Geometry-Aware Stability and Stiffness-Robust Fractional Time Integration of Partial Differential Equations by Dual-Reciprocity Boundary Elements and Implicit Adams–Moulton Schemes

Source abstract

A geometry-aware and stiffness-robust boundary element framework is developed for time-fractional partial differential equations governed by hereditary diffusion, reaction, and transport mechanisms. The formulation couples dual-reciprocity boundary reduction with an Adams predictor and an implicit Moulton correction in which the stiff diffusive boundary element operator is retained at the new time level. This construction moves stability assessment from an isolated scalar test equation to the spectrum, numerical range, and resolvent behavior of the assembled spatial operator, so that geometry, boundary conditions, collocation nonnormality, and stiffness enter the analysis directly. A graded start-up strategy addresses the loss of temporal regularity associated with fractional memory, while compressed history evaluation provides a scalable route for long-time simulations. The theoretical development establishes modal and resolvent-based stability conditions, consistency and convergence properties, and a practical post-assembly stability certificate. Verification on smooth curved and nonconvex domains demonstrates accurate boundary-reduced spatial approximation, robust temporal convergence, resistance to diffusion-driven stiffness, and reliable evolution in the presence of corner singularities. The resulting framework provides a unified and extensible computational architecture for fractional diffusion, reaction transport, thermal memory, and related multiphysics models in geometries for which global spectral discretizations are restrictive.