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Numerical Solutions for Weakly Singular Volterra Integral Equations Using Chebyshev and Legendre Pseudo-Spectral Galerkin Methods

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Abstract

In this paper we present and analyze Chebyshev and Legendre pseudo-spectral methods for the second kind Volterra integral equations with weakly singular kernel \((x-s)^{-\mu }, 0<\mu <1\). The proposed methods are based on the Gauss-type quadrature formula for approximating the integral operators involved in the equations. The present work is an extension of the earlier proposed spectral Jacobi–Galerkin method for the second kind Volterra integral equations with regular kernels (Xie et al. in J Sci Comput 53(2):414–434, [21]). A detailed convergence analysis is carried out, and several error estimates in \(L^{\infty } \) and \( L^2_{\omega }\) norms are obtained. Numerical examples are considered to verify the theoretical predictions.

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Author notes

  1. Tao Tang

    Present address: Department of Mathematics, South University of Science and Technology, Shenzhen, 518055, Guangdong, China

Authors and Affiliations

  1. College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350108, China

    Xianjuan Li

  2. Department of Mathematics & Institute for Computational and Theoretical Studies, Hong Kong Baptist University, Kowloon Tong, Hong Kong

    Tao Tang

  3. School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing, Xiamen University, Xiamen, 361005, China

    Chuanju Xu

Authors
  1. Xianjuan Li
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  2. Tao Tang
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  3. Chuanju Xu
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Corresponding author

Correspondence to Chuanju Xu.

Additional information

Xianjuan Li: supported by National NSF of China (11201077), NSF of Fujian Province (2012J01007), and Start-up fund of Fuzhou University (0460022456).

Tao Tang: partially supported by Hong Kong Research Grant Council GIF Grants and Hong Kong Baptist University FRG Grants.

Chuanju Xu: partially supported by National NSF of China (Grant Numbers 11471274, 11421110001, and 91130002).

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Li, X., Tang, T. & Xu, C. Numerical Solutions for Weakly Singular Volterra Integral Equations Using Chebyshev and Legendre Pseudo-Spectral Galerkin Methods. J Sci Comput 67, 43–64 (2016). https://doi.org/10.1007/s10915-015-0069-5

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  • DOI: https://doi.org/10.1007/s10915-015-0069-5

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