We investigate Turing pattern formation in the fractional Lengyel–Epstein model with the fractional Laplacian to describe anomalous superdiffusion. Pattern complexity is quantified using the box-counting fractal dimension. Numerical experiments compare classical and fractional diffusion and examine the effects of the parameters b, c, and the fractional order s. The results show that the fractal dimension converges after pattern formation and systematically varies with the model parameters. 

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.