A numerical technique based on the shifted Legendre polynomials for solving the time-fractional coupled KdV equations

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• 作者：A. H. Bhrawy ; E. H. Doha ; S. S. Ezz-Eldien ; M. A. Abdelkawy
• 关键词：Coupled KdV equation ; Operational matrix ; Gauss quadrature ; Collocation spectral method ; Caputo derivative
• 刊名：Calcolo
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：53
• 期：1
• 页码：1-17
• 全文大小：732 KB
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• 作者单位：A. H. Bhrawy (1) (2)
  E. H. Doha (3)
  S. S. Ezz-Eldien (4)
  M. A. Abdelkawy (2)
  
  1. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
  2. Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt
  3. Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
  4. Department of Basic Science, Institute of Information Technology, Modern Academy, Cairo, Egypt
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Theory of Computation
  
• 出版者：Springer Milan
• ISSN：1126-5434

文摘

The time-fractional coupled Korteweg–de Vries (KdV) system is a generalization of the classical coupled KdV system and obtained by replacing the first order time derivatives by fractional derivatives of orders \(\nu _1\) and \(\nu _2\), \((0<\nu _1,\nu _2\le 1).\) In this paper, an accurate and robust numerical technique is proposed for solving the time-fractional coupled KdV equations. The shifted Legendre polynomials are introduced as basis functions of the collocation spectral method together with the operational matrix of fractional derivatives (described in the Caputo sense) in order to reduce the time-fractional coupled KdV equations into a problem consisting of a system of algebraic equations that greatly simplifies the problem. In order to test the efficiency and validity of the proposed numerical technique, we apply it to solve two numerical examples.