Fractional Allen Cahn Equation Research Articles

On high-order schemes for the space-fractional conservative Allen–Cahn equations with local and local-nonlocal operators

In this study, we focus on two fractional conservative Allen–Cahn equations with a nonlocal space-independent operator (called the RSLM operator) and a local-nonlocal space–time dependent operator (called the BBLM operator), respectively. Recently, scholars have found that the fractional Allen–Cahn equation is better than the classical equation for describing the interface thickness. Subsequently, the fractional conservative Allen–Cahn equation was used for conservative fluid dynamics. We find that two fractional conservative Allen–Cahn equations both hold the mass conservation, energy stability, and maximum bound principle. In this study, we construct second- and third-order numerical algorithms for two fractional conservative Allen–Cahn equations, and prove the mass conservation and energy decrease laws for them. To our knowledge, for the conservative Allen–Cahn equation, there are only a few works for high-order schemes with proof of their energy stability. We then present several numerical examples whose results show that the convergence orders of the numerical solutions reach the second- and third-order in time as expected. We also simulate some phase transition behaviors to be mass conservative and energy stable, as described by the fractional conservative Allen–Cahn equations, and have some interesting discoveries. It’s found that the BBML formulation can better maintain small features, but the RSML formulation is better at capturing the profile of the interface. We also compare two fractional conservation equations with the fractional Cahn–Hilliard equation, whose changes are similar to those of the RSML formulation. In particular, we propose the variable-step BDF2 and BDF3 methods, and prove the mass conservation and energy stability under a time-ratio constraint.

Numerical energy dissipation for time fractional volume-conserved Allen–Cahn model based on the ESAV and R-ESAV approaches

In this study, we consider volume-conserved numerical schemes for the volume-conserved time fractional Allen–Cahn equation. We start with the L1 scheme based on a modified exponential scalar auxiliary variable (ESAV) approach for handling the nonlinear potential term. Then we introduce the fast L1 scheme to significantly reduce the CPU time and to make the long-term computation possible. Further to reduce the error between the discrete energy and the original energy of the ESAV approach, we introduce the fast L1 scheme based on a relaxed modified ESAV (R-ESAV) idea. We rigorously show the discrete energy dissipation properties for the above three schemes, including the discrete energy boundedness law, the discrete fractional energy dissipation law, and the discrete weighted energy dissipation law. Theoretical findings are validated by a few numerical experiments.

Convergence analysis of the maximum principle preserving BDF2 scheme with variable time-steps for the space fractional Allen–Cahn equation

In this work, a fully implicit discrete scheme preserving the maximum principle is analyzed for solving the space fractional Allen–Cahn equation with variable step BDF2 in time. The convergence analysis in L∞ norm is rigorously proved under the same requirement of time-step ratios for the maximum principle by means of the kernels recombination technique. Furthermore, the unique solvability of the proposed scheme is theoretically guaranteed depending on the property of convex functions. The energy decay law with the consideration of Riesz fractional derivative is also proved under sufficient restrictions on time-steps and time-step ratios. Numerical experiments are given to validate the second order accuracy in time, the maximum principle as well as the energy decay law of the proposed numerical scheme. In particular, comparisons of errors and time levels for the adaptive time-stepping strategy and uniform time steps are conducted to demonstrate the efficiency of adaptive time-stepping strategy.

An efficient technique based on barycentric interpolation collocation method for the time fractional Allen-Cahn equation

An efficient technique based on barycentric interpolation collocation method for the time fractional Allen-Cahn equation

Second-order linear adaptive time-stepping schemes for the fractional Allen–Cahn equation

Second-order linear adaptive time-stepping schemes for the fractional Allen–Cahn equation

A High-Order Discrete Energy Decay and Maximum-Principle Preserving Scheme for Time Fractional Allen–Cahn Equation

A High-Order Discrete Energy Decay and Maximum-Principle Preserving Scheme for Time Fractional Allen–Cahn Equation

An adaptive unconditional maximum principle preserving and energy stability scheme for the space fractional Allen-Cahn equation

An adaptive unconditional maximum principle preserving and energy stability scheme for the space fractional Allen-Cahn equation

Exact special solutions of space–time fractional Cahn–Allen equation by beta and M-truncated derivatives

In this paper, we consider the nonlinear space–time fractional form of Cahn–Allen equation (FCAE) with beta and M-truncated derivatives. Cahn–Allen equation (CAE) is commonly used in many problems of physics and engineering, such as, solidification problems, phase separation in iron alloys and others. We apply the improved [Formula: see text]-expansion method (ITEM). We obtain four types of traveling wave solutions, including, trigonometric, hyperbolic, rational and exponential function solutions. We demonstrate some of the extracted solutions using definitions of the beta (BD) and M-truncated derivatives (MTD) to understand their dynamical behavior. We observe the fractional effects of the aforementioned derivatives on the related physical phenomena up to possible extent.

The Allen-Cahn equation with a time Caputo-Hadamard derivative: Mathematical and Numerical Analysis

<abstract><p>In this paper, we investigate the local discontinuous Galerkin (LDG) finite element method for the fractional Allen-Cahn equation with Caputo-Hadamard derivative in the time domain. First, the regularity of the solution is analyzed, and the results indicate that the solution of this equation generally possesses initial weak regularity in the time dimension. Due to this property, a logarithmic nonuniform L1 scheme is adopted to approximate the Caputo-Hadamard derivative, while the LDG method is used for spatial discretization. The stability and convergence of this fully discrete scheme are proven using a discrete fractional Gronwall inequality. Numerical examples demonstrate the effectiveness of this method.</p></abstract>

Convergence of level sets in fractional Laplacian regularization

The use of the fractional Laplacian in image denoising and regularization of inverse problems has enjoyed a recent surge in popularity, since for discontinuous functions it can behave less aggressively than methods based on H 1 norms, while being linear and computable with fast spectral numerical methods. In this work, we examine denoising and linear inverse problems regularized with fractional Laplacian in the vanishing noise and regularization parameter regime. The clean data is assumed piecewise constant in the first case, and continuous and satisfying a source condition in the second. In these settings, we prove results of convergence of level set boundaries with respect to Hausdorff distance, and additionally convergence rates in the case of denoising and indicatrix clean data. The main technical tool for this purpose is a family of barriers constructed by Savin and Valdinoci for studying the fractional Allen–Cahn equation. To help put these fractional methods in context, comparisons with the total variation and classical Laplacian are provided throughout.

A new linearized maximum principle preserving and energy stability scheme for the space fractional Allen-Cahn equation

A new linearized maximum principle preserving and energy stability scheme for the space fractional Allen-Cahn equation

Global solutions of nonlinear fractional diffusion equations with time-singular sources and perturbed orders

In a Hilbert space, we consider a class of nonlinear fractional equations having the Caputo fractional derivative of the time variable t and the space fractional function of the self-adjoint positive unbounded operator. We consider various cases of global Lipschitz and local Lipschitz source with time-singular coefficient. These sources are generalized of the well–known fractional equations such as the fractional Cahn–Allen equation, the fractional Burger equation, the fractional Cahn–Hilliard equation, the fractional Kuramoto–Sivashinsky equation, etc. Under suitable assumptions, we investigate the existence, uniqueness of maximal solution, and stability of solution of the problems with respect to perturbed fractional orders. We also establish some global existence and prove that the global solution can be approximated by known asymptotic functions as \(t\rightarrow \infty \).

Regarding on the Results for the Fractional Clannish Random Walker’s Parabolic Equation and the Nonlinear Fractional Cahn-Allen Equation

In this research, the Ψ , Φ -expansion scheme has been implemented for the exact solutions of the fractional Clannish Random Walker’s parabolic (FCRWP) equation and the nonlinear fractional Cahn-Allen (NFCA) equation. Some new solutions of the FCRWP equation and the NFCA equation have been obtained by using this method. The diverse variety of exact outcomes such as intersection between rough wave and kinky soliton wave profiles, intersection between lump wave and kinky soliton wave profiles, soliton wave profiles, kink wave profiles, intersection between lump wave and periodic wave profiles, intersection between rough wave and periodic wave profiles, periodic wave profiles, and kink wave profiles are taken. Comparing our developed answers and that got in previously written research papers presents the novelty of our investigation. The above techniques could also be employed to get exact solutions for other fractional nonlinear models in physics, applied mathematics, and engineering.

Preconditioned Pseudo-Spectral Gradient Flow for Computing the Steady-State of Space Fractional Cahn-Allen Equations With Variable Coefficients

The aim of this paper is to propose some efficient and accurate numerical methods to compute the steady-state of variable coefficients space fractional Cahn-Allen equations. The approach combines an adaptive time stepping semi-implicit gradient flow method to minimize the fractional energy functional and pseudo-spectral approximation schemes. Based on the use of a preconditioned GMRES, the space fractional Cahn-Allen equation is then solved efficiently. The full methodology is supported by the numerical solution of a one-dimensional problem.

V-shaped traveling fronts of fractional Allen–Cahn equations

This paper is concerned with the existence and stability of V-shaped traveling fronts for the fractional Allen–Cahn equation in R2. By constructing a pair of super- and subsolutions, we establish the existence of V-shaped traveling fronts for each admissible speed. Furthermore, we show the uniqueness and stability of V-shaped traveling fronts.

The Adaptive Second-Order Stable Schemes for the Fractional Allen–Cahn Equation

The Adaptive Second-Order Stable Schemes for the Fractional Allen–Cahn Equation

Extraction of new exact soliton solutions and Painlevé-test for fractional Cahn–Allen equation

Extraction of new exact soliton solutions and Painlevé-test for fractional Cahn–Allen equation

On the existence of cylindrically symmetric traveling fronts of fractional Allen-Cahn equation in $\mathbb{R}^{3}$

This paper is concerned with the existence of cylindrically symmetric traveling fronts for bistable fractional Allen-Cahn equation in $\mathbb{R}^{3}$. By constructing a sequence of pyramidal traveling fronts and taking its limit, we obtain a traveling front with cylindrically symmetric type.

Metastable speeds in the fractional Allen–Cahn equation

We study numerically the one-dimensional Allen–Cahn equation with the spectral fractional Laplacian (−Δ)α/2 on intervals with homogeneous Neumann boundary conditions. In particular, we are interested in the speed of sharp interfaces approaching and annihilating each other. This process is known to be exponentially slow in the case of the classical Laplacian. Here we investigate how the width and speed of the interfaces change if we vary the exponent α of the fractional Laplacian. For the associated model on the real line we derive asymptotic formulas for the interface speed and time–to–collision in terms of α and a scaling parameter ε. We use a numerical approach via a finite-element method based upon extending the fractional Laplacian to a cylinder in the upper-half plane, and compute the interface speed, time–to–collapse and interface width for α∈(0.3,2]. A comparison shows that the asymptotic formulas for the interface speed and time–to–collision give a good approximation for large intervals.

An efficient second order stabilized scheme for the two dimensional time fractional Allen-Cahn equation

In this paper, we give a stabilized second order scheme for the time fractional Allen-Cahn equation. The scheme uses the fractional backward difference formula (FBDF) for the time fractional derivative and the Legendre spectral method for the space approximation. The nonlinear terms are treated implicitly with a second order stabilized term. Based on the fractional Grönwall inequality, we strictly prove that the proposed scheme converges to second order accuracy in time and spectral accuracy in space. To save computation time and storage, a fast evaluation is developed. Finally, we give some numerical examples to show the configurations of phase field evolution and verify the effectiveness of the proposed methods.