A robust numerical scheme and analysis for a class of multi-term time-fractional advection–diffusion equation with variable coefficients

One of the most significant mathematical models for transport processes is the advection–diffusion model. Numerous investigations demonstrate that this model is no longer accurate for heterogeneous diffusing particles found in anomalous diffusion in nature. Fractional partial differential equations have been developed to address this issue. Additionally, it is thought that these fractional advection–diffusion equations offer a far better explanation of issues with heat transmission, such as thermal pollution in river systems and the leaching of salts in soils. For the purpose of solving a class of multi-term time-fractional advection–diffusion equations with variable coefficients, we offer a numerical method in this study that operates on a uniform mesh. The classical L1–2 scheme is used to discretize the fractional time derivative defined in the Caputo sense, and the quintic B-spline approach is used to discretize the spatial derivatives. Analysis of the suggested method’s stability and convergence reveals fourth-order precision in space and \((3-\max (\beta _{i}))\) $(3-max({\beta }_i))$ order accuracy in time, where \(\beta _{i} (0<\beta _{i}\le 1)\) ${\beta }_i(0<{\beta }_i\le 1)$ denotes multi-fractional orders. To demonstrate the effectiveness and precision of the procedure validating the theoretical results, several numerical examples are provided.