A fractional-adaptive Gegenbauer spectral collocation method for time-fractional Allen-Cahn equations

The time-fractional Allen-Cahn equation with Caputo derivative order σ ∈ (0, 1] generalizes phase-field models to capture memory effects and anomalous diffusion in materials science. Its numerical solution faces the critical challenge of weak initial singularities that degrade the accuracy of conventional methods. To overcome this, we develop a spectral collocation scheme that combines fractional Gegenbauer functions in time and matches the solution behavior at t = 0 combined with shifted Gegenbauer polynomials in space. Our approach uses differentiation matrices for the fractional derivative computation and handles nonlinearity through collocation at Gegenbauer roots. Numerical results demonstrate spectral accuracy for both smooth solutions and challenging singular cases, validating our rigorous treatment of initial singularities.