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An Efficient Collocation Method for a Class of Boundary Value Problems Arising in Mathematical Physics and Geometry

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  • A. H. Bhrawy
  • A. S. Alofi
  • R. A. Van Gorder

Abstract

We present a numerical method for a class of boundary value problems on the unit interval which feature a type of power‐law nonlinearity. In order to numerically solve this type of nonlinear boundary value problems, we construct a kind of spectral collocation method. The spatial approximation is based on shifted Jacobi polynomials Jn(α,β)(r) with α, β ∈ (−1, ∞), r ∈ (0,1) and n the polynomial degree. The shifted Jacobi‐Gauss points are used as collocation nodes for the spectral method. After deriving the method for a rather general class of equations, we apply it to several specific examples. One natural example is a nonlinear boundary value problem related to the Yamabe problem which arises in mathematical physics and geometry. A number of specific numerical experiments demonstrate the accuracy and the efficiency of the spectral method. We discuss the extension of the method to account for more complicated forms of nonlinearity.

Suggested Citation

  • A. H. Bhrawy & A. S. Alofi & R. A. Van Gorder, 2014. "An Efficient Collocation Method for a Class of Boundary Value Problems Arising in Mathematical Physics and Geometry," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:425648
    DOI: 10.1155/2014/425648
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    References listed on IDEAS

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    1. D. Baleanu & A. H. Bhrawy & T. M. Taha, 2013. "Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    2. Fukang Yin & Junqiang Song & Yongwen Wu & Lilun Zhang, 2013. "Numerical Solution of the Fractional Partial Differential Equations by the Two-Dimensional Fractional-Order Legendre Functions," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-13, November.
    3. D. Baleanu & A. H. Bhrawy & T. M. Taha, 2013. "Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-10, June.
    4. Fukang Yin & Junqiang Song & Yongwen Wu & Lilun Zhang, 2013. "Numerical Solution of the Fractional Partial Differential Equations by the Two‐Dimensional Fractional‐Order Legendre Functions," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
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