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A Jacobi Dual‐Petrov Galerkin‐Jacobi Collocation Method for Solving Korteweg‐de Vries Equations

Author

Listed:
  • Ali H. Bhrawy
  • M. M. Al-Shomrani

Abstract

The present paper is devoted to the development of a new scheme to solve the initial‐boundary value Korteweg‐de Vries equation which models many physical phenomena such as surface water waves in a channel. The scheme consists of Jacobi dual‐Petrov Galerkin‐Jacobi collocation method in space combined with Crank‐Nicholson‐leap‐frog method in time such that at each time step only a sparse banded linear algebraic system needs to be solved. Numerical results are presented to show that the proposed numerical method is accurate and efficient for Korteweg‐de Vries equations and other third‐order nonlinear equations.

Suggested Citation

  • Ali H. Bhrawy & M. M. Al-Shomrani, 2012. "A Jacobi Dual‐Petrov Galerkin‐Jacobi Collocation Method for Solving Korteweg‐de Vries Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    • Handle: RePEc:wly:jnlaaa:v:2012:y:2012:i:1:n:418943
    DOI: 10.1155/2012/418943
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    References listed on IDEAS

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    1. A. H. Bhrawy & A. S. Alofi & S. I. El-Soubhy, 2011. "Spectral Shifted Jacobi Tau and Collocation Methods for Solving Fifth‐Order Boundary Value Problems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
    2. E. H. Doha & W. M. Abd-Elhameed, 2012. "Efficient Solutions of Multidimensional Sixth‐Order Boundary Value Problems Using Symmetric Generalized Jacobi‐Galerkin Method," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    3. E. H. Doha & W. M. Abd-Elhameed, 2012. "Efficient Solutions of Multidimensional Sixth-Order Boundary Value Problems Using Symmetric Generalized Jacobi-Galerkin Method," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-19, May.
    4. Wazwaz, Abdul-Majid, 2006. "Two reliable methods for solving variants of the KdV equation with compact and noncompact structures," Chaos, Solitons & Fractals, Elsevier, vol. 28(2), pages 454-462.
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