Parameter estimation using Taylor wavelet collocation method for inverse time-fractional advection-diffusion equations with variable coefficients

In this study, the Taylor wavelet collocation method (TWCM) is introduced to obtain an effective numerical solution for variable-coefficient inverse time-fractional advection–diffusion equations. The model is designed to represent the spatiotemporal evolution of pollutant concentrations in heterogeneous media, including both soil and atmospheric environments. Accordingly, the proposed approach is particularly effective for estimating unknown model quantities, specifically the state field and the unknown time-dependent boundary function, from prescribed initial and boundary data together with an overspecified interior observation, under varying environmental conditions. By applying TWCM, the inverse problem is reduced to a finite-dimensional linear algebraic system for the Taylor-wavelet coefficients. For noisy interior observations, Tikhonov regularization is incorporated to enhance the stability of the discrete reconstruction. Comparative analyses demonstrate that TWCM significantly outperforms established numerical methods, including the rationalized Haar wavelet (RHW), radial basis function (RBF) and finite-difference benchmarks. The results highlight the robustness and effectiveness of TWCM in addressing complex inverse problems governed by time-fractional advection–diffusion equations and demonstrate its potential to provide valuable insights for environmental monitoring and management.