation, execute the underlying steps:Step A. Expand the solutionux,t of model (13) about t0=0 as followsux,t=∑i=0∞Cixtiα/i!αi,t≥t0.Step B. Give a definition of thenth-truncated solution of ux,t in view of the initial condition (14) as followsunx,t=C0x+∑i=1nCixtiα/i!αi.Step C. Do truncate thenth residual error of Rsx,t soRsnx,t=∂tαunx,t+Nun,un2,⋯,unxx2+un6xx,t,whereunkx=∂kunx,t/∂xk.Step D. Invoke the series solution obtained in Step B to thenth-truncated residual error obtained in Step C as followsRsnx,t=∂tαC0x+∑i=1nCixtiα/i!αi+NC0x+∑i=1nCixtiα/i!αi,C0x+∑i=1nCixtiα/i!αi2,⋯,C0x+∑i=1nCixtiα/i!αixx2+C0x+∑i=1nCixtiα/i!αi6x.Step E. Employ∂tn−1α for every n=1,2,3,⋯ to the obtained equation in Step D to get∂tn−1αRsmx,t=∂tnαC0x+∑i=1nCixtiα/i!αi+∂tn−1αNC0x+∑i=1nCixtiα/i!αi,C0x+∑i=1nCixtiα/i!αi2,⋯,C0x+∑i=1nCixtiα/i!αixx2+∂tn−1αC0x+∑i=1nCixtiα/i!αi6x.Step F. To obtain the first few terms forCix with the aid of ∂tn−1αRsnx,0=0 execute the following subroutine:F1. In Step E, put n=1, computeRs1x,t and find solution for Rs1x,0=0 to get C1x.F2. Once again, in Step E, set n=2, compute∂tαRs2x,t and find solution for ∂tαRs2x,0=0 to get C2x.F3. Once again, n Step E, setn=3, compute ∂t2αRs3x,tand find solution for ∂t2αRs3x,0=0 to get C3x.F4. Proceed for arbitrary orderk by setting n=k, computing ∂tk−1αRskx,t, and establishing the new equation ∂tk−1αRskx,0=0 to get the kth coefficients Ckx.Step G. Keep the new components in an infinite series form. In fact, the closed form of the solution can be established in such way, that is,ux,t=limk⟶∞ukx,t, if the relation of the pattern is very regular. If it was not the case, the solutions ukx,t can be approximately obtained. Then, Stop.Lemma 9.
Letux,t be the solution of PDEs (13) and (14) that has nth order partial derivatives in conformable sense at any point t∈t0,t0+r1/α and the fractional expansion of equation (15) at t0=0. If there exist ηx>0 so that ∂tn+1αux,ξ≤ηx for all 0<ξ<t, then, the remainder term holds the underlying inequality
(19)Pkx,t≤ηxn+1!αn+1tn+1α,in which Pkx,t=∑k=n+1∞∂tkαux,ξ/αkk!tkα.Corollary 10.
Letux,t and unx,t be respectively the analytic and approximate solutions of PDEs (13) and (14). If there exists ξ∈0,1 so that un+1x,t≤ξunx,t for each x,t∈a,b×t0,T, and f0x<∞ for x∈a,b. Then, unx,t converges to ux,t as soon as n⟶∞.Proof.
Sinceun+1x,t≤ξunx,t for each x,t∈a,b×t0,T, then, u1x,t≤ξu0x,t=ξf0x, and then, u2x,t≤ξ2f0x. Subsequently, we have unx,t≤ξnf0x. This leads to ∑k=n+1∞ukx,t≤f0x∑k=n+1∞λk. Thus, it can be observed that
(20)ux,t−unx,t=∑k=n+1∞ukx,t≤∑k=n+1∞ukx,t≤∑k=n+1∞λkf0x=λn+11−λf0x⟶0forn⟶∞.


## 4. Numerical Experiments and Discussion

Temporal fractional evolution equations are efficient approaches for modeling nonlinear waves and knowing the basic physics, phase separation properties, and evolutionary dynamics that govern these equations. The fractional Gardner and Cahn-Hilliard equations are unidirectional temporal nonlinear parabolic partial differential equations, describe nonlinear wave propagation phenomena, and phase-field separation models [13, 24]. It balances the dispersion and nonlinearity effects of the soliton dynamics. In this portion, the conformable power series algorithm in view of the residual error functions is applied to solve the homogeneous third-order time-fractional Gardner equation as well as the homogeneous fourth and sixth-order time-fractional Cahn-Hilliard equations, which are very common species of higher-order fractional temporal evolution. The simulation of such models is investigated as well. More representative results are introduced with physical interpretations for various fractional parameters to hold up the theoretical framework and produce visualization of wave function behavior. Moreover, different comparisons are produced to justify the effect of our new method. Calculations are performed by Mathematica 12.2 computing system [51].


### 4.1. Solution of Nonlinear Third-Order Fractional Gardner Equation

The one-dimensional nonlinear third-order fractional Gardner equation (FGE) considered in this portion can be presented in view of the conformable time derivative as follows [13, 14]:
(21)∂tαu=−6u−λ2u2ux−uxxx,0<α≤1,along with the underlying initial condition
(22)ux,0=12+12tanhx2,where λ is constant, λ≠0, x∈a,b,t≥0,u=ux,t is a sufficiently differentiable function representing the wave-profile scaling spatiotemporal duration of wave propagation in dispersed media. Typically, the nonlinear terms in this model refer to wave steepening, and uxxx refers to wave scattering. The aforementioned equation defines an indispensable model for different nonlinear physical applications in plasma, surface tension, hydrodynamics, etc. [14]. The exact solution of the posed model when α=1 and λ=1 is given by
(23)ux,t=12+12tanhx−t2.According the CFRS algorithm, the fractional series solutionux,t of the FGE (21) about t=0 can be established as follows
(24)ux,t=∑i=0∞Cixtiααii!,provided that C0x=ux,0=1/21+tanhx/2. Subsequently, the nth fractional series unx,t in view of the initial condition (22) can be truncated as follows
(25)unx,t=12+12tanhx2+∑i=1nCixtiααii!,and the residual error function Rsx,t can be expressed as
(26)Rsx,t=∂tαu+6u−λ2u2ux+uxxx,0<α≤1,in which Rsx,t=0=∂tn−1αRsx,t,n=1,2,3,⋯, for each x∈a,b and t≥0.In this direction as well, thenth truncated error Rsnx,t of Rsx,t can be expressed as
(27)Rsnx,t=∂tαun+6un−λ2un2unx+unxxx,provided that Rsnx,t⟶Rsx,t as n⟶∞, and ∂tn−1αRsnx,tt=0≡0 for each n=1,2,3,⋯.By viewing the representation of the truncated series (25) and minimizing the residual error (27) of the governing equation, the unknown coefficients Cix can be computed for each value of i=1,2,⋯,n in order to obtain the nth approximate solution unx,t. To begin with, the first fractional series solution at n=1 assumes the form
(28)u1x,t=12+12tanhx2+1αC1xtα,as well as the first residual function assumes the form
(29)Rs1x,t=∂tαu1+6u1−λ2u12u1x+u1xxx.Consequently, puttingu1x,t into Rs1x,t to get
(30)Rs1x,t=C1x+6α3αC0x+C1xtααC0′x+C1′xtαα−λ2αC0x+C1xtα+1ααC03x+C13xtα.Herein, with the aid ofRs1x,tt=0=0, it yields
(31)C1x+6C0x1−λ2C0xC0′x+C03x=0,which implies
(32)C1x=−181+4−3λ2coshx+31−λ2sinhxsech4x2.Hence, the first series solutionu1x,t can be read as
(33)u1x,t=12+12tanhx2−18α1+4−3λ2coshx+31−λ2sinhxsech4x2tα.Sequentially, the second truncated seriesu2x,t can be computed for n=2 in (27) such that
(34)Rs2x,t=∂tαu2+6u2−λ2u22u2x+u2xxx,where u2x,t=u1x,t+1/2α2C2xt2α. By employing the conformable operator ∂tα on both sides of equation (34), we get that
(35)∂tαRs2x,t=C2x+6∂tαu2−λ2u22u2x+1αC13x+C23xtα.Consequently, solving the term∂tαRs2x,tt=0=0 in the aforementioned equation with the help of Mathematica's symbolic architecture [51] leads to
(36)C2x=−164sech7x2241−λ2coshx2−622−37λ2+15λ4cosh3x2+24−42λ2+18λ4cosh5x2+206−204λ2sinhx2−129−222λ2+90λ4sinh3x2+25−42λ2+18λ4sinh5x2.Hence, the solutionu2x,t can be given as
(37)u2x,t=12+12tanhx2−18α1+4−3λ2coshx+31−λ2sinhxsech4x2tα−1128α2sech7x2241−λ2coshx2−622−37λ2+15λ4cosh3x2+24−42λ2+18λ4cosh5x2+206−204λ2sinhx2−129−222λ2+90λ4sinh3x2+25−42λ2+18λ4sinh5x2t2α.Continuing in this manner, the truncated seriesu3x,t of the series expansion (25) may be computed by applying n=3 in (27), allowing ∂t2α to act on both sides of the new relevant equation then establishing ∂2αRs3x,t/∂t2t=0=0 with the aid of Mathematica. To finish our process, we can assume that u2x,t is our approximate solution of the FGE (21) along with condition (22). Moreover, the values of Cnx for each n≥3 may be counted likewise. In what follows, the achieved n terms in the form of an infinite series leads to the solution ux,t of the FGEs (21)–(22). Especially, the solution of FGEs (21) and (22) at α=1 and λ=1 can be written in the form
(38)ux,t=12+12tanhx2−14sech2x2t−csch3xsinh4x2t2+1482−coshxsech4x2t3−1384sinh3x2−11sinhx2sech5x2t4+⋯,which agrees with the analytical solution acquired by q-homotopy analysis transform method (q-HATM), fractional natural decomposition method (FNDM) [13], and q-homotopy analysis method (q-HAM) [14], so that
(39)ux,t=12+12tanhx−t2.In what follows, some graphic representations achieved by the presented algorithm for FGEs (21) and (22) are displayed in Figures 1 and 2. At lease three-dimensional surface plots of the exact solution whereas the fourth approximate solution are depicted in Figure 1 for diverse values of α with λ=1 over a large enough spatio temporal domain −20,20×0,3. In Figure 2, the moving and evolutionary dynamics of fractional wave function of FGEs (21) and (22) are provided in 2D graph over −10,10 versus t at λ=1 based on different values of α that are given as α=1,0.75,0.5, and 0.25, respectively. From these graphs, it can be observed the tremendous influence of the fractional parameters α on the solutions’ consistency with respect to the time t. By measuring the absolute error u−u3, the achieved numerical solutions of FGEs (21) and (22) are given in Table 1 for distinct values of x and fixed t=0.2 when α=1 and λ=1 and compared with absolute errors obtained in [13] as well. The efficiency of our method is straightforward from these results. Moreover, Table 2 provides the absolute errors between the exact solutions and the third approximate solutions of FGEs (21) and(22) for different values of fractional order α such as α=1,0.95,0.75,0.5,0.25 at t=0.2 and λ=1. These results show good agreement between the solutions when the fractional values differ.Figure 1
Surface wave behavior ofu4x,t of FGEs (21) and (22) with λ=1 for diverse α: (a) exact, (b) α=0.75, (c) α=0.5, and (d) α=0.25.
(a)(b)(c)(d)Figure 2
Elevation of wave surface ofu4x,t of FGEs (21) and (22) with λ=1 and fixed t for various values of α, in which exact blue, α=1 red, α=0.75 green, α=0.5 yellow, and α=0.25 gray.
(a)(b)(c)(d)Table 1
Comparison of numerical outcomes for FGEs (21) and (22) with =0.2,α=1, and λ=1.
xiux,tu3x,tu−u3u−u3u−1FNDM [13]q-HATM [13]0.10.4750210.4750209.95627×10−72.09596×10−69.95627×10−79.95627×10−70.20.5000000.4999972.61331×10−65.22661×10−62.61331×10−62.61331×10−60.30.5249790.5249754.12217×10−67.85207×10−64.12217×10−64.12217×10−60.40.5498340.5498295.46303×10−69.93579×10−65.46303×10−65.46303×10−60.50.5744430.5744366.58827×10−61.14690×10−56.58827×10−66.58827×10−6Table 2
Absolute errorsu−u3 for different values of fractional order α at t=0.2,λ=1 of FGEs (21) and (22).
xiα=1α=0.95α=0.75α=0.5α=0.250.19.95627×10−71.33546×10−63.56984×10−68.10067×10−62.05633×10−50.22.61331×10−63.01435×10−65.77678×10−63.93725×10−51.30242×10−40.34.12217×10−65.86542×10−62.67443×10−53.11823×10−51.65983×10−40.45.46303×10−66.13259×10−63.87654×10−51.92139×10−43.65432×10−40.56.58827×10−68.43010×10−61.65438×10−51.00045×10−43.76543×10−4


### 4.2. Solution of Nonlinear Fourth-Order Time-Fractional Cahn-Hilliard Equation

The one-dimensional nonlinear fourth-order fractional Cahn-Hilliard (FCH4) equation considered in this portion can be presented in terms of the conformable time derivative as follows [24, 25]:
(40)∂tαu=μux+6uux2+3u2−1uxx−uxxxx,0<α≤1,along with the underlying initial condition
(41)ux,0=tanhx2,where μ is aconstant, μ≠0, x∈a,b,t≥0,u=ux,t is sufficiently differentiable function representing the wave-profile scaling spatiotemporal duration of wave propagation in dispersed media. Herein, the nonlinear terms in this model refer to chemical potential dynamics, and uxxxx refers to wave scattering. This equation has various applications in topology optimization, surface reconstruction, phase separation, phase ordering dynamics, magneto-acoustic propagation in plasma, multiphase incompressible fluid flows, image inpainting, and so forth [25, 32]. The exact solution of the posed model when α=1 and μ=1 is given by
(42)ux,t=tanhx+t2.By performing the CFRS algorithm, the fractional series solutionux,t of the FCH4 equation (40) about t=0 can be constructed as follows
(43)ux,t=∑i=0∞Cixtiααii!,provided that C0x=ux,0 =tanhx/2. Subsequently, the nth fractional series unx,t in view of the initial condition (41) can be truncated by
(44)unx,t=tanhx2+∑i=1nCixtiααii!,and the error function Rsx,t can be expressed as
(45)Rsx,t=∂tαu−μux−6uux2−3u2−1uxx+uxxxx,0<α≤1,in which Rsx,t=0=∂tn−1αRsx,t,n=1,2,3,⋯, for each x∈a,b and t≥0.In this orientation as well, thenth truncated error Rsnx,t of Rsx,t can be expressed as
(46)Rsnx,t=∂tαun−μunx−6ununx2−3un2−1unxx+unxxxx,provided that Rsnx,t⟶Rsx,t as n⟶∞, and ∂tn−1αRsnx,tt=0≡0 for each n=1,2,3,⋯.By viewing the representation of the truncated series (44) and minimizing the residual error (46) of the governing equation, the unknown coefficients Cix can be computed for each value of i=1,2,⋯,n in order to obtain the nth approximate solution unx,t. To begin with, the first fractional series solution at n=1 has the form
(47)u1x,t=tanhx2+1αC1xtα,as well as the first residual function has the form
(48)Rs1x,t=∂tαu1−μu1x−6u1u1x2−3u12−1u1xx+u1xxxx.Consequently, puttingu1x,t into Rs1x,t to get
(49)Rs1x,t=C1x−μC0′x+C1′xtαα−3αC0x+C1xtαα32αC0′x+C1′xtα2+αC0x+C1xtααC0′′x+C1′′xtα+C0′′x+C1′′xtαα+C04x+C14xtαα.Hence, by usingRs1x,tt=0=0, it yields
(50)C1x−μC0′x−6C0xC0′x2+C0″x−3C0″xC0x2+C04x=0,which yields
(51)C1x=μ2sech2x2.Hence, the solutionu1x,t is obtained as
(52)u1x,t=tanhx2+μ2αsech2x2tα.Sequentially, we can compute the seriesu2x,t by assuming n=2 in the nth truncated error (46) such that
(53)Rs2x,t=∂tαu2−μu2x−6u2u2x2−3u22−1u2xx+u2xxxx,where u2x,t=tanhx/2+μ/2αsech2x/2tα+1/2α2C2xt2α. Then, by employing the conformable differential operator ∂tα on both sides of equation (53), we get that
(54)∂tαRs2x,t=C2x−μC1′x−μC2′xtαα−∂tα6u2u2x2+3u22−1u2xx+C1″x+C2″xtαα+C14x+C24xtαα.Now, solving the term∂tαRs2x,tt=0=0 in the above equation with the help of Mathematica's symbolic architecture [51] leads to
(55)C2x=−μ2tanhx2sech2x2.Hence, the solutionu2x,t can be given by
(56)u2x,t=tanhx2+μ2αsech2x2tα−μ2tanhx2sech2x2t2α2α2.Similarly, the truncated seriesu3x,t of the series expansion (44) can be calculate by assuming n=3 in (46), then, by solving the term ∂2αRs3x,t/∂t2t=0=0 with the aid of Mathematica's symbolic architecture [51], we get
(57)C3x=μ32Cosh2x−2sech4x2,which reveals that u3x,t has the form
(58)u3x,t=tanhx2+μ2αsech2x2tα−μ22α2tanhx2sech2x2t2α+μ362α3Cosh2x−2sech4x2t3α.Proceeding likewise, the solutionu4x,t will have the form
(59)u4x,t=tanhx2+μ2αsech2x2tα−μ22α2tanhx2sech2x2t2α+μ362α3Cosh2x−2sech4x2t3α−μ448α4sinh3x2−11sinhx2sech5x2t4α.To close this method, it is assumed thatu4x,t is an approximate solution, and Cnx, n≥5 can be followed likewise. Later, by gathering the terms, ux,t of the posed model (40) and (41) may be predicted. In particular, the solution of FCH4 equation (40) and (41) at α=1 and μ=1 can be written in the form
(60)ux,t=tanhx2+12sech2x2t−12tanhx2sech2x2t2+162Cosh2x−2sech4x2t3−148sinh3x2−11sinhx2sech5x2t4+⋯,which agrees with the analytical solution acquired by homotopy perturbation method (HPM) [25], q-HAM, and new iterative method (NIM) [24], so that
(61)ux,t=tanhx+t2.In the following, 3D graphical simulation ofu4x,t of FCH4 model (40) and (41) with respect to different fractional parameter α are shown in Figure 3 for μ=1 over −20,20×0,2. In Figure 4, 3D surface plots of FCH4 model (40) and (41) are depicted with fix α=0.75 versus μ such that μ=1 and μ=0.75 over the spatiotemporal domain −6,6×0,3. Further, the obtained absolute errors u−u4 are reported in Table 3 and compared to those results provided in [24] at μ=1 and α=1. The superiority of the present method follows from those results.
(i)
Exponential wave solution of FCH4 equationFigure 3
Surface wave behavior ofu4x,t of the FCH4 model (40) and (41) with μ=1 for diverse α: (a) exact, (b) α=0.75, (c) α=0.5, and (d) α=0.25.
(a)(b)(c)(d)Figure 4
Surface plots of FCH4 model (40) and (41) with fix α=0.75 versus μ: (a) μ=1 and (b) μ=0.75.
(a)(b)Table 3
Comparison of numerical results for FCH4 model (21)-(22) with λ=1 and α=1.
tixiux,tu4x,tu−u4u−u4u−1q-HAM [24]NIM [24]0.010.00.0070710.0070712.35697×10−123.33332×10−102.356975×10−121.151971×10−70.10.0776250.0776252.25475×10−122.90466×10−112.823765×10−101.810671×10−70.20.1474110.1474111.96920×10−121.33586×10−115.749512×10−116.167394×10−80.30.2157580.2157581.53980×10−127.13667×10−123.757261×10−111.165205×10−90.050.00.0353410.0353417.36197×10−92.08315×10−77.713501×10−84.940148×10−50.10.1056700.1056707.00209×10−96.62637×10−81.124520×10−68.990891×10−50.20.1749580.1749586.07535×10−93.47246×10−82.387229×10−73.218897×10−50.30.2425550.2425554.70990×10−91.94178×10−81.516340×10−74.548965×10−70.080.00.0565080.0565087.71350×10−81.36502×10−67.361971×10−91.306675×10−50.10.1265960.1265967.30464×10−85.77003×10−71.736922×10−72.224480×10−50.20.1954430.1954436.30668×10−83.22687×10−73.622408×10−87.794449×10−60.30.2624150.2624154.85727×10−81.85099×10−72.328496×10−81.257660×10−70.100.00.0705930.0705922.35226×10−73.33214×10−62.352262×10−79.109940×10−50.10.1404860.1404862.22113×10−71.58103×10−62.722916×10−61.740220×10−40.20.2090060.2090061.91136×10−79.14497×10−75.848640×10−76.321236×10−50.30.2755340.2755341.46559×10−75.31908×10−73.686350×10−78.108096×10−7This segment is an attempt to gain an effective approximate solution to FCH4 equation (40) with the initial condition [24]
(62)ux,0=ⅇλx,where λ is an arbitrary constant with λ≠0.According the CFRS algorithm, thenth fractional series solution unx,t of the FCH4 equation (40) about t=0 in view of the initial condition (62) can be expressed as
(63)unx,t=eλx+∑i=1nCixtiααii!.With the aid of thenth truncated error Rsnx,t of (46), the unknown coefficients Cix of the series expansion (63) can be computed for each value of i=1,2,⋯,n. To achieve this goal, let the first fractional series solution of FCH4 equations (40) and (62) at n=1 takes the form
(64)u1x,t=ⅇλx+1αC1xtα.Now, substituteu1x,t into Rs1x,t and then solve Rs1x,tt=0=0 to get
(65)C1x=λμ−λ1−9ⅇ2λx+λ2ⅇλx.Hence, the solutionu1x,t is
(66)u1x,t=ⅇλx+λαμ−λ1−9ⅇ2λx+λ2ⅇλxtα.Sequentially, substituteu2x,t into the second truncated residual error Rs2x,t, apply the conformable operator ∂tα on both sides of the resulting equation, and solve ∂tαRs2x,tt=0=0 with the aid of Mathematica's symbolic architecture [51] to get
(67)C2x=λ2675λ2ⅇ4λx+λ+λ3−μ2−54λ2λ+7λ3−μⅇ2λxⅇλx,which implies that the second series solution is
(68)u2x,t=ⅇλx1+λαμ−λ1−9ⅇ2λx+λ2tα+λ22α2675λ2ⅇ4λx+λ+λ3−μ2−54λ2λ+7λ3−μⅇ2λxt2α.In the same fashion, the third and fourth series solutions of FCH4 equations (40) and (62) can be obtained successively as follows
(69)u3x,t=ⅇλx1+λαμ−λ1−9ⅇ2λx+λ2tα+λ22α2675λ2ⅇ4λx+λ+λ3−μ2−54λ2λ+7λ3−μⅇ2λxt2α+λ36α3123039λ3ⅇ6λx−675λ241λ+713λ3−15μⅇ4λx+μ−λ1+λ23+81λ3μ2−12μλ+7λ3+λ213+194λ2+757λ4ⅇ2λxt3α,(70)u4x,t=u3x,t+λ424α4972λμ−2λ+7λ3μ2−4μλ+7λ3+λ25+82λ2+365λ4ⅇ2λx+1350λ275μ2−10λμ41+713λ2+λ2613+22898λ2+226477λ4ⅇ4λx−15876λ3732λ+23484λ3−217μⅇ6λx+39110121λ4ⅇ8λx+λ+λ3−μ4ⅇλxt4α.To close the process, we assume thatu4x,t is the approximate solution. Following the same procedure, the values of Cnx,n≥5 can be also computed. Thus, the expression of the series solution ux,t of the FCH4 equation (40) along with condition (62) at α=1 and μ=1 can be written in the form
(71)ux,t=ⅇλx1+λ1−λ1−9ⅇ2λx+λ2t+λ22!λ+λ3−12−54λ2λ+7λ3−1ⅇ2λx+675λ2ⅇ4λxt2+λ33!123039λ3ⅇ6λx−675λ241λ+713λ3−15ⅇ4λx+1−λ1+λ23+81λ3−12λ+7λ3+λ213+194λ2+757λ4ⅇ2λxt3+λ44!972λ1−2λ+7λ31−4λ+7λ3+λ25+82λ2+365λ4ⅇ2λx+1350λ275−10λ41+713λ2+λ2613+22898λ2+226477λ4ⅇ4λx−15876λ3732λ+23484λ3−217ⅇ6λx+39110121λ4ⅇ8λx+λ+λ3−14t4+⋯.In the following, the 3D behaviors of surface wave functionu4x,t of FCH4 model (40) and (62) are displayed in Figure 5 for the parameters μ=1 and λ=−0.05 with respect to α=1 and α=0.75 on −10,10×0,1. While the fractional level curves of u3x,t for FCH4 model (40) and (62) are shown in Figure 6 compared to the third approximate solutions obtained in [24] for fix t=1 on −15,15 for various α values when λ=−0.05 and λ=0.05. Error estimate for the third approximate solutions of FCH4 model (40) and (62) is provided in Table 4 by computing the absolute errors u3−uqHAM and u3−uNIM based on the results achieved by q-HAM and NIM [24] for α=1,μ=1, and λ=0.01. From this comparison, it is evident that the results obtained by CFRS are in good agreement with those presented in the literature.Figure 5
Surface plots of FCH4 model (40)-(62) at μ=1 and λ=−0.05 for diverse α: (a) α=1 and (b) α=0.75.
(a)(b)Figure 6
Fractional level curves ofu3x,t of FCH4 model (40) and (62) with t=1 and μ=1: blue for q-HAM [24], red α=1, pink α=0.75, green α=0.5, yellow α=0.25, gray α=0.1; (a) λ=−0.05 and (b) λ=0.05.
(a)(b)Table 4
Comparison of absolute errors of FCH4 model (40) and (62) with =1,μ=1, and λ=0.01.
tix=1x=5x=20u3−uqHAMu3−uNIMu3−uqHAMu3−uNIMu3−uqHAMu3−uNIM0.58.8741×10−97.7088×10−81.0333×10−88.7056×10−81.8620×10−81.4221×10−71.07.0993×10−86.1670×10−78.2659×10−86.9645×10−71.4896×10−71.1377×10−61.52.3960×10−72.0814×10−62.7898×10−72.3505×10−65.0274×10−73.8396×10−62.05.6794×10−74.9336×10−66.6128×10−75.5716×10−61.1917×10−69.1012×10−62.51.1093×10−69.6360×10−61.2916×10−61.0882×10−52.3275×10−61.7776×10−5


### 4.3. Solution of Nonlinear Sixth-Order Time-Fractional Cahn-Hilliard Equation

The one-dimensional nonlinear sixth-order fractional Cahn-Hilliard (FCH6) equation considered in this portion can be presented in terms of the conformable time derivative as follows [24]:
(72)∂tαu=μuux−18uuxx2−36ux2uxx−24uuxuxxx−3u2−1uxxxx+uxxxxxx,with the condition
(73)ux,0=tanhx2,0<α≤1,μ is a constant, μ≠0, x∈a,b,t≥0,u=ux,t is sufficiently differentiable function representing the wave-profile scaling spatiotemporal duration of wave propagation in dispersed media. Herein, the nonlinear terms in this model refer to chemical potential dynamics, and uxxxxxx refers to wave scattering. This equation has applications in topology optimization, surface reconstruction, phase separation, phase ordering dynamics, magneto-acoustic propagation in plasma, multiphase incompressible fluid flows, image inpainting, and so forth [32].According the CFRS algorithm, thenth fractional series solution unx,t of the FCH6 equation (72) about t=0 in view of the initial condition (73) can be expressed as
(74)unx,t=tanhx2+∑i=1nCixtiααii!,and the residual error function Rsx,t is
(75)Rsx,t=∂tαu−μuux+18uuxx2+36ux2uxx+24uuxuxxx+3u2−1uxxxx−uxxxxxx.For this purpose, thenth truncated error of Rsx,t can be expressed in the form
(76)Rsnx,t=∂tαun−μununx+18ununxx2+36unx2unxx+24ununxunxxx+3un2−1unxxxx−unxxxxxx.Thus, by minimizing the residual error (76) of the governing equation, the unknown coefficients Cix of series expansion (74) for each value of i=1,2,⋯,n can be computed. Subsequently, the series solution at n=1 has the form
(77)u1x,t=tanhx2+1αC1xtα,whereas the first residual function has the form
(78)Rs1x,t=∂tαu1−μu1u1x+18u1u1xx2+36u1x2u1xx+24u1u1xu1xxx+3u12−1u1xxxx−u1xxxxxx.Now, puttingu1x,t into Rs1x,t to get
(79)Rs1x,t=C1x−μα2αC0x+C1xtααC0′x+C1′xtα+36α3αC0′x+C1′xtα2αC0′′x+C1′′xtα+18α3αC0x+C1xtααC0′′x+C1′′xtα2+24α3αC0x+C1xtααC0′x+C1′xtααC03x+C13xtα+3α3αC0x+C1xtα2αC04x+C14xtα−1ααC04x+C14xtα−1ααC06x+C16xtα.By utilizing the factRs1x,tt=0=0, it yields
(80)C1x+36C0′x2C0′′x+C0x18C0′′x2−C0′xμ−24C03x+3C0x2C04x−C04x−C06x=0,which implies that
(81)C1x=μ2sech2x2tanhx2.Therefore, the solutionu1x,t is
(82)u1x,t=tanhx2+μ2αsech2x2tanhx2tα.Sequentially, the second truncated seriesu2x,t can be obtained by setting n=2 in (76) such that
(83)Rs2x,t=∂tαu2−μu2u2x+18u2u2xx2+36u2x2u2xx+24u2u2xu2xxx+3u22−1u2xxxx−u2xxxxxx,where u2x,t=tanhx/2+μ/2αsech2x/2tanhx/2tα+1/2α2C2xt2α. By employing the operator ∂tα on both sides of equation (83), we get
(84)∂tαRs2x,t=C2x−μC0x+C1xtαα+C2xt2α2α2C0′x+C1′xtαα+C2′xt2α2α2−∂tα18u2u2xx2+36u2x2u2xx+24u2u2xu2xxx+3u22−1u2xxxx−C06x−C16xtαα−C26xt2α2α2.Solving the term∂tαRs2x,tt=0=0 with the aid of Mathematica's symbolic architecture [51] leads to
(85)C2x+36C0′x2C1′xC0″x+C0′xC1″x+C0x36C0″xC1″x−C1′xμ−24C03x+24C0′xC13x−C1xC0′xμ−24C03x+63C0″x2+C0xC04x−C14x+3C0x2C14x−C16x=0,which implies that
(86)C2x=μ2ζ1xsech4x2tanhx2,in which ζ1x=μ3−cosh2x−32sech4x/2249−163cosh2x+8cosh22x.Therefore, the series solutionu2x,t can be given as
(87)u2x,t=tanhx2+μ2αsech2x2tanhx2tα+μ4α2ζ1xsech4x2tanhx2t2α.Following the same procedure, the seriesu3x,t of (74) can be computed through setting n=3 in (76) and solving the term ∂2αRs3x,t/∂t2t=0=0 to get C3x as follows
(88)C3x=μ8ζ2xsech6x2tanhx2,where
(89)ζ2x=2μ235−24cosh2x+cosh22x−182−28600273+33907584cosh2x−7525233cosh22x+585152cosh32x−12286cosh42x+32cosh52xsech8x2−48μ2499−96cosh2x−20−89+184cosh2xsech4x2,which implies that u3x,t has the form
(90)u3x,t=tanhx2+μ2αsech2x2tanhx2tα+μ4α2ζ1xsech4x2tanhx2t2α+μ48α3ζ2xsech6x2tanhx2t3α.To close the process, we suppose thatu3x,t is the approximate solution. Then, Cnx, n≥4, can be computed similarly. Anyhow, the n-term sequential solution can be written in the form Unx,t=∑knukx,t as well as the solution ux,t of the FCH6 equation (72) along with condition (73) can be predicted as ux,t=limn⟶∞Unx,t.In Figure7, the behaviors of surface wave function u3x,t for FCH6 model (72) and (73) are presented in 3D with μ=0.01 for diverse α such that α=0.75 and α=0.5. While the surface plots of absolute error for FCH6 model (72) and (73) based on the results obtained in [24] at μ=0.01 are depicted in Figures 8 and 9 for α=0.75 and α=0.5, respectively. From these graphs, it is evident that the achieved results are in good agreement with those obtained in [24].Figure 7
Surface wave behavior ofu3x,t of FCH6 model (72) and (73) with μ=0.01 for diverse α: (a) α=0.75 and (b) α=0.5.
(a)(b)Figure 8
Surface of absolute error for FCH6 model (72) and (73) at μ=0.01 and α=0.75: (a) u2−uqHAM and (b) u2−uNIM.
(a)(b)Figure 9
Surface of absolute error for FCH6 model (72) and (73) at μ=0.01 and α=0.5: (a) u2−uqHAM and (b) u2−uNIM.
(a)(b)


## 4.1. Solution of Nonlinear Third-Order Fractional Gardner Equation

The one-dimensional nonlinear third-order fractional Gardner equation (FGE) considered in this portion can be presented in view of the conformable time derivative as follows [13, 14]:
(21)∂tαu=−6u−λ2u2ux−uxxx,0<α≤1,along with the underlying initial condition
(22)ux,0=12+12tanhx2,where λ is constant, λ≠0, x∈a,b,t≥0,u=ux,t is a sufficiently differentiable function representing the wave-profile scaling spatiotemporal duration of wave propagation in dispersed media. Typically, the nonlinear terms in this model refer to wave steepening, and uxxx refers to wave scattering. The aforementioned equation defines an indispensable model for different nonlinear physical applications in plasma, surface tension, hydrodynamics, etc. [14]. The exact solution of the posed model when α=1 and λ=1 is given by
(23)ux,t=12+12tanhx−t2.According the CFRS algorithm, the fractional series solutionux,t of the FGE (21) about t=0 can be established as follows
(24)ux,t=∑i=0∞Cixtiααii!,provided that C0x=ux,0=1/21+tanhx/2. Subsequently, the nth fractional series unx,t in view of the initial condition (22) can be truncated as follows
(25)unx,t=12+12tanhx2+∑i=1nCixtiααii!,and the residual error function Rsx,t can be expressed as
(26)Rsx,t=∂tαu+6u−λ2u2ux+uxxx,0<α≤1,in which Rsx,t=0=∂tn−1αRsx,t,n=1,2,3,⋯, for each x∈a,b and t≥0.In this direction as well, thenth truncated error Rsnx,t of Rsx,t can be expressed as
(27)Rsnx,t=∂tαun+6un−λ2un2unx+unxxx,provided that Rsnx,t⟶Rsx,t as n⟶∞, and ∂tn−1αRsnx,tt=0≡0 for each n=1,2,3,⋯.By viewing the representation of the truncated series (25) and minimizing the residual error (27) of the governing equation, the unknown coefficients Cix can be computed for each value of i=1,2,⋯,n in order to obtain the nth approximate solution unx,t. To begin with, the first fractional series solution at n=1 assumes the form
(28)u1x,t=12+12tanhx2+1αC1xtα,as well as the first residual function assumes the form
(29)Rs1x,t=∂tαu1+6u1−λ2u12u1x+u1xxx.Consequently, puttingu1x,t into Rs1x,t to get
(30)Rs1x,t=C1x+6α3αC0x+C1xtααC0′x+C1′xtαα−λ2αC0x+C1xtα+1ααC03x+C13xtα.Herein, with the aid ofRs1x,tt=0=0, it yields
(31)C1x+6C0x1−λ2C0xC0′x+C03x=0,which implies
(32)C1x=−181+4−3λ2coshx+31−λ2sinhxsech4x2.Hence, the first series solutionu1x,t can be read as
(33)u1x,t=12+12tanhx2−18α1+4−3λ2coshx+31−λ2sinhxsech4x2tα.Sequentially, the second truncated seriesu2x,t can be computed for n=2 in (27) such that
(34)Rs2x,t=∂tαu2+6u2−λ2u22u2x+u2xxx,where u2x,t=u1x,t+1/2α2C2xt2α. By employing the conformable operator ∂tα on both sides of equation (34), we get that
(35)∂tαRs2x,t=C2x+6∂tαu2−λ2u22u2x+1αC13x+C23xtα.Consequently, solving the term∂tαRs2x,tt=0=0 in the aforementioned equation with the help of Mathematica's symbolic architecture [51] leads to
(36)C2x=−164sech7x2241−λ2coshx2−622−37λ2+15λ4cosh3x2+24−42λ2+18λ4cosh5x2+206−204λ2sinhx2−129−222λ2+90λ4sinh3x2+25−42λ2+18λ4sinh5x2.Hence, the solutionu2x,t can be given as
(37)u2x,t=12+12tanhx2−18α1+4−3λ2coshx+31−λ2sinhxsech4x2tα−1128α2sech7x2241−λ2coshx2−622−37λ2+15λ4cosh3x2+24−42λ2+18λ4cosh5x2+206−204λ2sinhx2−129−222λ2+90λ4sinh3x2+25−42λ2+18λ4sinh5x2t2α.Continuing in this manner, the truncated seriesu3x,t of the series expansion (25) may be computed by applying n=3 in (27), allowing ∂t2α to act on both sides of the new relevant equation then establishing ∂2αRs3x,t/∂t2t=0=0 with the aid of Mathematica. To finish our process, we can assume that u2x,t is our approximate solution of the FGE (21) along with condition (22). Moreover, the values of Cnx for each n≥3 may be counted likewise. In what follows, the achieved n terms in the form of an infinite series leads to the solution ux,t of the FGEs (21)–(22). Especially, the solution of FGEs (21) and (22) at α=1 and λ=1 can be written in the form
(38)ux,t=12+12tanhx2−14sech2x2t−csch3xsinh4x2t2+1482−coshxsech4x2t3−1384sinh3x2−11sinhx2sech5x2t4+⋯,which agrees with the analytical solution acquired by q-homotopy analysis transform method (q-HATM), fractional natural decomposition method (FNDM) [13], and q-homotopy analysis method (q-HAM) [14], so that
(39)ux,t=12+12tanhx−t2.In what follows, some graphic representations achieved by the presented algorithm for FGEs (21) and (22) are displayed in Figures 1 and 2. At lease three-dimensional surface plots of the exact solution whereas the fourth approximate solution are depicted in Figure 1 for diverse values of α with λ=1 over a large enough spatio temporal domain −20,20×0,3. In Figure 2, the moving and evolutionary dynamics of fractional wave function of FGEs (21) and (22) are provided in 2D graph over −10,10 versus t at λ=1 based on different values of α that are given as α=1,0.75,0.5, and 0.25, respectively. From these graphs, it can be observed the tremendous influence of the fractional parameters α on the solutions’ consistency with respect to the time t. By measuring the absolute error u−u3, the achieved numerical solutions of FGEs (21) and (22) are given in Table 1 for distinct values of x and fixed t=0.2 when α=1 and λ=1 and compared with absolute errors obtained in [13] as well. The efficiency of our method is straightforward from these results. Moreover, Table 2 provides the absolute errors between the exact solutions and the third approximate solutions of FGEs (21) and(22) for different values of fractional order α such as α=1,0.95,0.75,0.5,0.25 at t=0.2 and λ=1. These results show good agreement between the solutions when the fractional values differ.Figure 1
Surface wave behavior ofu4x,t of FGEs (21) and (22) with λ=1 for diverse α: (a) exact, (b) α=0.75, (c) α=0.5, and (d) α=0.25.
(a)(b)(c)(d)Figure 2
Elevation of wave surface ofu4x,t of FGEs (21) and (22) with λ=1 and fixed t for various values of α, in which exact blue, α=1 red, α=0.75 green, α=0.5 yellow, and α=0.25 gray.
(a)(b)(c)(d)Table 1
Comparison of numerical outcomes for FGEs (21) and (22) with =0.2,α=1, and λ=1.
xiux,tu3x,tu−u3u−u3u−1FNDM [13]q-HATM [13]0.10.4750210.4750209.95627×10−72.09596×10−69.95627×10−79.95627×10−70.20.5000000.4999972.61331×10−65.22661×10−62.61331×10−62.61331×10−60.30.5249790.5249754.12217×10−67.85207×10−64.12217×10−64.12217×10−60.40.5498340.5498295.46303×10−69.93579×10−65.46303×10−65.46303×10−60.50.5744430.5744366.58827×10−61.14690×10−56.58827×10−66.58827×10−6Table 2
Absolute errorsu−u3 for different values of fractional order α at t=0.2,λ=1 of FGEs (21) and (22).
xiα=1α=0.95α=0.75α=0.5α=0.250.19.95627×10−71.33546×10−63.56984×10−68.10067×10−62.05633×10−50.22.61331×10−63.01435×10−65.77678×10−63.93725×10−51.30242×10−40.34.12217×10−65.86542×10−62.67443×10−53.11823×10−51.65983×10−40.45.46303×10−66.13259×10−63.87654×10−51.92139×10−43.65432×10−40.56.58827×10−68.43010×10−61.65438×10−51.00045×10−43.76543×10−4


## 4.2. Solution of Nonlinear Fourth-Order Time-Fractional Cahn-Hilliard Equation

The one-dimensional nonlinear fourth-order fractional Cahn-Hilliard (FCH4) equation considered in this portion can be presented in terms of the conformable time derivative as follows [24, 25]:
(40)∂tαu=μux+6uux2+3u2−1uxx−uxxxx,0<α≤1,along with the underlying initial condition
(41)ux,0=tanhx2,where μ is aconstant, μ≠0, x∈a,b,t≥0,u=ux,t is sufficiently differentiable function representing the wave-profile scaling spatiotemporal duration of wave propagation in dispersed media. Herein, the nonlinear terms in this model refer to chemical potential dynamics, and uxxxx refers to wave scattering. This equation has various applications in topology optimization, surface reconstruction, phase separation, phase ordering dynamics, magneto-acoustic propagation in plasma, multiphase incompressible fluid flows, image inpainting, and so forth [25, 32]. The exact solution of the posed model when α=1 and μ=1 is given by
(42)ux,t=tanhx+t2.By performing the CFRS algorithm, the fractional series solutionux,t of the FCH4 equation (40) about t=0 can be constructed as follows
(43)ux,t=∑i=0∞Cixtiααii!,provided that C0x=ux,0 =tanhx/2. Subsequently, the nth fractional series unx,t in view of the initial condition (41) can be truncated by
(44)unx,t=tanhx2+∑i=1nCixtiααii!,and the error function Rsx,t can be expressed as
(45)Rsx,t=∂tαu−μux−6uux2−3u2−1uxx+uxxxx,0<α≤1,in which Rsx,t=0=∂tn−1αRsx,t,n=1,2,3,⋯, for each x∈a,b and t≥0.In this orientation as well, thenth truncated error Rsnx,t of Rsx,t can be expressed as
(46)Rsnx,t=∂tαun−μunx−6ununx2−3un2−1unxx+unxxxx,provided that Rsnx,t⟶Rsx,t as n⟶∞, and ∂tn−1αRsnx,tt=0≡0 for each n=1,2,3,⋯.By viewing the representation of the truncated series (44) and minimizing the residual error (46) of the governing equation, the unknown coefficients Cix can be computed for each value of i=1,2,⋯,n in order to obtain the nth approximate solution unx,t. To begin with, the first fractional series solution at n=1 has the form
(47)u1x,t=tanhx2+1αC1xtα,as well as the first residual function has the form
(48)Rs1x,t=∂tαu1−μu1x−6u1u1x2−3u12−1u1xx+u1xxxx.Consequently, puttingu1x,t into Rs1x,t to get
(49)Rs1x,t=C1x−μC0′x+C1′xtαα−3αC0x+C1xtαα32αC0′x+C1′xtα2+αC0x+C1xtααC0′′x+C1′′xtα+C0′′x+C1′′xtαα+C04x+C14xtαα.Hence, by usingRs1x,tt=0=0, it yields
(50)C1x−μC0′x−6C0xC0′x2+C0″x−3C0″xC0x2+C04x=0,which yields
(51)C1x=μ2sech2x2.Hence, the solutionu1x,t is obtained as
(52)u1x,t=tanhx2+μ2αsech2x2tα.Sequentially, we can compute the seriesu2x,t by assuming n=2 in the nth truncated error (46) such that
(53)Rs2x,t=∂tαu2−μu2x−6u2u2x2−3u22−1u2xx+u2xxxx,where u2x,t=tanhx/2+μ/2αsech2x/2tα+1/2α2C2xt2α. Then, by employing the conformable differential operator ∂tα on both sides of equation (53), we get that
(54)∂tαRs2x,t=C2x−μC1′x−μC2′xtαα−∂tα6u2u2x2+3u22−1u2xx+C1″x+C2″xtαα+C14x+C24xtαα.Now, solving the term∂tαRs2x,tt=0=0 in the above equation with the help of Mathematica's symbolic architecture [51] leads to
(55)C2x=−μ2tanhx2sech2x2.Hence, the solutionu2x,t can be given by
(56)u2x,t=tanhx2+μ2αsech2x2tα−μ2tanhx2sech2x2t2α2α2.Similarly, the truncated seriesu3x,t of the series expansion (44) can be calculate by assuming n=3 in (46), then, by solving the term ∂2αRs3x,t/∂t2t=0=0 with the aid of Mathematica's symbolic architecture [51], we get
(57)C3x=μ32Cosh2x−2sech4x2,which reveals that u3x,t has the form
(58)u3x,t=tanhx2+μ2αsech2x2tα−μ22α2tanhx2sech2x2t2α+μ362α3Cosh2x−2sech4x2t3α.Proceeding likewise, the solutionu4x,t will have the form
(59)u4x,t=tanhx2+μ2αsech2x2tα−μ22α2tanhx2sech2x2t2α+μ362α3Cosh2x−2sech4x2t3α−μ448α4sinh3x2−11sinhx2sech5x2t4α.To close this method, it is assumed thatu4x,t is an approximate solution, and Cnx, n≥5 can be followed likewise. Later, by gathering the terms, ux,t of the posed model (40) and (41) may be predicted. In particular, the solution of FCH4 equation (40) and (41) at α=1 and μ=1 can be written in the form
(60)ux,t=tanhx2+12sech2x2t−12tanhx2sech2x2t2+162Cosh2x−2sech4x2t3−148sinh3x2−11sinhx2sech5x2t4+⋯,which agrees with the analytical solution acquired by homotopy perturbation method (HPM) [25], q-HAM, and new iterative method (NIM) [24], so that
(61)ux,t=tanhx+t2.In the following, 3D graphical simulation ofu4x,t of FCH4 model (40) and (41) with respect to different fractional parameter α are shown in Figure 3 for μ=1 over −20,20×0,2. In Figure 4, 3D surface plots of FCH4 model (40) and (41) are depicted with fix α=0.75 versus μ such that μ=1 and μ=0.75 over the spatiotemporal domain −6,6×0,3. Further, the obtained absolute errors u−u4 are reported in Table 3 and compared to those results provided in [24] at μ=1 and α=1. The superiority of the present method follows from those results.
(i)
Exponential wave solution of FCH4 equationFigure 3
Surface wave behavior ofu4x,t of the FCH4 model (40) and (41) with μ=1 for diverse α: (a) exact, (b) α=0.75, (c) α=0.5, and (d) α=0.25.
(a)(b)(c)(d)Figure 4
Surface plots of FCH4 model (40) and (41) with fix α=0.75 versus μ: (a) μ=1 and (b) μ=0.75.
(a)(b)Table 3
Comparison of numerical results for FCH4 model (21)-(22) with λ=1 and α=1.
tixiux,tu4x,tu−u4u−u4u−1q-HAM [24]NIM [24]0.010.00.0070710.0070712.35697×10−123.33332×10−102.356975×10−121.151971×10−70.10.0776250.0776252.25475×10−122.90466×10−112.823765×10−101.810671×10−70.20.1474110.1474111.96920×10−121.33586×10−115.749512×10−116.167394×10−80.30.2157580.2157581.53980×10−127.13667×10−123.757261×10−111.165205×10−90.050.00.0353410.0353417.36197×10−92.08315×10−77.713501×10−84.940148×10−50.10.1056700.1056707.00209×10−96.62637×10−81.124520×10−68.990891×10−50.20.1749580.1749586.07535×10−93.47246×10−82.387229×10−73.218897×10−50.30.2425550.2425554.70990×10−91.94178×10−81.516340×10−74.548965×10−70.080.00.0565080.0565087.71350×10−81.36502×10−67.361971×10−91.306675×10−50.10.1265960.1265967.30464×10−85.77003×10−71.736922×10−72.224480×10−50.20.1954430.1954436.30668×10−83.22687×10−73.622408×10−87.794449×10−60.30.2624150.2624154.85727×10−81.85099×10−72.328496×10−81.257660×10−70.100.00.0705930.0705922.35226×10−73.33214×10−62.352262×10−79.109940×10−50.10.1404860.1404862.22113×10−71.58103×10−62.722916×10−61.740220×10−40.20.2090060.2090061.91136×10−79.14497×10−75.848640×10−76.321236×10−50.30.2755340.2755341.46559×10−75.31908×10−73.686350×10−78.108096×10−7This segment is an attempt to gain an effective approximate solution to FCH4 equation (40) with the initial condition [24]
(62)ux,0=ⅇλx,where λ is an arbitrary constant with λ≠0.According the CFRS algorithm, thenth fractional series solution unx,t of the FCH4 equation (40) about t=0 in view of the initial condition (62) can be expressed as
(63)unx,t=eλx+∑i=1nCixtiααii!.With the aid of thenth truncated error Rsnx,t of (46), the unknown coefficients Cix of the series expansion (63) can be computed for each value of i=1,2,⋯,n. To achieve this goal, let the first fractional series solution of FCH4 equations (40) and (62) at n=1 takes the form
(64)u1x,t=ⅇλx+1αC1xtα.Now, substituteu1x,t into Rs1x,t and then solve Rs1x,tt=0=0 to get
(65)C1x=λμ−λ1−9ⅇ2λx+λ2ⅇλx.Hence, the solutionu1x,t is
(66)u1x,t=ⅇλx+λαμ−λ1−9ⅇ2λx+λ2ⅇλxtα.Sequentially, substituteu2x,t into the second truncated residual error Rs2x,t, apply the conformable operator ∂tα on both sides of the resulting equation, and solve ∂tαRs2x,tt=0=0 with the aid of Mathematica's symbolic architecture [51] to get
(67)C2x=λ2675λ2ⅇ4λx+λ+λ3−μ2−54λ2λ+7λ3−μⅇ2λxⅇλx,which implies that the second series solution is
(68)u2x,t=ⅇλx1+λαμ−λ1−9ⅇ2λx+λ2tα+λ22α2675λ2ⅇ4λx+λ+λ3−μ2−54λ2λ+7λ3−μⅇ2λxt2α.In the same fashion, the third and fourth series solutions of FCH4 equations (40) and (62) can be obtained successively as follows
(69)u3x,t=ⅇλx1+λαμ−λ1−9ⅇ2λx+λ2tα+λ22α2675λ2ⅇ4λx+λ+λ3−μ2−54λ2λ+7λ3−μⅇ2λxt2α+λ36α3123039λ3ⅇ6λx−675λ241λ+713λ3−15μⅇ4λx+μ−λ1+λ23+81λ3μ2−12μλ+7λ3+λ213+194λ2+757λ4ⅇ2λxt3α,(70)u4x,t=u3x,t+λ424α4972λμ−2λ+7λ3μ2−4μλ+7λ3+λ25+82λ2+365λ4ⅇ2λx+1350λ275μ2−10λμ41+713λ2+λ2613+22898λ2+226477λ4ⅇ4λx−15876λ3732λ+23484λ3−217μⅇ6λx+39110121λ4ⅇ8λx+λ+λ3−μ4ⅇλxt4α.To close the process, we assume thatu4x,t is the approximate solution. Following the same procedure, the values of Cnx,n≥5 can be also computed. Thus, the expression of the series solution ux,t of the FCH4 equation (40) along with condition (62) at α=1 and μ=1 can be written in the form
(71)ux,t=ⅇλx1+λ1−λ1−9ⅇ2λx+λ2t+λ22!λ+λ3−12−54λ2λ+7λ3−1ⅇ2λx+675λ2ⅇ4λxt2+λ33!123039λ3ⅇ6λx−675λ241λ+713λ3−15ⅇ4λx+1−λ1+λ23+81λ3−12λ+7λ3+λ213+194λ2+757λ4ⅇ2λxt3+λ44!972λ1−2λ+7λ31−4λ+7λ3+λ25+82λ2+365λ4ⅇ2λx+1350λ275−10λ41+713λ2+λ2613+22898λ2+226477λ4ⅇ4λx−15876λ3732λ+23484λ3−217ⅇ6λx+39110121λ4ⅇ8λx+λ+λ3−14t4+⋯.In the following, the 3D behaviors of surface wave functionu4x,t of FCH4 model (40) and (62) are displayed in Figure 5 for the parameters μ=1 and λ=−0.05 with respect to α=1 and α=0.75 on −10,10×0,1. While the fractional level curves of u3x,t for FCH4 model (40) and (62) are shown in Figure 6 compared to the third approximate solutions obtained in [24] for fix t=1 on −15,15 for various α values when λ=−0.05 and λ=0.05. Error estimate for the third approximate solutions of FCH4 model (40) and (62) is provided in Table 4 by computing the absolute errors u3−uqHAM and u3−uNIM based on the results achieved by q-HAM and NIM [24] for α=1,μ=1, and λ=0.01. From this comparison, it is evident that the results obtained by CFRS are in good agreement with those presented in the literature.Figure 5
Surface plots of FCH4 model (40)-(62) at μ=1 and λ=−0.05 for diverse α: (a) α=1 and (b) α=0.75.
(a)(b)Figure 6
Fractional level curves ofu3x,t of FCH4 model (40) and (62) with t=1 and μ=1: blue for q-HAM [24], red α=1, pink α=0.75, green α=0.5, yellow α=0.25, gray α=0.1; (a) λ=−0.05 and (b) λ=0.05.
(a)(b)Table 4
Comparison of absolute errors of FCH4 model (40) and (62) with =1,μ=1, and λ=0.01.
tix=1x=5x=20u3−uqHAMu3−uNIMu3−uqHAMu3−uNIMu3−uqHAMu3−uNIM0.58.8741×10−97.7088×10−81.0333×10−88.7056×10−81.8620×10−81.4221×10−71.07.0993×10−86.1670×10−78.2659×10−86.9645×10−71.4896×10−71.1377×10−61.52.3960×10−72.0814×10−62.7898×10−72.3505×10−65.0274×10−73.8396×10−62.05.6794×10−74.9336×10−66.6128×10−75.5716×10−61.1917×10−69.1012×10−62.51.1093×10−69.6360×10−61.2916×10−61.0882×10−52.3275×10−61.7776×10−5


## 4.3. Solution of Nonlinear Sixth-Order Time-Fractional Cahn-Hilliard Equation

The one-dimensional nonlinear sixth-order fractional Cahn-Hilliard (FCH6) equation considered in this portion can be presented in terms of the conformable time derivative as follows [24]:
(72)∂tαu=μuux−18uuxx2−36ux2uxx−24uuxuxxx−3u2−1uxxxx+uxxxxxx,with the condition
(73)ux,0=tanhx2,0<α≤1,μ is a constant, μ≠0, x∈a,b,t≥0,u=ux,t is sufficiently differentiable function representing the wave-profile scaling spatiotemporal duration of wave propagation in dispersed media. Herein, the nonlinear terms in this model refer to chemical potential dynamics, and uxxxxxx refers to wave scattering. This equation has applications in topology optimization, surface reconstruction, phase separation, phase ordering dynamics, magneto-acoustic propagation in plasma, multiphase incompressible fluid flows, image inpainting, and so forth [32].According the CFRS algorithm, thenth fractional series solution unx,t of the FCH6 equation (72) about t=0 in view of the initial condition (73) can be expressed as
(74)unx,t=tanhx2+∑i=1nCixtiααii!,and the residual error function Rsx,t is
(75)Rsx,t=∂tαu−μuux+18uuxx2+36ux2uxx+24uuxuxxx+3u2−1uxxxx−uxxxxxx.For this purpose, thenth truncated error of Rsx,t can be expressed in the form
(76)Rsnx,t=∂tαun−μununx+18ununxx2+36unx2unxx+24ununxunxxx+3un2−1unxxxx−unxxxxxx.Thus, by minimizing the residual error (76) of the governing equation, the unknown coefficients Cix of series expansion (74) for each value of i=1,2,⋯,n can be computed. Subsequently, the series solution at n=1 has the form
(77)u1x,t=tanhx2+1αC1xtα,whereas the first residual function has the form
(78)Rs1x,t=∂tαu1−μu1u1x+18u1u1xx2+36u1x2u1xx+24u1u1xu1xxx+3u12−1u1xxxx−u1xxxxxx.Now, puttingu1x,t into Rs1x,t to get
(79)Rs1x,t=C1x−μα2αC0x+C1xtααC0′x+C1′xtα+36α3αC0′x+C1′xtα2αC0′′x+C1′′xtα+18α3αC0x+C1xtααC0′′x+C1′′xtα2+24α3αC0x+C1xtααC0′x+C1′xtααC03x+C13xtα+3α3αC0x+C1xtα2αC04x+C14xtα−1ααC04x+C14xtα−1ααC06x+C16xtα.By utilizing the factRs1x,tt=0=0, it yields
(80)C1x+36C0′x2C0′′x+C0x18C0′′x2−C0′xμ−24C03x+3C0x2C04x−C04x−C06x=0,which implies that
(81)C1x=μ2sech2x2tanhx2.Therefore, the solutionu1x,t is
(82)u1x,t=tanhx2+μ2αsech2x2tanhx2tα.Sequentially, the second truncated seriesu2x,t can be obtained by setting n=2 in (76) such that
(83)Rs2x,t=∂tαu2−μu2u2x+18u2u2xx2+36u2x2u2xx+24u2u2xu2xxx+3u22−1u2xxxx−u2xxxxxx,where u2x,t=tanhx/2+μ/2αsech2x/2tanhx/2tα+1/2α2C2xt2α. By employing the operator ∂tα on both sides of equation (83), we get
(84)∂tαRs2x,t=C2x−μC0x+C1xtαα+C2xt2α2α2C0′x+C1′xtαα+C2′xt2α2α2−∂tα18u2u2xx2+36u2x2u2xx+24u2u2xu2xxx+3u22−1u2xxxx−C06x−C16xtαα−C26xt2α2α2.Solving the term∂tαRs2x,tt=0=0 with the aid of Mathematica's symbolic architecture [51] leads to
(85)C2x+36C0′x2C1′xC0″x+C0′xC1″x+C0x36C0″xC1″x−C1′xμ−24C03x+24C0′xC13x−C1xC0′xμ−24C03x+63C0″x2+C0xC04x−C14x+3C0x2C14x−C16x=0,which implies that
(86)C2x=μ2ζ1xsech4x2tanhx2,in which ζ1x=μ3−cosh2x−32sech4x/2249−163cosh2x+8cosh22x.Therefore, the series solutionu2x,t can be given as
(87)u2x,t=tanhx2+μ2αsech2x2tanhx2tα+μ4α2ζ1xsech4x2tanhx2t2α.Following the same procedure, the seriesu3x,t of (74) can be computed through setting n=3 in (76) and solving the term ∂2αRs3x,t/∂t2t=0=0 to get C3x as follows
(88)C3x=μ8ζ2xsech6x2tanhx2,where
(89)ζ2x=2μ235−24cosh2x+cosh22x−182−28600273+33907584cosh2x−7525233cosh22x+585152cosh32x−12286cosh42x+32cosh52xsech8x2−48μ2499−96cosh2x−20−89+184cosh2xsech4x2,which implies that u3x,t has the form
(90)u3x,t=tanhx2+μ2αsech2x2tanhx2tα+μ4α2ζ1xsech4x2tanhx2t2α+μ48α3ζ2xsech6x2tanhx2t3α.To close the process, we suppose thatu3x,t is the approximate solution. Then, Cnx, n≥4, can be computed similarly. Anyhow, the n-term sequential solution can be written in the form Unx,t=∑knukx,t as well as the solution ux,t of the FCH6 equation (72) along with condition (73) can be predicted as ux,t=limn⟶∞Unx,t.In Figure7, the behaviors of surface wave function u3x,t for FCH6 model (72) and (73) are presented in 3D with μ=0.01 for diverse α such that α=0.75 and α=0.5. While the surface plots of absolute error for FCH6 model (72) and (73) based on the results obtained in [24] at μ=0.01 are depicted in Figures 8 and 9 for α=0.75 and α=0.5, respectively. From these graphs, it is evident that the achieved results are in good agreement with those obtained in [24].Figure 7
Surface wave behavior ofu3x,t of FCH6 model (72) and (73) with μ=0.01 for diverse α: (a) α=0.75 and (b) α=0.5.
(a)(b)Figure 8
Surface of absolute error for FCH6 model (72) and (73) at μ=0.01 and α=0.75: (a) u2−uqHAM and (b) u2−uNIM.
(a)(b)Figure 9
Surface of absolute error for FCH6 model (72) and (73) at μ=0.01 and α=0.5: (a) u2−uqHAM and (b) u2−uNIM.
(a)(b)


## 5. Concluding Remarks

In this paper, we have investigated the fractional parabolic partial differential models of Gardner and Cahn-Hilliard equations in conformable sense. Using the fractional residual method, the approximate solution has been successfully acquired of the posed problems without imposing any unsanctified restrictions. Numerical simulation has been carried out to highlight the ability of the suggested method. In this context, it can be concluded that the implemented approximation algorithm is a superior tool for computational purposes, it is computer oriented, it is relatively better compared to the existing numerical methods, and it is a straightforward and simple methodology that needs a few iterations to get accurate solutions. From the graphic representations, it is noticed that the solution behavior is harmonious for different fractional values and consistent with the integer value. In future work, multivariate series expansion based on residual error can be employed for multidimensional fractional evolution models in terms of the conformable derivative.



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