Comparison study on the different dynamics between the Allen–Cahn and the Cahn–Hilliard equations

Li, Yibao; Jeong, Darae; Kim, Hyun Dong; Lee, Chaeyoung; Kim, Junseok

doi:10.1016/j.camwa.2018.09.034

Cited by 17 publications

(5 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Herein, the nonlinear terms in this model refer to chemical potential dynamics, and u xxxx 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…”

Section: Solution Of Nonlinear Fourth-order Time-fractionalmentioning

confidence: 99%

“…where μ is constant parameter with μ ≠ 0: Herein, the nonlinear terms denote the chemical potential of the model, while u xxxx and u xxxxxx 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…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Al‐Smadi

Al‐Omari

Karaca

et al. 2022

Journal of Function Spaces

View full text Add to dashboard Cite

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 α 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.

show abstract

Section: Solution Of Nonlinear Fourth-order Time-fractionalmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Al‐Smadi

Al‐Omari

Karaca

et al. 2022

Journal of Function Spaces

View full text Add to dashboard Cite

show abstract

“…For positive integers k x and k y , let us consider Φ(x, y, t) � α(t)cos(k x x)cos(k y y) as a two-dimensional benchmark solution for (18), where α(t) is an amplitude. Substituting above Φ(x, y, t) into (18), then…”

Section: Two-dimensional Ac Equationmentioning

confidence: 99%

“…In this paper, we propose appropriate benchmark problems to verify the accuracy of the numerical methods based on the theoretical basis through linear stability analysis using simple initial conditions. First, we perform a linear stability analysis and then take a growth mode solution [18] as the benchmark problem, which is closely related to the dynamics of the original governing equations. e layout of this paper is as follows.…”

Section: Introductionmentioning

confidence: 99%

Benchmark Problems for the Numerical Schemes of the Phase‐Field Equations

Hwang

Lee

Kwak

et al. 2022

Discrete Dynamics in Nature and Society

Self Cite

View full text Add to dashboard Cite

In this study, we present benchmark problems for the numerical methods of the phase-field equations. To find appropriate benchmark problems, we first perform a linear stability analysis and then take a growth mode solution as the benchmark problem, which is closely related to the dynamics of the original governing equations. As concrete examples, we perform convergence tests of the numerical methods of the Allen–Cahn (AC) and Cahn–Hilliard (CH) equations using the proposed benchmark problems. The one- and two-dimensional computational experiments confirm the accuracy and efficiency of the proposed scheme as the benchmark problems.

show abstract

“…In this paper, we propose an efficient multi-reconstruction method from point clouds. The proposed method is based on the Allen-Cahn (AC) equation [7], which has the motion of the mean curvature flow [8][9][10][11]. Under the mean curvature flow, the reconstructed surface is smooth.…”

Section: Introductionmentioning

confidence: 99%

Multi-Reconstruction from Points Cloud by Using a Modified Vector-Valued Allen–Cahn Equation

Wang

Shi

2021

Mathematics

View full text Add to dashboard Cite

The Poisson surface reconstruction algorithm has become a very popular tool of reconstruction from point clouds. If we reconstruct each region separately in the process of multi-reconstruction, then the reconstructed objects may overlap with each other. In order to reconstruct multicomponent surfaces without self-intersections, we propose an efficient multi-reconstruction algorithm based on a modified vector-valued Allen–Cahn equation. The proposed algorithm produces smooth surfaces and closely preserves the original data without self-intersect. Based on operator splitting techniques, the numerical scheme is divided into one linear equation and two nonlinear equations. The linear equation is discretized using an implicit method, and the resulting discrete system of equation is solved by a fast Fourier transform. The two nonlinear equations are solved analytically due to the availability of a closed-form solution. The numerical scheme has merit in that it can be straightforwardly applied to a graphics processing unit, allowing for accelerated implementation that performs much faster than central processing unit alternatives. Various experimental, numerical results demonstrate the effectiveness and robustness of the proposed method.

show abstract

Comparison study on the different dynamics between the Allen–Cahn and the Cahn–Hilliard equations

Cited by 17 publications

References 28 publications

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

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

Benchmark Problems for the Numerical Schemes of the Phase‐Field Equations

Multi-Reconstruction from Points Cloud by Using a Modified Vector-Valued Allen–Cahn Equation

Contact Info

Product

Resources

About