Estimate of the fractional advection-diffusion equation with a time-fractional term based on the shifted Legendre polynomials

In this paper, we present a well-organized strategy to estimate the fractional advection-diffusion equations, which is an important class of equations that arises in many application fields. Thus,  Lagrange square interpolation is applied in the discretization of the fractional temporal derivative, and the weighted and shifted Legendre polynomials as operators are exploited to discretize the spatial fractional derivatives of the space-fractional term in multi-termtime fractional advection-diffusion model. The privilege of the numerical method is the orthogonality of Legendre polynomials and its operational matrices which reduces time computation and increases speed. A second-order implicit technique is given, and its stability and convergence are investigated. Finally, we propose three numerical examples to check the validity and numerical results    to illustrate the precision and efficiency of the new approach.