Deterministic and hybrid numerical schemes for variable-order space–time fractional advection–diffusion problems

Variable-order fractional partial differential equations provide a flexible framework for transport problems with nonlocal spatial interactions and memory effects. In this paper, we propose deterministic and hybrid numerical schemes for one-dimensional variable-order space–time fractional advection–diffusion problems on bounded domains. The method combines shifted Legendre collocation in space, an L1 approximation for the variable-order Caputo derivative, and a Grünwald–Letnikov treatment of the spatial fractional operator. The deterministic scheme uses the full Grünwald–Letnikov approximation. The hybrid scheme evaluates the leading terms exactly and approximates the remaining tail by Monte Carlo importance sampling. The methods are tested on polynomial, oscillatory, published variable-order, and localized smooth benchmarks. The results show that the deterministic scheme is more robust and consistently accurate. The hybrid scheme remains effective when the nonlocal tail is sampled efficiently, and its variability decreases as the sample size increases.