High‐Order Spectral and Pseudospectral Schemes for Nonlinear Fractional PDEs on Complex Geometries

A unified high‐order spectral–pseudospectral framework was proposed for the efficient numerical solution of nonlinear fractional partial differential equations (FPDEs) on irregular and curved domains. The method combined eigenfunction‐based spectral discretizations with grid‐based pseudospectral operators to accurately resolve smooth solutions, strong nonlinearities, and nonlocal fractional operators. Complex geometries were treated using domain mappings, spectral elements, and a flexible fictitious‐domain embedding with penalty or masking strategies that preserved spectral convergence. Stable time integration was achieved through exponential time‐differencing and energy‐stable implicit–explicit schemes tailored to fractional diffusion, dispersive, and reaction–diffusion dynamics. The framework was applied to fractional diffusion equations, fractional Schrödinger models, Fisher‐type systems, and the Schnakenberg activator–inhibitor model, where long‐range interactions and fine‐scale pattern formation were successfully captured. Rigorous stability and convergence estimates were established in fractional Sobolev norms. Numerical experiments on benchmark problems with complex geometries confirmed spectral accuracy, excellent agreement with reference solutions, and high computational efficiency. The approach was shown to naturally extend to three‐dimensional and manifold domains through FFT‐accelerated, matrix‐free implementations.