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 and Continuous Dynamical Systems - S, 2019, 12 (3) : 685-702. doi: 10.3934/dcdss.2019043
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. 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.

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

[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. 

[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.

[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. 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.

[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.

[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. 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.

[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.

show all references

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. 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.

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

[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. 

[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.

[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. 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.

[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.

[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. 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.

[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.

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 $N$
[1]

A. Pedas, G. Vainikko. Smoothing transformation and piecewise polynomial projection methods for weakly singular Fredholm integral equations. Communications on Pure and Applied Analysis, 2006, 5 (2) : 395-413. doi: 10.3934/cpaa.2006.5.395

[2]

T. Diogo, N. B. Franco, P. Lima. High order product integration methods for a Volterra integral equation with logarithmic singular kernel. Communications on Pure and Applied Analysis, 2004, 3 (2) : 217-235. doi: 10.3934/cpaa.2004.3.217

[3]

Xiaomeng Li, Qiang Xu, Ailing Zhu. Weak Galerkin mixed finite element methods for parabolic equations with memory. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 513-531. doi: 10.3934/dcdss.2019034

[4]

Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, 2021, 29 (3) : 2489-2516. doi: 10.3934/era.2020126

[5]

Oleksandr Boichuk, Victor Feruk. Boundary-value problems for weakly singular integral equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1379-1395. doi: 10.3934/dcdsb.2021094

[6]

Zhong-Qing Wang, Jing-Xia Wu. Generalized Jacobi rational spectral methods with essential imposition of Neumann boundary conditions in unbounded domains. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 325-346. doi: 10.3934/dcdsb.2012.17.325

[7]

Adriano De Cezaro, Johann Baumeister, Antonio Leitão. Modified iterated Tikhonov methods for solving systems of nonlinear ill-posed equations. Inverse Problems and Imaging, 2011, 5 (1) : 1-17. doi: 10.3934/ipi.2011.5.1

[8]

Yinhua Xia, Yan Xu, Chi-Wang Shu. Efficient time discretization for local discontinuous Galerkin methods. Discrete and Continuous Dynamical Systems - B, 2007, 8 (3) : 677-693. doi: 10.3934/dcdsb.2007.8.677

[9]

Mohammad Asadzadeh, Anders Brahme, Jiping Xin. Galerkin methods for primary ion transport in inhomogeneous media. Kinetic and Related Models, 2010, 3 (3) : 373-394. doi: 10.3934/krm.2010.3.373

[10]

Qiumei Huang, Xiuxiu Xu, Hermann Brunner. Continuous Galerkin methods on quasi-geometric meshes for delay differential equations of pantograph type. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5423-5443. doi: 10.3934/dcds.2016039

[11]

Jie Shen, Li-Lian Wang. Laguerre and composite Legendre-Laguerre Dual-Petrov-Galerkin methods for third-order equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1381-1402. doi: 10.3934/dcdsb.2006.6.1381

[12]

Shan Li, Shi-Mi Yan, Zhong-Qing Wang. Efficient Legendre dual-Petrov-Galerkin methods for odd-order differential equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1543-1563. doi: 10.3934/dcdsb.2019239

[13]

Z. K. Eshkuvatov, M. Kammuji, Bachok M. Taib, N. M. A. Nik Long. Effective approximation method for solving linear Fredholm-Volterra integral equations. Numerical Algebra, Control and Optimization, 2017, 7 (1) : 77-88. doi: 10.3934/naco.2017004

[14]

Mario Ahues, Filomena D. d'Almeida, Alain Largillier, Paulo B. Vasconcelos. Defect correction for spectral computations for a singular integral operator. Communications on Pure and Applied Analysis, 2006, 5 (2) : 241-250. doi: 10.3934/cpaa.2006.5.241

[15]

Takeshi Saito, Kazuyuki Yagasaki. Chebyshev spectral methods for computing center manifolds. Journal of Computational Dynamics, 2021, 8 (2) : 165-181. doi: 10.3934/jcd.2021008

[16]

Dajana Conte, Raffaele D'Ambrosio, Beatrice Paternoster. On the stability of $\vartheta$-methods for stochastic Volterra integral equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2695-2708. doi: 10.3934/dcdsb.2018087

[17]

Elena V. Kudryashova, Volker Reitmann. Contraction analysis of Volterra integral equations via realization theory and frequency-domain methods. Journal of Computational Dynamics, 2022  doi: 10.3934/jcd.2022020

[18]

Hong Lu, Ji Li, Mingji Zhang. Spectral methods for two-dimensional space and time fractional Bloch-Torrey equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3357-3371. doi: 10.3934/dcdsb.2020065

[19]

Suna Ma, Huiyuan Li, Zhimin Zhang. Novel spectral methods for Schrödinger equations with an inverse square potential on the whole space. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1589-1615. doi: 10.3934/dcdsb.2018221

[20]

Wenxiong Chen, Shijie Qi. Direct methods on fractional equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1269-1310. doi: 10.3934/dcds.2019055

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (410)
  • HTML views (205)
  • Cited by (0)

Other articles
by authors

[Back to Top]