ANALYTICAL EIGENFUNCTION EXPANSION FOR TEMPERED TIME-FRACTIONAL ADVECTION DIFFUSION IN BOUNDED DOMAINS

This work develops an analytical–computational approach for solving the tempered time-fractional advection–diffusion equation on a rectangular domain with homogeneous Dirichlet boundary conditions. The model, based on the Caputo tempered fractional derivative, captures anomalous transport, transitioning from nonlocal, memory–driven subdiffusion to standard diffusion. A closed–form series solution is obtained by combining an eigenfunction expansion of sine spatial modes with a Laplace transform in time, in which the temporal dynamics are governed by tempered Mittag–Leffler kernels. The solution demonstrates consistency with established models: As the fractional order $ \alpha \to 1 $, it converges to the classical diffusion equation with an effective sink. When the tempering parameter $ \theta = 0 $, it reduces to the standard Caputo formulation. The framework efficiently transforms the original boundary-value problem into a sum of quickly converging modes, thereby enabling accurate numerical calculations using a limited number of terms from the eigenfunction expansion. Theoretical and numerical analyses, including surface plots and error tables, confirm the convergence and high accuracy of the method. Parametric analysis shows that the fractional order $ \alpha $ mainly controls the memory effects and spatial spread characteristics of subdiffusion, whereas the tempering parameter $ \theta $ introduces an exponential truncation that accelerates temporal decay. The effectiveness of the method is illustrated through a street canyon application, where pollutant build-up, delayed dispersion, and subsequent clearance under limited ventilation are captured. This study not only underscores the framework's relevance for urban air quality modeling but also establishes it as a valuable benchmark for validating numerical solvers and reduced-order models.