University of Guilan Journal of Mathematical Modeling 2345-394X 11 4 2023 12 01 Estimate of the fractional advection-diffusion equation with a time-fractional term based on the shifted Legendre polynomials 731 744 7070 10.22124/jmm.2023.24479.2191 EN Yones Esmaeelzade Aghdam Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785 -136, Iran Hamid Mesgarani Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785 -136, Iran Zeinab Asadi Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785 -136, Iran Journal Article 2023 05 09 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-term<br />time 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.  https://jmm.guilan.ac.ir/article_7070_3e4a0b0b76be636451737d4f01b2ad7c.pdf