# American Institute of Mathematical Sciences

June  2019, 12(3): 685-702. doi: 10.3934/dcdss.2019043

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

 1 Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan, China 2 Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan, China

* Corresponding author: Yin Yang

Received  April 2017 Revised  August 2017 Published  September 2018

Fund Project: 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).

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.

Citation: Yin Yang, Yunqing Huang. Spectral Jacobi-Galerkin methods and iterated methods for Fredholm integral equations of the second kind with weakly singular kernel. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 685-702. doi: 10.3934/dcdss.2019043
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##### References:
Example 6.1 Errors of spectral Legendre-Galerkin method (left) and spectral Chebyshev-Galerkin method (right) versus $N$
Example 6.2 Errors of spectral Legendre-Galerkin method (left) and spectral Chebyshev-Galerkin method (right) versus $N$
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