This study investigates phase separation using the time-fractional Allen–Cahn equation with the Caputo derivative on geometric computational domains. Numerical simulations with NURBS-based collocation reveal how varying fractional orders affect phase-field behavior across different surfaces and materials. The results show phenomena such as Ostwald ripening and curve shortening flow on polyhedral and conical geometries.

This study explores the catenoid, a minimal surface formed between two circular rings, to support mathematics education. It examines the catenoid’s mathematical properties and demonstrates its creation through experiments and metaverse simulations. The work highlights how the catenoid connects geometry, calculus, and virtual learning for effective educational use.