Numerical solution of fractional differential equations with a Tau method based on Legendre and Bernstein polynomials |
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Authors: | JA Rad S Kazem M Shaban K Parand A Yildirim |
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Institution: | 1. Department of Computer Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, , Tehran 19839, Iran;2. Department of Applied Mathematics, Imam Khomeini International University, , Ghazvin 34149‐16818, Iran;3. Department of Physics, Shahid Beheshti University, , G.C., Tehran, Iran;4. Department of Mathematics, Ege University, , Bornova, Izmir 35100, Turkey |
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Abstract: | 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. |
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Keywords: | Bernstein polynomials operational matrices fractional‐order differential equations Tau method Legendre polynomials |
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