Analytical and numerical results for the Swift–Hohenberg equation

doi:10.1016/j.amc.2010.01.041

Applied Mathematics and Computation

Volume 216, Issue 1, 1 March 2010, Pages 221-226

https://doi.org/10.1016/j.amc.2010.01.041 Get rights and content

Abstract

The problem of the Swift–Hohenberg equation is considered in this paper. Using homotopy analysis method (HAM) the series solution is developed and its convergence is discussed and documented here for the first time. In particular, we focus on the roles of the eigenvalue parameter $α$ and the length parameter l on the large time behaviour of the solution. For a given time t, we obtain analytical expressions for eigenvalue parameter $α$ and length l which show how different values of these parameters may lead to qualitatively different large time profiles. Also, the results are presented graphically. The results obtained reveal many interesting behaviors that warrant further study of the equations related to non-Newtonian fluid phenomena, especially the shear-thinning phenomena. Shear thinning reduces the wall shear stress.

Introduction

Density gradient-driven fluid convection arises in geophysical fluid flows in the atmosphere, oceans and in the earth’s mantle. The Rayleigh–Benard convection is a prototype model for fluid convection, aiming at predicting spatio-temporal convection patterns. The mathematical model for the Rayleigh–Benard convection involves the Navier–Stokes equations coupled with the transport equation for temperature. When the Rayleigh number is near the onset of convection, the Rayleigh–Benard convection model may be approximately reduced to an amplitude or order parameter equation, as derived by Swift and Hohenberg [1].

The Swift–Hohenberg (SH) equation is defined as $\frac{\partial u}{\partial t} = α u - {(1 + \frac{\partial^{2}}{\partial x^{2}})}^{2} u - u^{3},$ where $α \in R$ is a parameter. This equation plays a central role in pattern formation. We view the Swift–Hohenberg equation as a model equation for a large class of higher-order parabolic model equations arising in a wide range of applications, such as the extended Fisher–Kolmogorov equation in statistical mechanics [2], [3], and a sixth-order equation introduced by Caginalp and Fife [4] in phase field models [5]. Details of the physics of the Swift–Hohenberg equation can be found in Refs. [6], [7].

In this paper, we study the Cauchy–Dirichlet problem for the Swift–Hohenberg equation on the interval (0, l). It is observed that the length l is also an important parameter in addition to the parameter $α$ . Thus, writing the SH equation in a more conventional form, we consider the problem $u_{t} = - u_{xxxx} - 2 u_{xx} - (1 - α) u - u^{3},$ $u = 0 and u_{xx} = 0 at x = 0, l for all t > 0,$ $u (x, 0) = u_{0} (x) for all 0 < x < l .$ The above boundary conditions were chosen so that solutions can be extended as periodic functions over the real line. Here $u_{0} (x)$ is a smooth function that vanishes at $x = 0$ and $x = l$ . In this paper we assume that $u_{0} (x)$ is symmetric with respect to the center of the domain, $x = l / 2$ . That is, $u_{0} (l - x) = u_{0} (x)$ for all $0 < x < l$ .

In this paper, we use the homotopy analysis method (HAM) of Liao [8] and homotopy Padé approximation method to obtain analytical solutions for Swift–Hohenberg equation. The HAM was developed in 1992 by Liao [9], [10], [11], [12]. This method has been successfully applied to solve many types of nonlinear problems in science and engineering. We aim in this work to effectively employ the HAM to establish exact solutions for the Swift–Hohenberg equation with boundary and initial conditions. By the present method, approximate analytical results can be obtained with only a few iterations. The HAM contains the auxiliary parameter ℏ which, unlike, other numerical methods, provides us with a simple way to adjust and control the convergence region of solution series for large values of x and t. Therefore, the HAM handles linear and nonlinear problems without any assumptions and/or restrictions on the parameters.

As far as the homotopy analysis method (HAM) is concerned, there are papers in the literature, such as Liao [8] and Wu et al. [13]. They used this method to solve solitary waves governed by Camassa-Holm and Vakhnenko, equation respectively. Hayat et al. [14] obtained the series solution for magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell (UCM) fluid over a porous stretching sheet. Allan [15] also used this method for the Lorenz equation. Numerical solutions to the Swift–Hohenberg equation were obtained by in [16], [17], while eigenvalues of the Swift–Hohenberg equation for piece-wise constant potentials were considered by Caceres [18]. However, the application of the HAM to the Swift–Hohenberg equation has not been considered until now. We thus extend the existing literature by applying the HAM in order to construct analytical solutions, which are shown to agree with the existing numerical results.

Section snippets

The homotopy analysis method

Consider the nonlinear differential equation $\tilde{N} [u (x, t)] = 0,$ where $\tilde{N}$ is a (in general nonlinear) partial differential operator and $u (x, t)$ is a solution. To solve Eq. (2.1) by the HAM, we first need to construct the following family of equations: $(1 - q) L [θ (x, t; q) - u_{0} (x, t)] = q ℏ \tilde{N} [θ (x, t; q)],$ where L is a properly selected auxiliary linear operator satisfying $L (0) = 0,$ $ℏ \neq 0$ is an auxiliary parameter, and is an initial approximation. Obviously, Eq. (2.2) gives $θ (x, t; 0) = u_{0} (x),$ when q = 0. Similarly, when q = 1, Eq.

Solution of Swift–Hohenberg equation by homotopy analysis method

It is known that if $α ⩽ 0$ , then the steady case of our problem has only the trivial solution (see [21]), and hence, for every $x \in (0, l), u (x, t) \to 0$ as $t \to \infty$ , for $α > 0$ the situation is much more complicated, as stated in [21]. Therefore, we only consider the case $α > 0$ . We now apply the HAM to Eqs. (1.1), (1.2), (1.3), (1.4) to establish analytical solutions for this problem. We choose the linear operator as $L [θ (x, t; q)] = \frac{\partial θ (x, t; q)}{\partial t} .$ Hence, $L [c] = 0,$ where c is an arbitrary constant in t. We define the nonlinear

Convergence of the series solution and discussion of results

Liao [9], [10], [11], [12] showed that the series solution always converges to a solution of the problem under consideration. In general, admissible values of the convergence control parameter ℏ will depend on both $α$ and l. To find the optimal value of ℏ, an error analysis is performed. Let $R_{k} [u_{k}]$ denote the residual error of the kth–order homotopy-series approximation for fixed k, $α$ andl; the integral of the residual error is defined as $E_{k} (ℏ) = \int_{t_{0}}^{t_{1}} (\int_{0}^{l} {R_{k} [u_{k} (x, t)]}^{2} dx) dt .$ It is not difficult to

Acknowledgements

The authors would like to thank Editor-in-Chief M. Scott and an anonymous referee for constructive comments which led to the definite improvement of the paper.

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View full text

Analytical and numerical results for the Swift–Hohenberg equation

Abstract

Introduction

Section snippets

The homotopy analysis method

Solution of Swift–Hohenberg equation by homotopy analysis method

Convergence of the series solution and discussion of results

Acknowledgements

Appl. Math. Lett.

Int. J. Non-Linear Mech.

Int. J. Non-Linear Mech.

Chaos Solitons Fract.

Phys. Lett. A

Chaos Solitons Fract.

Math. Comput. Modell.

Chaos Solitons Fract.

Physica D

Hydrodynamic fluctuations at the convective instability

Phys. Rev. A

Bistable systems with propagating fronts leading to pattern formation

Phys. Rev. Lett.

Propagating fronts near a Lifschitz point

Phys. Rev. Lett.