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
References:
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K. E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, 1997, Cambridge. doi: 10.1017/CBO9780511626340. Google Scholar

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H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press, 2004, Cambridge.Google Scholar

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C. Canuto, M. Y. Hussaini and A. Quarteroni, Spectral Methods Funda- mentals in Single Domains, Springer-Verlag, 2006. Google Scholar

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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. Google Scholar

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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. Google Scholar

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M. A. Golberg and C. S. Chen, Discrete Projection Methods for Integral Equations, Computational Mechanics, Southampton, 1997. Google Scholar

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B. Guo, Spectral Methods and Their Applications, World Scientific, Singapore, 1998. doi: 10.1142/3662. Google Scholar

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D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981. Google Scholar

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P. Nevai, Mean convergence of Lagrange interpolation: Ⅲ, Trans. Am. Math. Soc., 282 (1984), 669-698. doi: 10.1090/S0002-9947-1984-0732113-4. Google Scholar

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D. L. Ragozin, Constructive polynomial approximation on spheres and projective spaces, Trans. Amer. Math. Soc., 162 (1971), 157-170. doi: 10.2307/1995746. Google Scholar

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T. TangX. Xu and J. Cheng, On Spectral methods for Volterra integral equation and the convergence analysis, J. Comput. Math., 26 (2008), 825-837. Google Scholar

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X. TaoZ. 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. Google Scholar

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Z. WanY. 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. Google Scholar

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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. Google Scholar

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Z. XieX. 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. Google Scholar

[23]

Y. YangY. 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. Google Scholar

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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. Google Scholar

[25]

Y. Yang, Jacobi spectral Galerkin methods for fractional integro-differential equations, Calcolo, 52 (2015), 519-542. doi: 10.1007/s10092-014-0128-6. Google Scholar

[26]

Y. YangY. ChenY. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

References:
[1]

P. M. Anselone, Collectively Compact Operator Approximation Theory and Applications to Integral Equations, Prentice Hall, 1971, Englewood Cliffs. Google Scholar

[2]

K. E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, 1997, Cambridge. doi: 10.1017/CBO9780511626340. Google Scholar

[3]

H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press, 2004, Cambridge.Google Scholar

[4]

C. Canuto, M. Y. Hussaini and A. Quarteroni, Spectral Methods Funda- mentals in Single Domains, Springer-Verlag, 2006. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

J. DouglasT. 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. Google Scholar

[9]

M. A. Golberg and C. S. Chen, Discrete Projection Methods for Integral Equations, Computational Mechanics, Southampton, 1997. Google Scholar

[10]

B. Guo, Spectral Methods and Their Applications, World Scientific, Singapore, 1998. doi: 10.1142/3662. Google Scholar

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981. Google Scholar

[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. Google Scholar

[13]

A. Kufner and L. E. Persson, Weighted Inequalities of Hardy Type, World Scientific, New York, 2003. doi: 10.1142/5129. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

D. L. Ragozin, Constructive polynomial approximation on spheres and projective spaces, Trans. Amer. Math. Soc., 162 (1971), 157-170. doi: 10.2307/1995746. Google Scholar

[18]

T. TangX. Xu and J. Cheng, On Spectral methods for Volterra integral equation and the convergence analysis, J. Comput. Math., 26 (2008), 825-837. Google Scholar

[19]

X. TaoZ. 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. Google Scholar

[20]

Z. WanY. 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. Google Scholar

[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. Google Scholar

[22]

Z. XieX. 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. Google Scholar

[23]

Y. YangY. 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. Google Scholar

[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. Google Scholar

[25]

Y. Yang, Jacobi spectral Galerkin methods for fractional integro-differential equations, Calcolo, 52 (2015), 519-542. doi: 10.1007/s10092-014-0128-6. Google Scholar

[26]

Y. YangY. ChenY. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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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