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 several 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.
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()-expansion method, generalized Kudryashov method, tanh-coth method, and exp 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] where is nontrivial constant parameter, signifies the order of time-dependent derivatives of fractional order, and is wave-profile function scaling spatiotemporal durations of and . Typically, represents the variance of with time and fixed location, and the nonlinear terms and refer to wave steepening while the linear dispersive term refers to wave effects. Therefore, it has a vital role regarding interactions of dispersion and nonlinearity in soliton theory. Hereinafter, 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]: and where is constant parameter with Herein, the nonlinear terms denote the chemical potential of the model, while and 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 where is a smooth analytic function of .
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 -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 Section 3, 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 function on , the conformable derivative of at is given by where is understood to mean
Definition 2 (see [46]). Given a real-valued function on , if is -differentiable; then, the -fractional integral is given by provided that the integral is Riemann improper.
The following results are some of the basic characteristics gained in terms of . For additional properties, we refer to [47–50] and the references therein.
Lemma 3. Let and the functions , be -differentiable at a point Then, for all real constants the following properties hold: (i)(ii)(iii) (iv)(v)(vi)If is differentiable, then it also holds that
Lemma 4 (see [46]). Given the real-valued functions and on , let , be first order differentiable and -differentiable and let be first order differentiable on the range of . So, the use of the known chain rule yield
Definition 5 (see [47]). Given a real-valued function on , the th order conformable partial derivative at a point is defined as
Definition 6 (see [47]). Given a real-valued function on , the th order conformable integral is defined as
Definition 7 (see [47]). The fractional series expansion at can be defined as follows where is the th unknown coefficient, , and is the radius of convergence.
Theorem 8 (see [47]). Given a real-valued function on , let has many conformable partial derivatives at any point with the following fractional series expansion at : Then, can be calculated by in which is the th conformable partial derivative of about so that (-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 along with the condition
is the order of conformable time-fractional index, , is a given analytical function, and is an unknown sufficiently differentiable wave-profile function. Herein, indicates the nonlinear operator from a Banach space to itself in terms of and over a one-dimensional spatiotemporal domain.
Based on the proposed algorithm, the solution of (13) has following form of fractional series expansion at : provided that . So, the -term truncated series solution of in view of the initial condition (14) can be described as
Basically, the residual error of model (13) is defined as and thus the -term truncated residual of is expressed by where , , and for each
To demonstrate the main steps of the residual series algorithm in finding out the values of unknown parameters of the -term truncated solution (16), set and equate to zero at , so that can be acquired. Thereafter by applying the operator on both sides of the resulting relevant equation for , and solving , the coefficient can be acquired as well. Continuing likewise, the rest of coefficients for each of the fractional series expansion (16) can be acquired. To complete the presentation and clarification, the underlying algorithm is dedicated.
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Lemma 9. Let be the solution of PDEs (13) and (14) that has th order partial derivatives in conformable sense at any point and the fractional expansion of equation (15) at . If there exist so that for all , then, the remainder term holds the underlying inequality in which
Corollary 10. Let and be respectively the analytic and approximate solutions of PDEs (13) and (14). If there exists so that for each and for . Then, converges to as soon as .
Proof. Since for each , then, , and then, . Subsequently, we have . This leads to . Thus, it can be observed that
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]: along with the underlying initial condition where is constant, , 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 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 and is given by
According the CFRS algorithm, the fractional series solution of the FGE (21) about can be established as follows provided that Subsequently, the th fractional series in view of the initial condition (22) can be truncated as follows and the residual error function can be expressed as in which for each and
In this direction as well, the th truncated error of can be expressed as provided that as , and for each
By viewing the representation of the truncated series (25) and minimizing the residual error (27) of the governing equation, the unknown coefficients can be computed for each value of in order to obtain the th approximate solution . To begin with, the first fractional series solution at assumes the form as well as the first residual function assumes the form
Consequently, putting into to get
Herein, with the aid of it yields which implies
Hence, the first series solution can be read as
Sequentially, the second truncated series can be computed for in (27) such that where By employing the conformable operator on both sides of equation (34), we get that
Consequently, solving the term in the aforementioned equation with the help of Mathematica's symbolic architecture [51] leads to
Hence, the solution can be given as
Continuing in this manner, the truncated series of the series expansion (25) may be computed by applying in (27), allowing to act on both sides of the new relevant equation then establishing with the aid of Mathematica. To finish our process, we can assume that is our approximate solution of the FGE (21) along with condition (22). Moreover, the values of for each may be counted likewise. In what follows, the achieved terms in the form of an infinite series leads to the solution of the FGEs (21)–(22). Especially, the solution of FGEs (21) and (22) at and can be written in the form which agrees with the analytical solution acquired by -homotopy analysis transform method (-HATM), fractional natural decomposition method (FNDM) [13], and -homotopy analysis method (-HAM) [14], so that
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 with over a large enough spatio temporal domain . In Figure 2, the moving and evolutionary dynamics of fractional wave function of FGEs (21) and (22) are provided in 2D graph over versus at based on different values of that are given as and , respectively. From these graphs, it can be observed the tremendous influence of the fractional parameters on the solutions’ consistency with respect to the time . By measuring the absolute error , the achieved numerical solutions of FGEs (21) and (22) are given in Table 1 for distinct values of and fixed when and 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 such as at and . These results show good agreement between the solutions when the fractional values differ.
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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]: along with the underlying initial condition where is aconstant, , 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 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 and is given by
By performing the CFRS algorithm, the fractional series solution of the FCH4 equation (40) about can be constructed as follows provided that Subsequently, the th fractional series in view of the initial condition (41) can be truncated by and the error function can be expressed as in which for each and
In this orientation as well, the th truncated error of can be expressed as provided that as , and for each
By viewing the representation of the truncated series (44) and minimizing the residual error (46) of the governing equation, the unknown coefficients can be computed for each value of in order to obtain the th approximate solution . To begin with, the first fractional series solution at has the form as well as the first residual function has the form
Consequently, putting into to get
Hence, by using it yields which yields
Hence, the solution is obtained as
Sequentially, we can compute the series by assuming in the th truncated error (46) such that where Then, by employing the conformable differential operator on both sides of equation (53), we get that
Now, solving the term in the above equation with the help of Mathematica's symbolic architecture [51] leads to
Hence, the solution can be given by
Similarly, the truncated series of the series expansion (44) can be calculate by assuming in (46), then, by solving the term with the aid of Mathematica's symbolic architecture [51], we get which reveals that has the form
Proceeding likewise, the solution will have the form
To close this method, it is assumed that is an approximate solution, and , can be followed likewise. Later, by gathering the terms, of the posed model (40) and (41) may be predicted. In particular, the solution of FCH4 equation (40) and (41) at and can be written in the form which agrees with the analytical solution acquired by homotopy perturbation method (HPM) [25], -HAM, and new iterative method (NIM) [24], so that
In the following, 3D graphical simulation of of FCH4 model (40) and (41) with respect to different fractional parameter are shown in Figure 3 for over In Figure 4, 3D surface plots of FCH4 model (40) and (41) are depicted with fix versus such that and over the spatiotemporal domain . Further, the obtained absolute errors are reported in Table 3 and compared to those results provided in [24] at and . The superiority of the present method follows from those results. (i)Exponential wave solution of FCH4 equation
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This segment is an attempt to gain an effective approximate solution to FCH4 equation (40) with the initial condition [24] where is an arbitrary constant with
According the CFRS algorithm, the th fractional series solution of the FCH4 equation (40) about in view of the initial condition (62) can be expressed as
With the aid of the th truncated error of (46), the unknown coefficients of the series expansion (63) can be computed for each value of . To achieve this goal, let the first fractional series solution of FCH4 equations (40) and (62) at takes the form
Now, substitute into and then solve to get
Hence, the solution is
Sequentially, substitute into the second truncated residual error , apply the conformable operator on both sides of the resulting equation, and solve with the aid of Mathematica's symbolic architecture [51] to get which implies that the second series solution is
In the same fashion, the third and fourth series solutions of FCH4 equations (40) and (62) can be obtained successively as follows
To close the process, we assume that is the approximate solution. Following the same procedure, the values of can be also computed. Thus, the expression of the series solution of the FCH4 equation (40) along with condition (62) at and can be written in the form
In the following, the 3D behaviors of surface wave function of FCH4 model (40) and (62) are displayed in Figure 5 for the parameters and with respect to and on . While the fractional level curves of for FCH4 model (40) and (62) are shown in Figure 6 compared to the third approximate solutions obtained in [24] for fix on for various values when and . Error estimate for the third approximate solutions of FCH4 model (40) and (62) is provided in Table 4 by computing the absolute errors and based on the results achieved by -HAM and NIM [24] for , and . From this comparison, it is evident that the results obtained by CFRS are in good agreement with those presented in the literature.
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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]: with the condition
is a constant, , 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 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, the th fractional series solution of the FCH6 equation (72) about in view of the initial condition (73) can be expressed as and the residual error function is
For this purpose, the th truncated error of can be expressed in the form
Thus, by minimizing the residual error (76) of the governing equation, the unknown coefficients of series expansion (74) for each value of can be computed. Subsequently, the series solution at has the form whereas the first residual function has the form
Now, putting into to get
By utilizing the fact it yields which implies that
Therefore, the solution is
Sequentially, the second truncated series can be obtained by setting in (76) such that where By employing the operator on both sides of equation (83), we get
Solving the term with the aid of Mathematica's symbolic architecture [51] leads to which implies that in which
Therefore, the series solution can be given as
Following the same procedure, the series of (74) can be computed through setting in (76) and solving the term to get as follows where which implies that has the form
To close the process, we suppose that is the approximate solution. Then, , can be computed similarly. Anyhow, the -term sequential solution can be written in the form as well as the solution of the FCH6 equation (72) along with condition (73) can be predicted as
In Figure 7, the behaviors of surface wave function for FCH6 model (72) and (73) are presented in 3D with for diverse such that and . While the surface plots of absolute error for FCH6 model (72) and (73) based on the results obtained in [24] at are depicted in Figures 8 and 9 for and , respectively. From these graphs, it is evident that the achieved results are in good agreement with those obtained in [24].
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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.
Data Availability
No data to be declared.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to this article. They read and approved the final manuscript.
Acknowledgments
This work has been carried out during sabbatical leave granted to the author Mohammed Al-Smadi from Al-Balqa Applied University (BAU) during the academic year 2021/2022.