On shifted Jacobi spectral approximations for solving fractional differential equations

• Published in: Applied mathematics and computation Vol. 219; no. 15; pp. 8042 - 8056
• Main Authors: Doha, E.H., Bhrawy, A.H., Baleanu, D., Ezz-Eldien, S.S.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.04.2013
• Subjects: Approximation; Caputo derivative; Derivatives; Differential equations; Initial value problems; Jacobi–Gauss–Lobatto quadrature; Mathematical analysis; Mathematical models; Multi-term fractional differential equations; Nonlinear fractional initial value problems; Nonlinearity; Shifted Jacobi polynomials; Spectra; Spectral methods; Multi-term fractional differential equations; Shifted Jacobi polynomials; Jacobi–Gauss–Lobatto quadrature; Caputo derivative; Spectral methods; Nonlinear fractional initial value problems
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2013.01.051

► A new formula of fractional-order derivatives of shifted Jacobi polynomials is proved. ► A Jacobi spectral tau approximation for solving linear FDEs with constant coefficients is proposed. ► A quadrature tau approximation is shown for linear FDEs with variable coefficients. ► A Jacobi collocation method for nonlinear multi-order FDEs is introduced. ► The advantages of using the proposed algorithms are discussed. In 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.