Numerical Analysis for the Normalized Time-Fractional Modica–Mortola Functional

In this work, we study the gradient flow of the normalized time-fractional Modica–Mortola (MM) functional and propose a numerical method combining operator splitting with finite differences, where an analytical solution is used for the nonlinear reaction term to capture the memory effects inherent in fractional dynamics. Numerical experiments show the effect of the normalized fractional order on interface evolution. The results indicate that the normalized time-fractional MM functional allows flexible control of interfacial dynamics through the fractional order and provides a numerical method for simulating fractional phase-field models.