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Zero interval limit perturbation expansion for the spectral entities of Hilbert-Schmidt operators combined with most dominant spectral component extraction: formulation and certain technicalities. (English) Zbl 1373.65093

Summary: A perturbation-expansion-at-zero-interval-limit based numeric al algorithm to calculate the eigenpairs of Hilbert-Schmidt integral operators having symmetric kernels is developed in the present work. We have developed the perturbation expansion only for the most dominant eigenvalue and relevant eigenfunction. The less important eigenpairs have been determined by using the most dominant spectral component extraction recursively over the kernel restrictions. The main lines of the formulation and certain related technicalities are presented here. The confirmation of the presented theory via certain illustrative implementations and the convergence discussion for the obtained perturbation series as well as the numerical comparison with some mostly considered methods are given in the next companion of this paper.

MSC:

65R20 Numerical methods for integral equations
45C05 Eigenvalue problems for integral equations
45P05 Integral operators
47A10 Spectrum, resolvent
47B07 Linear operators defined by compactness properties
47G10 Integral operators

Software:

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References:

[1] Gan, HH; Eu, BC, No article title, J. Chem. Phys. (1993) · doi:10.1063/1.466106
[2] Cancés, E.; Mennucci, B., No article title, J. Math. Chem. (1998) · Zbl 0905.65123 · doi:10.1023/A:1019133611148
[3] Buchukuri, T.; Chkadua, O.; Natroshvili, D., No article title, Integral Equ. Oper. Theory (2009) · Zbl 1183.35263 · doi:10.1007/s00020-009-1694-x
[4] Arnrich, S.; Bräuer, P.; Kalies, G., No article title, J. Math. Chem. (2015) · Zbl 1327.80010 · doi:10.1007/s10910-015-0531-5
[5] A. Townsend, L.N. Trefethen, Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. (2014) doi:10.1098/rspa.2014.0585
[6] S. Tuna, M. Demiralp, in AIP Conference Proceedings, vol. 1702 (2015), p. 170009. doi:10.1063/1.4938944
[7] Tunga, B.; Demiralp, M., No article title, J. Math. Chem. (2010) · Zbl 1293.62138 · doi:10.1007/s10910-010-9714-2
[8] Tunga, MA; Demiralp, M., No article title, J. Math. Chem. (2011) · doi:10.1063/1.3637819
[9] Tuna, S.; Tunga, B., No article title, J. Math. Chem. (2012) · Zbl 1312.41043 · doi:10.1007/s10910-013-0179-y
[10] Tunga, MA; Demiralp, M., No article title, J. Math. Chem. (2013) · Zbl 1310.92004 · doi:10.1007/s10910-013-0228-6
[11] Tunga, MA, No article title, Int. J. Comput. Math. (2014) · Zbl 1326.41007 · doi:10.1080/00207160.2014.941825
[12] E. Korkmaz Özay, M. Demiralp, J. Math. Chem. (2014). doi:10.1007/s10910-014-0396-z
[13] Tunga, MA, No article title, Int. J. Comput. Math. (2015) · Zbl 1326.41007 · doi:10.1080/00207160.2014.941825
[14] L. Debnath, P. Mikusiński, Introduction to Hilbert Spaces with Applications, 3rd edn. (Elsevier, Amsterdam, 2005)
[15] F. Chatelin, Spectral Approximation of Linear Operators (SIAM, Philedelphia, 2011) · Zbl 1214.01004 · doi:10.1137/1.9781611970678
[16] F.G. Tricomi, Integral Equations (Interscience Publishers, New York, 1957) · Zbl 0078.09404
[17] L.N. Trefethen, D. Bau, Numerical Linear Algebra (SIAM, Philadelphia, 1997) · Zbl 0874.65013 · doi:10.1137/1.9780898719574
[18] Corduneanu, C., No article title, Math. Syst. Theory (1967) · Zbl 0166.09801 · doi:10.1007/BF01705524
[19] T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin, 1995) · Zbl 0836.47009
[20] E.J. Hinch, Perturbation Methods (Cambridge University Press, Cambridge, 1991) · Zbl 0746.34001 · doi:10.1017/CBO9781139172189
[21] M.H. Holmes, Introduction to Perturbation Methods (Springer, New York, 1995) · Zbl 0830.34001 · doi:10.1007/978-1-4612-5347-1
[22] Sobol, IM, No article title, Math. Model. Comput. Exp., 1, 407-414 (1993)
[23] Rabitz, H.; Alış, ÖF; Shorter, J.; Shim, K., No article title, Comput. Phys. Commun. (1999) · Zbl 1015.68219 · doi:10.1016/S0010-4655(98)00152-0
[24] Alış, ÖF; Rabitz, H., No article title, J. Math. Chem. (2001) · Zbl 1051.93502 · doi:10.1023/A:1010979129659
[25] Tunga, MA; Demiralp, M., No article title, Appl. Math. Comput. (2005) · Zbl 1070.65009 · doi:10.1016/j.amc.2004.06.056
[26] Tunga, MA; Demiralp, M., No article title, Int. J. Comput. Math. (2008) · Zbl 1154.65010 · doi:10.1080/00207160701576095
[27] W.W. Bell, Special Functions for Scientists and Engineers (Dover, New York, 2004)
[28] S. Tuna, M. Demiralp, J. Math. Chem. (2017). doi:10.1007/s10910-017-0740-1
[29] W. Oevel, F. Postel, S. Wehmeier, J. Gerhard, The MuPAD Tutorial (Springer, Berlin, 2000) · Zbl 1007.68211 · doi:10.1007/978-3-642-98064-0
[30] E. Korkmaz Özay, M. Demiralp, J. Math. Chem. (2012). doi:10.1007/s10910-012-0018-6
[31] E. Korkmaz Özay, M. Demiralp, Combined small scale enhanced multivariance product representation, in Proceedings of the International Conference on Applied Computer Science (2010), p. 350356 · Zbl 1331.41042
[32] Tuna, S.; Demiralp, M., No article title, Mathematics (2017) · Zbl 1365.45010 · doi:10.3390/math5010002
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