AN EFFICIENT OPERATOR SPLITTING METHOD FOR A NORMALIZED TIME-FRACTIONAL ALLEN-CAHN EQUATION

초록

In this paper, we propose a normalized time-fractional Allen-Cahn (TFAC) equation, in which a time-fractional derivative replaces the conventional derivative. We apply an efficient operator splitting technique to discretize the normalized TFAC equation. Compared to the conventional AC equation, the normalized TFAC equation features a unique time scale. This unique time scale provides an intuitive perspective on the fractional time derivative, as it represents a weighted average of the temporal history of the derivative. Moreover, the total integration of the weighting function is always 1 at all times. To study the dynamic characteristics of the computational solutions of the normalized TFAC equation, we investigated the mean curvature motion with a circular initial condition under different cases. Additionally, we applied the equation to more complex shapes to observe the differences in evolution over time between the normalized TFAC equation and the traditional AC equation. The experimental results show that the parameter significantly influences the numerical solutions. By adjusting the parameter, we can control the evolution rate to achieve the desired behavior.

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