An operational matrix of fractional integration of the Laguerre polynomials and its application on a semi-infinite interval

Purpose: In this paper, we construct the operational matrix of fractional integration of arbitrary order for Laguerre polynomials. Methods: We introduce some necessary definitions and give some relevant properties of Laguerre polynomials. The fractional integration is described in the Riemann-Liouville sense. We develop a direct solution technique for solving the integrated forms of fractional differential equations (FDEs) on the half line using the Laguerre tau method based on operational matrix of fractional integration in the Riemann-Liouville sense. Results: In order to show the fundamental importance of the Laguerre operational matrix, we apply it together with the spectral Laguerre tau method for the numerical solution of general linear multi-term FDEs on a semi-infinite interval. Conclusions: The results obtained by the present methods reveal that the present method is very effective and convenient for linear FDEs. Illustrative examples are included to demonstrate the validity and applicability of the new technique for linear muti-term FDEs on a semi-infinite interval.