Time-fractional Allen-Cahn Equation Research Articles

Optimal H1 spatial convergence of a fully discrete finite element method for the time-fractional Allen-Cahn equation

A time-fractional Allen-Cahn problem is considered, where the spatial domain Ω is a bounded subset of $\mathbb {R}^{d}$ for some d ∈{1,2,3}. New bounds on certain derivatives of the solution are derived. These are used in the analysis of a numerical method (L1 discretization of the temporal fractional derivative on a graded mesh, with a standard finite element discretization of the spatial diffusion term, and Newton linearization of the nonlinear driving term), showing that the computed solution achieves the optimal rate of convergence in the Sobolev H1(Ω) norm. (Previous papers considered only convergence in L2(Ω).) Numerical results confirm our theoretical findings.

Adaptive linear second-order energy stable schemes for time-fractional Allen-Cahn equation with volume constraint

A time-fractional Allen-Cahn equation with volume constraint is first proposed by introducing a nonlocal time-dependent Lagrange multiplier. Adaptive linear second-order energy stable schemes are developed for the proposed model by combining invariant energy quadratization and scalar auxiliary variable approaches with the recent L1+ formula. The new developed methods are proved to be volume-preserving and unconditionally energy stable on arbitrary nonuniform time meshes. The accelerated algorithm and adaptive time strategy are employed in numerical implementation. Numerical results show that the proposed algorithms are computationally efficient in multi-scale simulations, and appropriate for accurately resolving the intrinsically initial singularity of solution and for efficiently capturing the fast dynamics away initial time.

A second-order and nonuniform time-stepping maximum-principle preserving scheme for time-fractional Allen-Cahn equations

In this work, we present a second-order nonuniform time-stepping scheme for the time-fractional Allen-Cahn equation. We show that the proposed scheme preserves the discrete maximum principle, and by using the convolution structure of consistency error, we present sharp maximum-norm error estimates which reflect the temporal regularity. As our analysis is built on nonuniform time steps, we may resolve the intrinsic initial singularity by using the graded meshes. Moreover, we propose an adaptive time-stepping strategy for large time simulations. Numerical experiments are presented to show the effectiveness of the proposed scheme. This seems to be the first second-order maximum principle preserving scheme for the time-fractional Allen-Cahn equation.

Simple maximum principle preserving time-stepping methods for time-fractional Allen-Cahn equation

Two fast L1 time-stepping methods, including the backward Euler and stabilized semi-implicit schemes, are suggested for the time-fractional Allen-Cahn equation with Caputo's derivative. The time mesh is refined near the initial time to resolve the intrinsically initial singularity of solution, and unequal time-steps are always incorporated into our approaches so that an adaptive time-stepping strategy can be used in long-time simulations. It is shown that the proposed schemes using the fast L1 formula preserve the discrete maximum principle. Sharp error estimates reflecting the time regularity of solution are established by applying the discrete fractional Gr\"{o}nwall inequality and global consistency analysis. Numerical experiments are presented to show the effectiveness of our methods and to confirm our analysis.

On finite series solutions of conformable time-fractional Cahn-Allen equation

AbstractThe aim of this article is to derive new exact solutions of conformable time-fractional Cahn-Allen equation. We have achieved this aim by hyperbolic function and expa function methods with the aid of symbolic computation using Mathematica. This idea seems to be very easy to employ with reliable results. The time fractional Cahn-Allen equation is reduced to respective nonlinear ordinary differential equation of fractional order. Also, we have depicted graphically the constructed solutions.

A fast Galerkin finite element method for a space–time fractional Allen–Cahn equation

In this paper, a space–time fractional Allen–Cahn equation is investigated to account for the memory effect of certain materials or the anomalous diffusion processes in heterogeneous distributed media for the mixture of two immiscible phases. Due to the nonlocal features of fractional derivatives, conventional solvers such as the Gauss elimination method require O(M2+MN) memory units and O(MN2+M3N) operations to solve the space–time fractional Allen–Cahn equation, where M is the number of spatial unknowns and N is the number of time steps. By exploring the special structure of the stiffness matrix and utilizing the “Invariant Energy Quadratization” approach, we developed an efficient linearized divide-and-conquer Galerkin finite element method without resorting to any lossy compression. This new method is efficient as it significantly reduces the memory requirement to O(MN) and the operation count to O(MN(log2N+logM)). We have developed a fast Galerkin finite element method with local refinement that allows accurate capture of the coarsening dynamics at the interface. Furthermore, we have investigated the modeling capacities and energy dissipation properties of the space–time fractional Allen–Cahn equation. Numerical experiments are presented to demonstrate the efficiency of the new methods and the flexibility of turnable sharpness and decay behavior of the space–time fractional Allen–Cahn equation.

Multi-solitary wave solutions to the general time fractional Sharma–Tasso–Olver equation and the time fractional Cahn–Allen equation

The space time fractional Sharma–Tasso–Olver (STO) equation and the time fractional Cahn–Allen (CA) equation are well-designated to fission and fusion phenomena including those for solitons, electromagnetic interactions, quantum relativistic atom theory, phase isolation in several components bass system, the relativistic energy-momentum relation. In this article, we have exerted the exact solution to STO and CA equation in the light of fractional derivative (e.g. modified Riemann–Liouville derivative). By using wave transformation the ODE of integer order is generated by the fractional partial differential equation (FPDE). We investigated the travelling wave solution to these equations through the recently established double-expansion method. The results obtained in view of hyperbolic, trigonometric and rational functions containing parameters. We have demonstrated that this method is convenient, effective and powerful tools for solving nonlinear fractional differential equations (NLFDEs).

On Energy Dissipation Theory and Numerical Stability for Time-Fractional Phase-Field Equations

For the time-fractional phase field models, the corresponding energy dissipation law has not been settled on both the continuous level and the discrete level. In this work, we shall address this open issue. More precisely, we prove for the first time that the time-fractional phase field models indeed admit an energy dissipation law of an integral type. In the discrete level, we propose a class of finite difference schemes that can inherit the theoretical energy stability. Our discussion covers the time-fractional gradient systems, including the time-fractional Allen-Cahn equation, the time-fractional Cahn-Hilliard equation, and the time-fractional molecular beam epitaxy models. Numerical examples are presented to confirm the theoretical results. Moreover, a numerical study of the coarsening rate of random initial states depending on the fractional parameter $\alpha$ reveals that there are several coarsening stages for both time-fractional Cahn-Hilliard equation and time-fractional molecular beam epitaxy model, while there exists a $-\alpha/3$ power law coarsening stage.

An iterative approximation for time-fractional Cahn–Allen equation with reproducing kernel method

In this article, we construct a novel iterative approach that depends on reproducing kernel method for Cahn–Allen equation with Caputo derivative. Representation of solution and convergence analysis are presented theoretically. Numerical results are given as tables and graphics with intent to show efficiency and power of method. The results demonstrate that approximate solution uniformly converges to exact solution for Cahn–Allen equation with fractional derivative.

New solitary wave solutions of the time-fractional Cahn-Allen equation via the improved ( $\frac{G'}{G}$ G G )-expansion method

An improved $(\frac{G'}{G})$ -expansion method is proposed for extracting more general solitary wave solutions of the nonlinear fractional Cahn-Allen equation. The temporal fractional derivative is taken in the sense of Jumarie’s fractional derivative. The results of this article are generalized and extended version of previously reported solutions.

On time fractional Cahn-Allen equation

In [1], Ozkan Guner et al. obtained some exact solutions of the time fractional Cahn-Allen equation. By using the method proposed in [10], we have tested these solutions and have found that they are not the solutions of this equation.

A reliable method for the space-time fractional Burgers and time-fractional Cahn-Allen equations via the FRDTM

We propose a new method called the fractional reduced differential transform method (FRDTM) to solve nonlinear fractional partial differential equations such as the space-time fractional Burgers equations and the time-fractional Cahn-Allen equation. The solutions are given in the form of series with easily computable terms. Numerical solutions are calculated for the fractional Burgers and Cahn-Allen equations to show the nature of solutions as the fractional derivative parameter is changed. The results prove that the proposed method is very effective and simple for obtaining approximate solutions of nonlinear fractional partial differential equations.

New approach for exact solutions of time fractional Cahn–Allen equation and time fractional Phi-4 equation

In this article, new exact analytical solutions of some nonlinear evolution equations (NLEEs) arising in science, engineering and mathematical physics, namely time fractional Cahn–Allen equation and time fractional Phi-4 equation are developed using tanh method by means of fractional complex transform. The obtained results are demonstrated by graphs for the new solutions.

Elliptic and solitary wave solutions for Bogoyavlenskii equations system, couple Boiti-Leon-Pempinelli equations system and Time-fractional Cahn-Allen equation

In this article and for the first time, we introduce and describe Khater method which is a new technique for solving nonlinear partial differential equations (PDEs.). We apply this method for each of the following models Bogoyavlenskii equation, couple Boiti-Leon-Pempinelli system and Time-fractional Cahn-Allen equation. Khater method is very powerful, Effective, felicitous and fabulous method to get exact and solitary wave solution of (PDEs.). Not only just like that but it considers too one of the general methods for solving that kind of equations since it involves some methods as we will see in our discuss of the results. We make a comparison between the results of this new method and another method.

A variety of exact solutions for the time fractional Cahn-Allen equation

In this paper, the nonlinear time fractional Cahn-Allen equation is studied by three distinct methods. These methods are also applied to derive a variety of travelling wave solutions with distinct physical structures for this nonlinear fractional equation. As a result, different types of exact solutions are obtained. The three methods demonstrate power, reliability and efficiency.