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This article develops an efficient solver based on collocation points for solving numerically a system of linear Volterra integral equations (VIEs) with variable coefficients. By using the Euler polynomials and the collocation points, this method transforms the system of linear VIEs into the matrix equation. The matrix equation corresponds to a system of linear equations with the unknown Euler coefficients. A small number of Euler polynomials is needed to obtain a satisfactory result. Numerical results with comparisons are given to confirm the reliability of the proposed method for solving VIEs with variable coefficients. MSC: 45D05; 45F05; 65D30 ª 2013 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.

In this paper, the operational matrix of Euler functions for fractional derivative of order b in the Caputo sense is derived. Via this matrix, we develop an efficient collocation method for solving nonlinear fractional Volterra integro-differential equations. Illustrative examples are given to demonstrate the validity and applicability of the proposed method, and the comparisons are made with the existing results.

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