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

Abstract

This research presents a computational framework for the numerical solution and stability analysis of time-fractional partial differential equations on complex two-dimensional domains using the dual reciprocity boundary element method combined with an implicit Adams-Bashforth-Moulton predictor-corrector time-integration strategy.The proposed formulation transforms the governing fractional partial differential equation into a boundary-element-based semidiscrete system, thereby retaining the geometric flexibility of boundary element methods while avoiding conventional domain discretization. The dual reciprocity technique is employed to approximate domain and source contributions, leading to a matrix fractional differential system whose stability characteristics can be examined through the spectrum of the resulting boundary element operator.A principal contribution of the study is the development of a stability framework that connects the eigenvalue distribution of the boundary element system matrix with the stability behavior of the fractional Adams-Bashforth-Moulton scheme. The analysis is complemented by nonnormal resolvent considerations to provide a more robust assessment of transient amplification and stiffness effects that may not be captured by eigenvalue information alone.The numerical methodology incorporates an implicit Adams-Moulton correction to improve robustness for stiff fractional systems, while an Adams-Bashforth predictor provides an efficient estimate of the next solution state. Graded temporal meshes are considered to improve accuracy in the presence of the weak initial-time singularities commonly associated with fractional-order models. Fast history-evaluation concepts are also discussed to reduce the computational and memory costs arising from the nonlocal character of the Caputo fractional derivative.The method is assessed using benchmark problems defined on both smooth and nonconvex geometries, including circular and L-shaped computational domains. Numerical investigations examine spatial convergence, temporal convergence, spectral stability, long-time decay behavior, sensitivity to the fractional order, and the influence of boundary discretization. The results demonstrate stable and accurate performance across a range of fractional orders and geometrical configurations, while confirming the characteristic memory-dependent behavior of anomalous diffusion processes.The accompanying Figshare repository contains the reproducibility material associated with the study, including source code, benchmark definitions, numerical input data, machine-readable result tables, figure-generation scripts, reusable boundary element matrices, regression tests, environment specifications, metadata files, and documentation required to reproduce the principal computational results reported in the manuscript.The repository is intended to support transparent verification, independent reproduction, further methodological development, and reuse of the proposed DRBEM-based fractional PDE framework in computational mathematics, fractional calculus, anomalous transport, and boundary element research.