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In this paper, a new set of functions called fractional-order Euler functions (FEFs) is constructed to obtain the solution of fractional integro-differential equations. The properties of the fractional-order Euler functions are utilized to construct the operational matrix of fractional integration. By using the matrix and the functions approximation, the fractional integro-differential equations are reduced to systems of algebraic equations. The convergence analysis of fractional-order Euler functions approximation is given. Illustrative examples are included to demonstrate the high precision and good performance of the new scheme.

Fractional-order Euler functions for solving fractional integro-differential equations with weakly singular kernel