Legendre spectral projection methods for Fredholm integral equations of first kind

References

[1] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic, Dordrecht, 1994. 10.1007/978-94-015-8289-6Search in Google Scholar

[2] C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer, Berlin, 2006. 10.1007/978-3-540-30726-6Search in Google Scholar

[3] Z. Chen, S. Cheng, G. Nelakanti and H. Yang, A fast multiscale Galerkin method for the first kind ill-posed integral equations via Tikhonov regularization, Int. J. Comput. Math. 87 (2010), no. 3, 565–582. 10.1080/00207160802155302Search in Google Scholar

[4] Z. Chen, Y. Xu and H. Yang, Fast collocation methods for solving ill-posed integral equations of the first kind, Inverse Problems 24 (2008), no. 6, Article ID 065007. 10.1088/0266-5611/24/6/065007Search in Google Scholar

[5] H. W. Engl and C. W. Groetsch, Projection-regularization methods for linear operator equations of the first kind, Special Program on Inverse Problems, Proc. Centre Math. Anal. Austral. Nat. Univ. 17, Austrailian National University, Canberra (1988), 17–31. Search in Google Scholar

[6] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Math. Appl. 375, Kluwer Academic, Dordrecht, 1996. 10.1007/978-94-009-1740-8Search in Google Scholar

[7] C. W. Groetsch, On a regularization-Ritz method for Fredholm equations of the first kind, J. Integral Equations 4 (1982), no. 2, 173–182. Search in Google Scholar

[8] C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First kind, Res. Notes in Math. 105, Pitman, Boston, 1984. Search in Google Scholar

[9] C. W. Groetsch, Uniform convergence of regularization methods for Fredholm integral equations of the first kind, J. Austral. Math. Soc. 39 (1985), 282–286. 10.1017/S1446788700022539Search in Google Scholar

[10] C. W. Groetsch, Convergence analysis of a regularized degenerate kernel method for Fredholm integral equations of the first kind, Integral Equations Operator Theory 13 (1990), no. 1, 67–75. 10.1007/BF01195293Search in Google Scholar

[11] C. W. Groetsch, Integral equations of the first kind, inverse problems and regularization: A crash course, J. Phys. Conf. Ser. 73 (2007), 1–32. 10.1088/1742-6596/73/1/012001Search in Google Scholar

[12] C. W. Groetsch and A. Neubauer, Regularization of ill-posed problems: Optimal parameter choice in finite dimensions, J. Approx. Theory 58 (1989), no. 2, 184–200. 10.1016/0021-9045(89)90019-1Search in Google Scholar

[13] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Appl. Math. Sci. 120, Springer, New York, 1996. 10.1007/978-1-4612-5338-9Search in Google Scholar

[14] X. Luo, X. Yang, X. Huang and F. Li, A fast multiscale Galerkin method for ill-posed integral equations with not exactly given input data via Tikhonov regularization, J. Inverse Ill-Posed Probl. 20 (2012), no. 3, 367–385. 10.1515/jip-2011-0020Search in Google Scholar

[15] P. Maaß, S. V. Pereverzev, R. Ramlau and S. G. Solodky, An adaptive discretization for Tikhonov–Phillips regularization with a posteriori parameter selection, Numer. Math. 87 (2001), no. 3, 485–502. 10.1007/PL00005421Search in Google Scholar

[16] K. Maleknejad and S. Sohrabi, Numerical solution of Fredholm integral equations of the first kind by using Legendre wavelets, Appl. Math. Comput. 186 (2007), no. 1, 836–843. 10.1016/j.amc.2006.08.023Search in Google Scholar

[17] M. T. Nair, Linear Operator Equations: Approximations and Regularzations, World Scientific, Hackensack, 2009. 10.1142/7055Search in Google Scholar

[18] B. Neggal, N. Boussetila and F. Rebbani, Projected Tikhonov regularization method for Fredholm integral equations of the first kind, J. Inequal. Appl. 2016 (2016), Paper No. 195. 10.1186/s13660-016-1137-6Search in Google Scholar

[19] G. A. Pereverzēva, Projection methods for solving Fredholm integral equations of the first kind with ( ϕ , β ) -differentiable kernels and random errors, Ukrainian Math. J. 51 (1999), no. 5, 793–798. 10.1007/BF02591713Search in Google Scholar

[20] M. P. Rajan, Convergence analysis of a regularized approximation for solving Fredholm integral equations of the first kind, J. Math. Anal. Appl. 279 (2003), no. 2, 522–530. 10.1016/S0022-247X(03)00027-1Search in Google Scholar

[21] M. P. Rajan, A modified convergence analysis for solving Fredholm integral equations of the first kind, Integral Equations Operator Theory 49 (2004), no. 4, 511–516. 10.1007/s00020-002-1213-9Search in Google Scholar

[22] E. S. Shoukralla and M. A. Markos, The economized monic Chebyshev polynomials for solving weakly singular Fredholm integral equations of the first kind, Asian-Eur. J. Math. 13 (2020), no. 1, Article ID 2050030. 10.1142/S1793557120500308Search in Google Scholar

[23] Y.-F. Wang and T.-Y. Xiao, Fast realization algorithms for determining regularization parameters in linear inverse problems, Inverse Problems 17 (2001), no. 2, 281–291. 10.1088/0266-5611/17/2/308Search in Google Scholar