efficient numerical approximation of distributed-order fractional pdes using gl1-2 time discretization and second-order riesz operators

This paper presents a novel and efficient fully discrete numerical scheme for distributed-order fractional partial differential equations involving both the caputo time-fractional derivative and the riesz space-fractional derivative. such equations frequently arise in the modeling of anomalous diffusion and transport phenomena, where accurate and stable computational methods are crucial. the temporal discretization is carried out using the second-order generalized l1 (gl1-2) scheme, which improves accuracy over traditional l1-based methods. for the spatial discretization, the riesz derivative is approximated by a second-order finite-difference method, ensuring robustness and precision. the resulting scheme provides a high-order numerical framework that can effectively address a wide class of distributed-order fractional models. a rigorous theoretical analysis is conducted, proving unconditional stability and optimal convergence rates via the energy method. the effectiveness of the scheme is further validated through two numerical experiments, which confirm the theoretical results and highlight the computational efficiency, accuracy, and practical applicability of the proposed approach.