en عمومى General پژوهشي Research paper <div style="text-align: justify;">In this work, the convection-diffusion integro-differential equation with a weakly singular kernel is discussed. The&nbsp; Legendre spectral tau method is introduced for finding the unknown function. The proposed method is based on expanding the approximate solution as the elements of a shifted Legendre polynomials. We reduce the problem to a set of algebraic equations by using operational matrices. Also the convergence analysis for&nbsp; shifted Legendre polynomials and error estimation for tau method have been discussed and approved with the exact solution. Finally, several numerical examples are given to demonstrate the high accuracy of the method.</div> Shifted Legendre tau method, Weakly singular kernel, Integro-differential equation, Convection-diffusion equation. 145 168 http://ijmsi.ir/browse.php?a_code=A-10-1568-1&slc_lang=en&sid=1 R. Pourgholi pourgholi@du.ac.ir 10031947532846008778 10031947532846008778 Yes School of Mathematics and Computer Science, A. Tahmasbi tahmasbi@du.ac.ir 10031947532846008779 10031947532846008779 No School of Mathematics and Computer Science, R. Azimi r.azimi@std.du.ac.ir 10031947532846008780 10031947532846008780 No School of Mathematics and Computer Science,