E. H., DohaA. H., BhrawyS. S., Ezz-Eldien2019-11-052019-11-052011https://doi.org/10.1016/j.apm.2011.05.011https://www.sciencedirect.com/science/article/pii/S0307904X11003052In this paper, we state and prove a new formula expressing explicitly the derivatives of shifted Chebyshev polynomials of any degree and for any fractional-order in terms of shifted Chebyshev polynomials themselves. We develop also a direct solution technique for solving the linear multi-order fractional differential equations (FDEs) with constant coefficients using a spectral tau method. The spatial approximation with its fractional-order derivatives (described in the Caputo sense) are based on shifted Chebyshev polynomials T(L,n)(x) with x is an element of (0,L), L > 0 and n is the polynomial degree. We presented a shifted Chebyshev collocation method with shifted Chebyshev-Gauss points used as collocation nodes for solving nonlinear multi-order fractional initial value problems. Several numerical examples are considered aiming to demonstrate the validity and applicability of the proposed techniques and to compare with the existing results. (C) 2011 Elsevier Inc. All rights reserved.enOctober University for Multi-term fractional differential equationsNonlinear fractional differential equationsTau methodCollocation methodShifted Chebyshev polynomialsGauss quadratureArticlehttps://doi.org/10.1016/j.apm.2011.05.011