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n this paper, we have implemented the finite element method for the numerical solution of a boundary and initial value problems, mainly on solving the one and two-dimensional advection-diffusion equation with constant parameters. In doing so, the basic idea is to first rewrite the problem as a variational equation, and then seek a solution approximation from the space of continuous piece-wise linear’s. This discretization procedure results in a linear system that can be solved by using a numerical algorithm for systems of these equations. The techniques are based on the finite element approximations using Galerkin’s method in space resulting system of the first order ODE’s and then solving this first order ODE’s using backward Euler descritization in time. For the two-dimensional problems, we use the ODE solver ODE15I to descritize time. The validity of the numerical model is verified using different
test examples. The computed results showed that the use of the current method is very applicable for the solution of the advection-diffusion equation.