Investigation and analysis of the numerical approach to solve the multi-term time-fractional advection-diffusion model

In this paper, a methodical approach is presented to approximate the multi-term fractional advection-diffusion model (MT-FAD). The Lagrange squared interpolation is used to discretize temporal fractional derivatives, and Legendre polynomials are shifted as an operator to discretize the spatial fractional derivatives. The advantage of these numerical techniques lies in the orthogonality of Legendre polynomials and its matrix operations. A quadratic implicit design as well as its stability and convergence analysis are evaluated. It should be noted that the theoretical proof obtained from this study represents the first results for these numerical schemes. Finally, we provide three numerical examples to verify the validity of the proposed methods and demonstrate their accuracy and effectiveness in comparison with previous studies shown in [W. P. Bu, X. T. Liu, Y. F. Tang, J. Y. Yang, Finite element multigrid method for multi-term time fractional advection diffusion equations, Int. J. Model. Simul. Sci. Comput., 6 (2015), 1540001].