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Numerical solution of fractional differential equations with a Tau method based on Legendre and Bernstein polynomials
Author(s) -
Rad J.A.,
Kazem S.,
Shaban M.,
Parand K.,
Yildirim A.
Publication year - 2014
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.2794
Subject(s) - mathematics , legendre polynomials , bernstein polynomial , classical orthogonal polynomials , algebraic equation , polynomial , associated legendre polynomials , orthogonal polynomials , degree (music) , transformation (genetics) , differential equation , gegenbauer polynomials , nonlinear system , jacobi polynomials , legendre function , discrete orthogonal polynomials , pure mathematics , mathematical analysis , biochemistry , chemistry , physics , quantum mechanics , acoustics , gene
In this paper, we state and prove a new formula expressing explicitly the integratives of Bernstein polynomials (or B‐polynomials) of any degree and for any fractional‐order in terms of B‐polynomials themselves. We derive the transformation matrices that map the Bernstein and Legendre forms of a degree‐ n polynomial on [0,1] into each other. By using their transformation matrices, we derive the operational matrices of integration and product of the Bernstein polynomials. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations. Copyright © 2013 John Wiley & Sons, Ltd.