A New Legendre Spectral Galerkin and Pseudo-Spectral Approximations for Fractional Initial Value Problems

• Published in: Abstract and Applied Analysis Vol. 2013; no. 2013; pp. 237 - 246-322
• Main Authors: Bhrawy, Ali H., Alghamdi, Mohammad Ali
• Format: Journal Article
• Language: English
• Published: Cairo, Egypt Hindawi Limiteds 01.01.2013; Hindawi Puplishing Corporation; Hindawi Publishing Corporation; John Wiley & Sons, Inc; Wiley
• Subjects: Algorithms; Approximation; Approximation theory; Boundary value problems; Differential equations; Functions (mathematics); Galerkin methods; Initial conditions; Mathematical analysis; Mathematical research; Numerical analysis; Polynomials; Spectra; Qatar
• ISSN: 1085-3375; 1687-0409
• DOI: 10.1155/2013/306746

We extend the application of the Galerkin method for treating the multiterm fractional differential equations (FDEs) subject to initial conditions. A new shifted Legendre-Galerkin basis is constructed which satisfies exactly the homogeneous initial conditions by expanding the unknown variable using a new polynomial basis of functions which is built upon the shifted Legendre polynomials. A new spectral collocation approximation based on the Gauss-Lobatto quadrature nodes of shifted Legendre polynomials is investigated for solving the nonlinear multiterm FDEs. The main advantage of this approximation is that the solution is expanding by a truncated series of Legendre-Galerkin basis functions. Illustrative examples are presented to ensure the high accuracy and effectiveness of the proposed algorithms are discussed.