×

Approximate solution for integral equations involving linear Toeplitz plus Hankel parts. (English) Zbl 1476.65338

Summary: We propose three approximate methods using regularization techniques and finite Hartley transforms for solving first-kind integral equations involving linear Toeplitz plus Hankel parts. Numerical examples are given for illustrating these new algorithms.

MSC:

65R20 Numerical methods for integral equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
65J15 Numerical solutions to equations with nonlinear operators
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alber, Y.; Ryazantseva, I., Nonlinear ill-posed problems of monotone type (2006), New York: Springer, New York · Zbl 1086.47003
[2] Anh, PK; Chung, CV, Parallel iterative regularization methods for solving systems of ill-posed equations, Appl Math Comput, 212, 542-550 (2009) · Zbl 1170.65035
[3] Anh, PK; Tuan, NM; Tuan, PD, The finite Hartley new convolutions and solvability of the integral equations with Toeplitrz plus Hankel kernels, J Math Anal Appl, 397, 537-549 (2013) · Zbl 1310.45003 · doi:10.1016/j.jmaa.2012.07.041
[4] Bauschke, HH; Combettes, PL, Convex analysis and monotone operator theory in Hilbert spaces (2011), New York: Springer, New York · Zbl 1218.47001 · doi:10.1007/978-1-4419-9467-7
[5] Castro, LP; Silva, AS; Saitoh, S., A reproducing kernel Hilbert space constructive approximation for integral equations with Toeplitz and Hankel kernels, Lib Math (N.S.), 34, 1, 1-22 (2014) · Zbl 1330.45005
[6] Castro, LP; Guerra, RC; Tuan, NM, New convolutions and their applicability to integral equations of Wiener-Hopf plus Hankel type, Math Methods Appl Sci, 43, 7, 4835-4846 (2020) · Zbl 1450.45002
[7] Didenko, VD; Silbermann, B., Generalized inverses and solution of equations with Toeplitz plus Hankel operators, Bol Soc Mat Mex (3), 22, 2, 645-667 (2016) · Zbl 1394.47034 · doi:10.1007/s40590-016-0101-2
[8] Groetsch, CW, Integral equations of the first kind, inverse problems and regularization: a crash course, J Phys Conf Ser, 73, 012001 (2007) · doi:10.1088/1742-6596/73/1/012001
[9] Hieu DV, Anh PK, Muu LD, Strodiot JJ (2021) Iterative regularization methods with new stepsize rules for solving variational inclusions. J Appl Math Comput. doi:10.1007/s12190-021-01534-9 · Zbl 1487.65075
[10] Kaltenbacher, B.; Neubauer, A.; Scherzer, O., Iterative regularization methods for nonlinear ill-posed problems (2008), Berlin: Walter de Gruyter, Berlin · Zbl 1145.65037 · doi:10.1515/9783110208276
[11] Ku, T-K; Kuo, C-CJ, Predconditioned iterative methods for solving Toeplitz plus Hankel systems, SIAM J Numer Anal, 30, 3, 824-845 (1993) · Zbl 0782.65046 · doi:10.1137/0730042
[12] Krein, SG, Functional analysis (1972), Groningen: Wolters-Noordhoff Publishing, Groningen · Zbl 0236.47001
[13] Liu, F.; Nashed, MZ, Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Anal, 6, 313-344 (1998) · Zbl 0924.49009 · doi:10.1023/A:1008643727926
[14] Nair MT (2009) Linear operator equations. World Scientific, Approximation and Regularization · Zbl 1208.47002
[15] Nair, MT; Pereverzev, SV, Regularized collocation method for Fredholm integral equations of the first kind, J Complex, 23, 454-467 (2007) · Zbl 1131.65113 · doi:10.1016/j.jco.2006.09.002
[16] Ng MK (1994) Fast iterative methods for solving Toeplitz-plus-Hankel least squares problems. Electron Trans Numer Anal 2:154-170 · Zbl 0852.65040
[17] Tsitsiklis JN, Levy BC (1981) Integral equations and resolvents of Toeplitz plus Hankel kernels, Technical Report LIDS-P-1170. M.I.T., silver edition. Laboratory for Information and Decision Systems
[18] Zygmund, A., Trigonometric Series (2002), London: Cambridge University Press, London · Zbl 1084.42003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.