This work is aim at providing a numerical technique for the Volterra integral equations using Galerkin method. For this purpose, an effective matrix formulation is proposed to solve linear Volterra integral equations of the first and second kind respectively using orthogonal polynomials as trial functions which are constructed in the interval [ ] 1,1 − with respect to the weight function () w x x 2 1 = +. The efficiency of the proposed method is tested on several numerical examples and compared with the analytic solutions available in the literature.

This paper presents Tau-collocation approximation approach for solving first and second orders ordinary differential equations. We use the method in the stimulation of numerical techniques for the approximate solution of linear initial value problems (IVP) in first and second order ordinary differential equations. The resulting numerical evidences show the method is adequate and effective.