# 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:
 [1] P. M. Anselone, Collectively Compact Operator Approximation Theory and Applications to Integral Equations, Prentice Hall, 1971, Englewood Cliffs. [2] K. E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, 1997, Cambridge. doi: 10.1017/CBO9780511626340. [3] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press, 2004, Cambridge. [4] C. Canuto, M. Y. Hussaini and A. Quarteroni, Spectral Methods Funda- mentals in Single Domains, Springer-Verlag, 2006. [5] Y. Chen and T. Tang, Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equation with a weakly singular kernel, Math. Comput., 79 (2010), 147-167. doi: 10.1090/S0025-5718-09-02269-8. [6] Y. Chen and T. Tang, Spectral methods for weakly singular Volterra integral equations with smooth solutions, J. Comput. Appl. Math., 233 (2009), 938-950. doi: 10.1016/j.cam.2009.08.057. [7] D. Colton and R. Kress, Inverse Coustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, Springer-Verlag, Heidelberg, 2nd Edition, 1998. doi: 10.1007/978-3-662-03537-5. [8] J. Douglas, T. Dupont and L. Wahlbin, The stability in Lq of the L2-projection into finite element function spaces, Numer. Math., 23 (1975), 193-197. doi: 10.1007/BF01400302. [9] M. A. Golberg and C. S. Chen, Discrete Projection Methods for Integral Equations, Computational Mechanics, Southampton, 1997. [10] B. Guo, Spectral Methods and Their Applications, World Scientific, Singapore, 1998. doi: 10.1142/3662. [11] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981. [12] S. Jie, T. Tao and L. Wang, Spectral Methods. Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, 41. Springer, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7. [13] A. Kufner and L. E. Persson, Weighted Inequalities of Hardy Type, World Scientific, New York, 2003. doi: 10.1142/5129. [14] G. Mastroianni and D. Occorsto, Optimal systems of nodes for Lagrange interpolation on bounded intervals: A survey, J. Comput. Appl. Math., 134 (2001), 325-341. doi: 10.1016/S0377-0427(00)00557-4. [15] P. Nevai, Mean convergence of Lagrange interpolation: Ⅲ, Trans. Am. Math. Soc., 282 (1984), 669-698. doi: 10.1090/S0002-9947-1984-0732113-4. [16] D. L. Ragozin, Polynomial approximation on compact manifolds and homogeneous spaces, Trans. Amer. Math. Soc., 150 (1970), 41-53. doi: 10.1090/S0002-9947-1970-0410210-0. [17] D. L. Ragozin, Constructive polynomial approximation on spheres and projective spaces, Trans. Amer. Math. Soc., 162 (1971), 157-170. doi: 10.2307/1995746. [18] T. Tang, X. Xu and J. Cheng, On Spectral methods for Volterra integral equation and the convergence analysis, J. Comput. Math., 26 (2008), 825-837. [19] X. Tao, Z. Xie and X. Zhou, Spectral Petrov-Galerkin methods for the second kind Volterra type integro-differential equations, Numer. Math. Theor. Meth. Appl., 4 (2011), 216-236. [20] Z. Wan, Y. Chen and Y. Huang, Legendre spectral Galerkin method for second-kind Volterra integral equations, Front. Math. China, 4 (2009), 181-193. doi: 10.1007/s11464-009-0002-z. [21] Y. Wei and Y. Chen, Convergence analysis of the spectral methods for weakly singular Volterra integro-differential equations with smooth solutions, Adv. Appl. Math. Mech., 4 (2012), 1-20. doi: 10.4208/aamm.10-m1055. [22] Z. Xie, X. Li and T. Tang, Convergence analysis of spectral Galerkin methods for Volterra type integral equations, J. Sci. Comput., 53 (2012), 414-434. doi: 10.1007/s10915-012-9577-8. [23] Y. Yang, Y. Chen and Y. Huang, Convergence analysis of the Jacobi spectral-collocation method for fractional integro-differential equations, Acta Math. Sci., 34 (2014), 673-690. doi: 10.1016/S0252-9602(14)60039-4. [24] Y. Yang, Jacobi spectral Galerkin methods for Volterra integral equations with weakly singular kernel, B. Korean Math. Soc., 53 (2016), 247-262. doi: 10.4134/BKMS.2016.53.1.247. [25] Y. Yang, Jacobi spectral Galerkin methods for fractional integro-differential equations, Calcolo, 52 (2015), 519-542. doi: 10.1007/s10092-014-0128-6. [26] Y. Yang, Y. Chen, Y. Huang and W. Yang, Convergence analysis of Legendre-collocation methods for nonlinear Volterra type integro Equations, Adv. Appl. Math. Mech., 7 (2015), 74-88. doi: 10.4208/aamm.2013.m163. [27] Y. Yang, Y. Chen, Y. Huang, H. Wei, Spectral collocation method for the time-fractional diffusion-wave equation and convergence analysis, Comput. Mathe. Appl., 73 (2017), 1218-1232. doi: 10.1016/j.camwa.2016.08.017. [28] Y. Yang, Y. Huang, Y. Zhou, Numerical solutions for solving time fractional Fokker-Planck equations based on spectral collocation methods, J. Comput. Appl. Math., 339 (2018), 389-404. doi: 10.1016/j.cam.2017.04.003.
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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