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Review

Discrete Legendre spectral Galerkin method for Urysohn integral equations

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Pages 465-489 | Received 13 Feb 2015, Accepted 13 Nov 2016, Published online: 12 Mar 2017

References

  • M. Ahues, A. Largillier, and B. Limaye. Spectral computations for bounded operators, CRC Press, Florida, 2010.
  • P.M. Anselone, Collectively Compact Operator Approximation Theory and Application to Integral Equations, Prentice-Hall, Englewood cliffs, NJ, 1971.
  • K. Atkinson and A. Bogomolny, The discrete Galerkin method for integral equations, Math. Comput. 48(178) (1987), pp. 595–616. doi: 10.1090/S0025-5718-1987-0878693-6
  • K.E. Atkinson and F.A. Potra, Projection and iterated projection methods for nonlinear integral equations, SIAM J. on Numer. Anal. 24(6) (1987), pp. 1352–1373. doi: 10.1137/0724087
  • K.E. Atkinson and F. Potra, The discrete Galerkin method for nonlinear integral equations, J. Integral Equ. Appl. 1(1) (1988), pp. 17–54. doi: 10.1216/JIE-1988-1-1-17
  • G. Ben-yu, Spectral Methods and Their Applications, World Scientific, Singapore, 1998.
  • P. Das and G. Nelakanti, Convergence analysis of discrete Legendre spectral projection methods for Hammerstein integral equations of mixed type, Appl. Math. Comput. 265 (2015), pp. 574–601.
  • P. Das, M.M. Sahani, and G. Nelakanti, Legendre spectral projection methods for Urysohn integral equations, Journal of Comput. Appl. Math. 263 (2014), pp. 88–102. doi: 10.1016/j.cam.2013.12.002
  • P. Das, G. Nelakanti, and G. Long, Discrete Legendre spectral projection methods for Fredholm–Hammerstein integral equations, J. Comput. Appl. Math. 278 (2015), pp. 293–305. doi: 10.1016/j.cam.2014.10.012
  • M.A. Golberg, Improved convergence rates for some discrete Galerkin methods, J. Integral Equ. Appl. 8(3) (1996), pp. 307–335. doi: 10.1216/jiea/1181075955
  • O. Hansen, K. Atkinson, and D. Chien, On the norm of the hyperinterpolation operator on the unit disc and its use for the solution of the nonlinear Poisson equation, IMA J. Numer. Anal. 29(2) (2009), pp. 257–283. doi: 10.1093/imanum/drm052
  • H. Kaneko, R.D. Noren, and Y. Xu, Regularity of the solution of Hammerstein equations with weakly singular kernel, Integral Equ. Operator Theory 13 (1990), pp. 660–670. doi: 10.1007/BF01732317
  • L. Schumaker, Spline Functions: Basis Theory, Cambridge University Press, New York, 2007.
  • J. Shen and T. Tang, Spectral and High-Order Methods with Applications, Science Press, Beijing, 2006.
  • I.H. Sloan, Polynomial interpolation and hyperinterpolation over general regions, J. Approx. Theory 83(2) (1995), pp. 238–254. doi: 10.1006/jath.1995.1119
  • I.H. Sloan and R.S. Womersley, The uniform error of hyperinterpolation on the sphere, Math. Res. 107 (1999), pp. 289–306.
  • T. Tang, X. Xu, and J. Cheng, On spectral methods for Volterra type integral equations and the convergence analysis, J. Comput. Math. 26(6) (2008), pp. 825–837.
  • G.M. Vainikko, Galerkin's perturbation method and the general theory of approximate methods for non-linear equations, USSR Comput. Math. Math. Phys. 7(4) (1967), pp. 1–41. doi: 10.1016/0041-5553(67)90140-1
  • Z. Wan, Y. Chen, and Y. Huang, Legendre spectral Galerkin method for second-kind Volterra integral equations, Front. Math. China 4(1) (2009), pp. 181–193. doi: 10.1007/s11464-009-0002-z
  • Z. Xie, X. Li, and T. Tang, Convergence analysis of spectral Galerkin methods for Volterra type integral equations, J. Scient. Comput. 53(2) (2012), pp. 414–434. doi: 10.1007/s10915-012-9577-8

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