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Spectral Jacobi-Galerkin methods and iterated methods for Fredholm integral equations of the second kind with weakly singular kernel

  • * Corresponding author: Yin Yang

    * Corresponding author: Yin Yang 

The work was supported by NSFC Project (11671342, 91430213, 11771369), Project of Scientific Research Fund of Hunan Provincial Science and Technology Department (2018JJ2374) and Key Project of Hunan Provincial Department of Education (17A210)

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  • We consider spectral and pseudo-spectral Jacobi-Galerkin methods and corresponding iterated methods for Fredholm integral equations of the second kind with weakly singular kernel. The Gauss-Jacobi quadrature formula is used to approximate the integral operator and the inner product based on the Jacobi weight is implemented in the weak formulation in the numerical implementation. We obtain the convergence rates for the approximated solution and iterated solution in weakly singular Fredholm integral equations, which show that the errors of the approximate solution decay exponentially in $L^∞$-norm and weighted $L^2$-norm. The numerical examples are given to illustrate the theoretical results.

    Mathematics Subject Classification: Primary: 65R20, 45J05; Secondary: 65N12.


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  • Figure 1.  Example 6.1 Errors of spectral Legendre-Galerkin method (left) and spectral Chebyshev-Galerkin method (right) versus $N$

    Figure 2.  Example 6.2 Errors of spectral Legendre-Galerkin method (left) and spectral Chebyshev-Galerkin method (right) versus $N$

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