Exploring fractional dynamics of fourth order parabolic partial differential equations with power la

Abstract

Introduction

The dynamical behavior of fourth-order parabolic equations has attracted global interest. These equations can be framed by non-linear functional equations defined by:

w = g + ℳ(w, wη, wξ, wξξ, wξ,ξ,ξ, wξξξξ) where g is the known function and ℳ is the non-linear operator.

*Corresponding author. *E-mail address: mawx@cas.usf.edu, atulpasrija_rs@pbi.ac.in This paper was recommended for publication in revised form by Editor-in-Chief Ahmet Selim Dalkilic Published by Yıldız Technical University Press, İstanbul, Turkey © Author. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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The first-order derivative wξ reflects the advective transport. The term wξξ accounts for instability at large scales. The inclusion of the third-order term wξξξ yields dispersion effect. The fourth-order term wξξξξ provides exponential damping out of very sharp spikes at small scales, and the non-linear term stabilizes the system dynamics by transferring energy between large and small scales. In the past few decades, theoretical study and practical simulation have been addressed via fractional operators [1-4]. These operators have been found to be better predictors than integer-order models. Fractional operators have emerged as a non-local tool for providing an excellent description of a real evolution process with memory and hereditary properties. It naturally arises in multiple conducts of wave phenomena, exhibiting chaos in wave motions, solitons, wrinkled flame front propagation, phase turbulence in reaction-diffusion systems, and chaotic drifting waves induced by photon collision [5-10]. However, the inclusion of fractional derivatives with variable and non-variable order demands highly influential analytic tools. For numerical treatment, it requires a large number of variables to cope with reality. The implementation of fractional operators can be found in a wide range of fields such as electromagnetism, bioengineering, signal processing, optimal control theory, and finance [7-14]. There are innumerable alternative definitions of fractional operators available in the literature [11-15]. Although the Liouville-Caputo (L-C) derivative and the RiemannLiouville (R-L) derivative are the most frequently used terminologies. With the definition of Caputo fractional derivative, a fractional derivative would be applicable for differentiable functions only. To deal with non-differentiable functions, R-L derivative is preferred, but it denies being zero for constant functions. To address non-local and non-singular kernel properties, the Caputo-Fabrizio (C-F) and Atangana Baleanu in Caputo sense (ABC) serve as extensions of the L-C operator. However, there are certain drawbacks when employing a non-singular fractional differential operator. The inability of a fractional differential operator with a non-singular kernel to satisfy the fundamental theorem of fractional calculus has been documented by Diethelm [16]. Another notion of local fractional derivatives extends the familiar limit definition of the classical derivative [17,18]. Because of their simplicity, the local fractional derivatives can meet a large number of wellknown aspects of classical derivatives, such as the chain rule, quotient rule, integration by parts, Gronwall’s inequality, fractional series representation of functions, and so on. However, the inclusion of the memory effect falls outside the scope of local fractional derivatives. In present study, renowned models of fourth order partial differential equations featured with Caputo derivative are considered as: • Fractional generalized Kuramoto-Sivashinsky (FGKS) equation:

Fractional Swift-Hohenberg (FSH) equation without dispersion: (2)

Fractional Swift-Hohenberg equation (FSH) with dispersion: (3)

or (5) where x and y are real constantss. For α = 1, Equation 1 represents the classical generalized Kuramoto-Sivashinsky (GKS) equation [19]. The variants of GKS equation have been extensively explored with different initial and boundary data by numerous analytic and numerical tools, such as homotopy perturbation method [20] and homotopy analysis method, which have been projected to treat space fractional GKS equation [21], residual power series method has been implemented with the projection of Riemann-Liouville integral and Caputo’s fractional framework [22], the amalgamation of the integral transform with variational iteration method [23] and q-HAM [24] are used to construct semi-analytic expansion solutions. The G’/G expansion method [25,26], exp-function method [27], modified Kudryashov method [28], tanh method and its extended version [29,30] are employed to interpret various soliton conductance. The non-integrability of K-S equation has been established via Painleve analysis [31]. The rich characterization of several dynamics of K-S model, such as multimodal, cellular, oscillatory, and chaotic solutions, have been studied so far [32-35]. In 1977, Jack Swift and Pierre Hohenberg have introduced and nurtured Swift-Hohenberg (S-H) equation as a universal model for the temperature and fluid velocity dynamics of convection [36]. S-H equation intrinsically occurs in nano-crystalline materials for expressing the average density of detachment during creation of shear microbands [37]. S-H equation addresses numerous issues such as patterns in Rayleigh-Bernard convections, patterns within thin vibrated granular layers, defect dynamics, spatial-temporal chaos and study of lasers [38-43]. S-H equation demonstrates the mechanism of amplitude of optical electrical field within the cavity embodying a non-linear medium [44]. Application of S-H equation rangs in the theory of pattern formation in fluid layers confined between horizontal well-conducting boundaries [45] Several researchers have conducted analytic and numerical experiments to unveil the dynamics of S-H equation. [46]

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provides evidence for the existence of quasi-patterns for the S-H equation. With inclusion of dispersive effect, in [47] authors have reported the achievement of spatial reversibility. In [48], investigators have proved the existence of non-stationary meromorphic solutions of S-H equation. Consequently, they have derived the simple periodic and elliptic non-stationary travelling wave solutions of S-H equation. Semi-analytic techniques [49-52] are eminently aided to interpret the behavior of S-H equation. The Cahn-Hilliard (C-H) equation is a non-linear, stiff, fourth-order parabolic equation. It shows how the conserved field has changed over time. In essence, the C-H equation has been created to simulate the spinodal breakdown of a binary system at a specific temperature. It records both a very gradual coarsening, which results in bulk phases with an interface separating them, and a quick phase separation, which creates a thin interface between two phases. The C-H equation has been solved by many researchers using various time discretization and spatial techniques. The C-H equation has been examined in 2D form by Grasselli et al. [53], who have established a number of findings about the well-posedness and large-time behavior of solutions. Analysis has been done in the presence of bounded absorbing sets in two distinct phase spaces. Additionally, the existence of exponential and global attractors has been proven. The proposed C-H equation has been solved by Lee et al. [54] utilizing the pseudo-spectral, finite element, and finite difference methods. Together with firstand second-order temporal algorithms, Chen et al. [55] have suggested a finite difference technique for the C-H system with a logarithmic Flory Huggins energy potential. For the C-H and Allen-Cahn equations, Zhang et al. [56] have suggested the optimal control issue. For optimum control issues, the optimality requirements have been applied. The C-H Biot model has been suggested by Storvik et al. [57] to describe flow through deformable porous media. Biot’s theory has controlled the coupling between flow and deformation, resulting in a three-way coupled system. In [58], Manzenaro et al. have addressed a C-H model with wall boundary conditions. Using a curvilinear hexahedral model, the model equations have been discretized by the discontinuous Galerkin spectral element approach. The first-order implicit-explicit technique, which separated the system into two C-H equations, has been used to explore the discretization of time. For the space-time fractional order C-H equation, Khan et al. [59] have presented the traveling wave solution using the auxiliary approach. For complex dynamics, Zhou and Xie [60] have presented a multicomponent three-dimensional C-H equation. Using the projected gradient approach, lagrangian multiples of Gibbs n simplex phase constraints have been computed. In its versatile framework, the present study incorporates potential models (FGKS, FSH and FCH) of thermal science, material science, biological science, hydrodynamics, study of lasers, oceanography and many more. The blend of these subjects forms the basis of vast studies in

engineering and natural sciences. Therefore, the choice of FGKS, FSH and FCH equations will engage the interest of a large part of the scientific community. Despite the numerous semi-analytic studies for L-C fractional GKS, S-H and C-H equations, the establishment of reliability of solutions for 0 < α <1 is extensively unexplored. While addressing this gap through proposing the first-time application of the generalized bivariate homotopy perturbation method (GB-HPM) for fourth-order equations, the present study demonstrates the competence of GB-HPM in terms of accuracy. The manuscript is organized in the given manner. Section 1 is based on the introduction of the theory and applications of the considered problems. In Section 2, definitions, properties and terminologies of L-C derivative and generalized bivariate (GB) transform are presented. In Section 3, existence and uniqueness of a continuous solution are shown for time fractional K-S, S-H and C-H equations. In Section 4, solution procedure of the proposed algorithms is explained for general non-linear initial value problems. In Section 5, the application of theory developed in the above sections has been presented through several test problems. These are analyzed via comparative and error analysis to test efficacy and robustness of the proposed scheme. In the last section the concluding remarks are presented based on the above study. Preliminary Results This section covers the definitions, properties, and terminologies of L-C derivative and GB transform, which are discussed in further detail throughout the paper. Definition 1. [15] The R-L fractional integral of order α is conceptualized as (6) Definition 2. [15] For a given function w(η): [0,∞) → R, the L-C fractional derivative of w of order n-1 < α ≤ n is defined by (7) Lemma 1. [15] For any n-1 < α ≤ n, n ∈ N. Then one hass (8)

In the subsequent development of fractional calculus, the operational features of integral transforms are extensively explored to construct solutions to fractional models [4,49,50,61]. However, each existing integral transform pursues some constraints, which stimulates interest in developing modified integral transforms. In this series of developments, a large amount of research work has been devoted to formulate several novel integral

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transforms [62-70]. In 2023, Arora et al. [70] proposed GB transform as the generalization of Laplace, Shehu, ARA and Formable transforms [63,64,68]. Mathematically, the GB transform of order m and its inverse can be defined as [70]:

where s > γ > 0, δ is real constant and 𝕊 is the Shehu transform [68]. Proposition 1. [70] GB transform holds the fundamental properties of integral transforms such as: • Linearity Property

Lemma 2. [72] The function N satisfies the Lipschitz condition. Proof: Let w and w* be two bounded functions. Using definition of N and triangular inequality gives

(13) where μ and λ are arbitrary constants. Theorem 1. [71] Formulation for application of GB transform over L-C derivative is given by

Now suppose that ∃ constants κ1, κ 2> 0 such that ∀(ξ,η ) ∥w∥ ≤ κ1 and ∥w*∥≤ κ2. κ = max{κ1, κ2}. Then their first derivative function (.)ξ fulfills Lipschitz condition and ∃ a non-negative number L1 such that (19)

(14) where n-1 < α ≤ n Existence and Uniqueness The present section illustrates the existence and uniqueness of continuous solutions for the considered fractional equations. The FGKS reads as: (15) Employing the fractional integral operator given in Equation 6 on Equation 15, gives

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Theorem 2. [72] The FGKS, FSH and FCH equations admit a unique continuous solution provided that (25) Proof: The expressions 16, 17 and 18 can be expressed in a unified form as (22) which recommends the consideration of recurrence relation as • • Let (23) Now the aim is to prove that is a continuous solution. For that end, set the algebraic difference of successive terms as Θk (ξ,η) = wk (ξ,η) - wk-1 (ξ,η).

Evaluating the limit k → ∞ for Equation 25 yields Ψk→ 0 and left-hand side gives

The above result implies the certainty of taking as a solution which is continuous. Finally for uniqueness consider w and w* be two distinct solutions for FKS, FSH and FCH equations then Equation 24 and Lipschitz condition of N implies

Application of Generalized Bivariate Homotopy Perturbation Method In this section, the fundamental idea behind the solution procedure of the proposed algorithms is elucidated. The general form of the fractional non-linear partial differential equation can be considered as:

(24) subject to the initial condition Following the recurrence procedure given in Equation 24 deduce

which implies the existence and continuity of the solution. is the solution Now to prove that of the Cauchy problems defined as FKS, FSH and FCH equations. Let

where w is the function of ξ and η. Ω is a bounded domain and T is finite time. 𝒫 and 𝒬 are linear and non-linear operators, respectively, and N(ξ,η) is the non-homogeneous term. Apply A1 on both sides of Equation 26 and use initial condition given in Equation 27, such that Equation 26 takes the following form:

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Comparing the coefficients of like powers of homotopy parameter i.e., θ, derives (29)

which is a recurrence relation for Equation 26, where 𝒢(ξ,η) denotes the term that arises from the non-homogeneous term and initial condition. Here, it is worth noting that for existing integral transforms ℐ[f.g] ≠ ℐ[f].ℐ[g] where, ℐ is an integral transform. Therefore, these integral transforms are insufficient for handling a large class of differential and integral equations due to the restrictions arising from non-linear terms. In the present scheme, GB transform is consolidated with homotopy perturbation method (HPM) [73] to overlap this difficulty. According to the expansion methodology, which is followed by semi-analytic tools, let the solution be represented by the following infinite series: (30) Homotopy is a continuous process that associates a family of objects, which represent intermediatory stages of deformation of an object to another object. HPM is a semi-analytic strategy that relies on the principles of homotopy and perturbation methods. Application of HPM redistributes the existing non-linearities with the aid of homotopy parameter i.e., θ. The smooth tracing of homotopy parameter θ in [0,1] is equivalent to the variance in the strength of inherited non-linearity. This enhances to deform an accessible solution into the desired one. To consolidate the limitations of GB transform for non-linear events, HPM offers the decomposition of non-linear terms such as: (31)

The approximate solution of nth-order for the Equation 26 can be obtained as: (32) Test Problems This section is devoted to extracting and discussing the semi-analytic results of the L-C fractional FGKS, FSH and FCH equations under various initial conditions. The comparative analysis of the proposed algorithms is developed using graphical and tabular representations of the results. Example 1. Consider FGKS without dispersion effect at x = -1, y = 0 and z = 1 for which initial condition is extracted from the analytic solution: (33) where s, κ and λ are real constants. The computational work carried out with choice of

produces components of the series solutions in terms of GB-HPM solution as

where ℋ𝓀 are considered to be the He’s polynomials [74], given by

Using Equation 30 and Equation 31 in Equation 29 parametric homotopy equation with homotopy parameter θ can be constructed as

The solution profile of GB-HPM solution at is drawn in Figure 1 and Figure

2. These graphs represent the behavior for integer order

K-S equation with α = 1 and the variation arises for different values of α in 3D and contour surfaces. The sensitivity of GB-HPM solution profiles with respect to the time

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Figure 1. Solution profile of GB-HPM solution for Example 1 when (a) α = 0.5 (b) α = 0.75 (c) α = 1 (d) ξ = 3.

fractional orders can be observed apparently through 2D simulations given in Figure 1(d) at ξ = 3 and contour surfaces in Figure 2. Fractional solution profile converges to the solution of integer order equation as α → 1. 3D surface plots of FGKS exhibit time evolution of flame-front. The contour surfaces effectively highlight the patterns in the data which reflects the heat distribution in the system. The fractional derivative reveals broader visualization of evolutionary process with its enhancing feature of controlling rate of change. Table 1 shows the numerical findings for different time fractional orders. The absolute error is found competent and aligned to the error deduced for q-HATM scheme [24]. However, the present scheme is free from analysis of any auxiliary parameter. Table 2 demonstrates the convergence phenomenon of series solution for fractional derivative α = 0.75. It provides the credibility of obtained solution for fractional phenomenon when exact solution is unknown. The convergence rate gets slower as time increases.

Example 2. Consider the FGKS equation with dispersion effect at x = 1, y = 4, and z = 1. The analytic solution is given by (34) where θ = κ(ξ - sη - s0) with arbitrary constant s and s0. With choice of

and using the initial conditions from Equation 34 at η = 0, the suggested algorithm gives the explicit expressions of components of series solutions in terms of GB-HPM solution as

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Figure 2. Contour surfaces of GB-HPM solution for Example 1when (a) α = 0.5 (b) α = 0.75 (c) α = 1.

and so on. For parameters s = 3, κ = 0.5 and s0 = -13, Figure 3 and Figure 4 exhibit the dynamics that have been produced

2. To visualize the variability in solution profile in temporal

domain at ξ = 3, 2D solution profiles are drawn in Figure 3(d) for different values of α. Figure 4 simulates the projection of solution profile for distinct orders of fractional derivative. In this case of incorporating dispersion effect the flame-fronts exhibit some sharpness. The contour surfaces strikingly illustrate the regions of sharp transitions or abrupt changes which implies the occurring of steep thermal gradient. Numerical values of the GB-HPM solution for severe fractional orders are tabulated in Table 3. In addition, error analysis in terms of absolute error of proposed solutions in comparison to q-HATM [24] reveals the supremacy of GB-HPM. This performance of accomplished results for FGKS equation with dispersion ensures the capability of the present method to capture every comprised effect of the system.

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Table 1. Absolute error for approximate solution of Example 1 at κ = 1/(2√19), s = 5, λ= -25 for different α ξ

Gb-Hpm

Table 2. Absolute error in consecutive approximate solutions of Example 1 at κ = 1/(2√19), s = 5, λ= -25 for α = 0.7 ξ

Example 3. Consider FSH equation without dispersion effect with initial data

Following the computational procedure of GB-HPM with choice of

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Figure 3. Solution profile of GB-HPM solution for Example 2 when (a) (a) α = 0.5 (b) α = 0.75 (c) α = 1 (d) ξ = 3.

and so on. Solution profile of S-H equation signifies the physical effect of bifurcation and dispersive parameters on probability density function w(ξ,η) for fractional Brownian and standard motions. For non-dispersive FSH equation, the graphical representation of GB-HPM solution in Figure 5, Figure 6 and Figure 7 simulate oscillatory and periodic solutions depending on the value of l. The behavior of obtained solutions varies harmonically in space and time. The variation of solution profiles at l = 3,6,8,10 for several fractional order displayed as 2D graphs is apparent in Figure 5, Figure 6 and Figure7. Behavior of obtained

solution is more sensitive with respect to the α at l = 3, but at l = 6,8,10 there are no changes. For α = x = 0.5 overshoots observed for probability density function w(ξ,η) increase with increase in η for l = 3. However, overshoots of w(ξ,η) show inverse relation with time η at l = 6,8,10. Table 4 demonstrates the numerical results of 3rd- order approximation for Example 3 at distinct fractional orders obtained by GB-HPM in comparison on of NTDM [49] and q-HATM [50] scheme. Obtained results are completely aligned with results of NTDM and agrees with q-HATM solution for atleast upto second significant digit. In comparison with NTDM scheme present method with alternative approach of He polynomials, overcomes the demerit of tedious calculations for Adomian polynomials [74]. In Table 5 the decreasing absolute error of consecutive approximate solutions ensures the convergence of the solution for α = 0.75. Therefore, credibility GB-HPM solution is independent from the condition of availability of exact solution. Example 4. Consider FSH equation without dispersion effect with initial data

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Figure 4. Contour surfaces of GB-HPM solution for Example 2 when (a) α = 0.5 (b) α = 0.75 (c) α = 1.

(36) where l is a real constant. Following the computational procedure of GB-HPM with choice of

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Table 3. Absolute error for approximate solution of Example 2 at κ = 0.5, s = 3, s0 = -13 for different α ξ

Gb-Hpm

Table 4. Numerical values of approximate solution of Example 3 at x = 0.9 and l = 10 for different α. ξ

Gb-Hpm

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Figure 5. Solution profile of GB-HPM solution for Example 3 when α = 0.5 and x = 0.5 at (a) l = 3 (b) l = 6 (c) l = 8 (d) l = 10.

Table 5. Absolute error in consecutive approximate solutions of Example 3 at x = 0.9 and l = 10 for α = 0.75 ξ

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Figure 6. Solution profile of GB-HPM solution for Example 3 when α = 0.5 and x = 1 at (a) l = 3 (b) l = 6 (c) l = 8 (d) l = 10.

Table 6. Numerical values of approximate solution of Example 4 at x = 0.5, y = 0.7 and l = 10 for different α ξ 1

Gb-Hpm

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Figure 7. Solution profile of GB-HPM solution for Example 3 when α = 1 and x = 1 at (a) l = 3 (b) l = 6 (c) l = 8 (d) l = 10. Table 7. Absolute error for approximate solution of Example 5 at x = 1 and α = 1 ξ

Gb-Hpm

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Figure 8. Solution profile of GB-HPM solution for Example 4 when x = 0, y = 0.7 and l = 3 at (a) α = 0.25 (b) α = 0.5 (c) α = 0.75 (d) α = 1. Table 8. Absolute error in consecutive approximate solutions of Example 5 at x =1 and α = 0.75 ξ

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Figure 9. Solution profile of GB-HPM solution for Example 4 when x = 0.5, y = 0.7 and l = 3 at (a) α = 0.25 (b) α = 0.5 (c) α = 0.75 (d) α = 1.

and so on. The graphical illustrations for obtained GB-HPM solution to analyze the effect of dispersion and bifurcation parameters (i.e., x and y, respectively) on probability density function w(ξ,η) have been presented in Figure 8, Figure 9, Figure 10 and Figure 11. From comparative visualization of presented figures for Example 4, it can be stated that w(ξ,η) has a significant effect of included physical parameters for l = 3. However, for l = 10 solution profile does not depict much difference with respect to the variation in x and y. The presented figures also exhibit the evolution of solution profile with an increase in time for different values of fractional order α. Numerical values tabulated in Table

6 again demonstrate the agreement of GB-HPM solution with NTDM [49] and q-HATM [50] solutions. Example 5. Consider FCH with the initial data and for x = 1 on setting α = 1 the corresponding exact solution is given by: (37) Following the criteria of proposed framework with choice of

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Figure 10. Solution profile of GB-HPM solution for Example 4 when x = 0, y = 0.7 and l = 10 at (a) α = 0.25 (b) α = 0.5 (c) α = 0.75 (d) α = 1.

Continuing the same procedure, the remaining components of the series solution Wn (ξ,η) (n ≥ 3) can be obtained.

Figure 12 depicts the geometrical configuration of the obtained semi-analytic solution at variant fractional orders. This Figure 12 suggests that the implementation of the proposed method is significantly efficient in finding a smooth wave solution to the specified problem. Surface plots of obtained solution describe the configuration of a kink wave. This wave represents a localized solution that rises or descends from one asymptotic state to another. In phase separation process, the solution profile of Cahn-Hilliard quantifies the transition of concentration of one of the two metallic components of the alloy. The present solution displays the stationary transition of phase in spinodal interval of |w| < 1, which directs to the unstable stationary state [75].

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Figure 11. Solution profile of GB-HPM solution for Example 4 when x = 0.5, y = 0.7 and l = 10 at (a) α = 0.25 (b) α = 0.5 (c) α = 0.75 (d) α = 1.

In Table 7, a quantitative response of the achieved results in comparison of RPS [76], HPM [77] and q-HAM [78] is presented. For x = 1 obtained solution has acquired alike accuracy to RPS and q-HAM. In [76], RPS is found superior to HPM for solving fractional C-H equation. For the case α = 1, the obtained solution converges to the solution obtained for the classical C-H equation, utilizing HPM and ADM [79]. Therefore, the GB transform can be considered a convergence booster of classical HPM for fractional equations. In addition, Table 8 reveals the convergence phenomenon with rapidly diminishing errors of consecutive approximations with increasing n for α = 0.75. The consistency in terms of accuracy of the approximate solution in

both cases of fractional and integer order equations confirms the efficiency of the proposed method.

Conclusion

In this article, the time-fractional generalized Kuramoto-Sivashinsky equation, Swift-Hohenberg equation, Cahn-Hilliard equation, and their special cases have been studied using the semi-analytic generalized bivariate homotopy perturbation method. Dynamics of several waveforms have been analyzed for different values of fractional order and physical parameters. Existence and uniqueness are presented for continuous solutions of fractional variants

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Figure 12. Solution profile of GB-HPM solution for Example 5 when x = 1 at (a) α = 0.5 (b) α = 0.75 (c) α = 1 (d) ξ = 3.

of Kuramoto-Sivashinsky, Swift-Hohenberg and CahnHilliard equations. The suggested method is free from the assumption of requiring major or minor physical parameters in the problem. The graphical illustrations and tabulated absolute error of given results in contrast to existing analytic results establish the worthiness of the proposed scheme. While addressing non-linear problems by generalized bivariate homotopy perturbation method, the assessment of He’s polynomials is much easier than the Adomain’s polynomials. Generalized bivariate homotopy perturbation method is found highly competent in targeting fractional order challenges. Another advantage is that this method does not require any discretization of variables or linearization of non-linear terms, which preserves actual non-linear effects, large computer memory and extensive time. Beyond this, generalized bivariate homotopy perturbation method does not require exact solution to prove its credibility. For aspects of further advancements, the application

of the proposed scheme to stochastic, hyperbolic and higher-dimensional systems can provide good supplements to the existing literature.

Acknowledgement

Shelly Arora is thankful to SERB-POWER for grant support to complete this research via grant number SPG/2022/001269.

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article.

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Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript

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