Doha, E. HBhrawy, AHBaleanu, DEzz-Eldien, S. S.2019-11-112019-11-112013Cited References in Web of Science Core Collection: 490096-3003https://doi.org/10.1016/j.amc.2013.01.051https://www.sciencedirect.com/science/article/abs/pii/S009630031300091XAccession Number: WOS:000318051700014In this paper, a new formula of Caputo fractional-order derivatives of shifted Jacobi polynomials of any degree in terms of shifted Jacobi polynomials themselves is proved. We discuss a direct solution technique for linear multi-order fractional differential equations (FDEs) subject to nonhomogeneous initial conditions using a shifted Jacobi tau approximation. A quadrature shifted Jacobi tau (Q-SJT) approximation is introduced for the solution of linear multi-order FDEs with variable coefficients. We also propose a shifted Jacobi collocation technique for solving nonlinear multi-order fractional initial value. problems. The advantages of using the proposed techniques are discussed and we compare them with other existing methods. We investigate some illustrative examples of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques. (C) 2013 Elsevier Inc. All rights reservedenUniversity for Multi-term fractional differential equationsNonlinear fractional initial value problemsSpectral methodsShifted Jacobi polynomialsJacobi-Gauss-Lobatto quadratureCaputo derivativeArticlehttps://doi.org/10.1016/j.amc.2013.01.051