Integral relations associated with the semi-infinite Hilbert transform and applications to singular integral equations
Authors:
Y. A. Antipov and S. M. Mkhitaryan
Journal:
Quart. Appl. Math. 76 (2018), 739-766
MSC (2010):
Primary 30E20, 42C05, 44A15, 44A20; Secondary 65D32
DOI:
https://doi.org/10.1090/qam/1508
Published electronically:
May 16, 2018
MathSciNet review:
3855829
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Abstract: Integral relations with the Cauchy kernel on a semi-axis for the Laguerre polynomials, the confluent hypergeometric function, and the cylindrical functions are derived. A part of these formulas is obtained by exploiting some properties of the Hermite polynomials, including their Hilbert and Fourier transforms and connections to the Laguerre polynomials. The relations discovered give rise to complete systems of new orthogonal functions. Free of singular integrals, exact and approximate solutions to the characteristic and complete singular integral equations in a semi-infinite interval are proposed. Another set of the Hilbert transforms in a semi-axis are deduced from integral relations with the Cauchy kernel in a finite segment for the Jacobi polynomials and the Jacobi functions of the second kind by letting some parameters involved go to infinity. These formulas lead to integral relations for the Bessel functions. Their application to a model problem of contact mechanics is given. A new quadrature formula for the Cauchy integral in a semi-axis based on an integral relation for the Laguerre polynomials and the confluent hypergeometric function is derived and tested numerically. Bounds for the remainder are found.
References
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References
- Y. A. Antipov, Weight functions of a crack in a two-dimensional micropolar solid, Quart. J. Mech. Appl. Math. 65 (2012), no. 2, 239–271. MR 2929304, DOI https://doi.org/10.1093/qjmam/hbr029
- Y. A. Antipov and A. V. Smirnov, Subsonic propagation of a crack parallel to the boundary of a half-plane, Math. Mech. Solids 18 (2013), no. 2, 153–167. MR 3179445, DOI https://doi.org/10.1177/1081286512462182
- Y.A. Antipov, Subsonic frictional cavitating penetration of a thin rigid body into an elastic medium, Quart. J. Mech. Appl. Math., 71 (2018), 221–243, DOI 10.1093/qjmam/hby003.
- H. Bateman, Higher Transcendental Functions, vol. 1, Bateman Manuscript Project, McGraw-Hill, New York, 1953.
- H. Bateman, Higher Transcendental Functions, vol. 2, Bateman Manuscript Project, McGraw-Hill, New York, 1953.
- H. Bateman, Tables of Integral Transforms, vol. 1, Bateman Manuscript Project, McGraw-Hill, New York, 1954.
- Joanne Elliott, On a class of integral equations, Proc. Amer. Math. Soc. 3 (1952), 566–572. MR 0049467, DOI https://doi.org/10.2307/2032589
- F. D. Gakhov, Boundary value problems, Translation edited by I. N. Sneddon, Pergamon Press, Oxford-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1966. MR 0198152
- L. A. Galin, Kontaktnye zadachi teorii uprugosti i vyazkouprugosti, “Nauka”, Moscow, 1980 (Russian). MR 598269
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products, Elsevier/Academic Press, Amsterdam, 2007. MR 2360010
- W. Koppelman and J. D. Pincus, Spectral representations for finite Hilbert transformations, Math. Z. 71 (1959), 399–407. MR 0107144, DOI https://doi.org/10.1007/BF01181411
- A. A. Korneĭčuk, Quadrature formulae for singular integrals, Ž. Vyčisl. Mat. i Mat. Fiz. 4 (1964), no. 4, suppl., 64–74 (Russian). MR 0175299
- M. J. Lighthill, Introduction to Fourier analysis and generalised functions, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1958. MR 0092119
- S. M. Mhitarjan, The spectral decompositions of integral operators which are analogous to the finite Hilbert transform, Mat. Issled. 4 (1969), no. vyp. 1 (11), 98–109 (Russian). MR 0259678
- Suren M. Mkhitaryan, Mushegh S. Mkrtchyan, and Eghine G. Kanetsyan, On a method for solving Prandtl’s integro-differential equation applied to problems of continuum mechanics using polynomial approximations, ZAMM Z. Angew. Math. Mech. 97 (2017), no. 6, 639–654. MR 3664367, DOI https://doi.org/10.1002/zamm.201600025
- N. I. Muskhelishvili, Singular integral equations, Wolters-Noordhoff Publishing, Groningen, 1972. Boundary problems of functions theory and their applications to mathematical physics; Revised translation from the Russian, edited by J. R. M. Radok; Reprinted. MR 0355494
- N. I. Muskhelishvili, Some basic problems of the mathematical theory of elasticity. Fundamental equations, plane theory of elasticity, torsion and bending, Translated from the Russian by J. R. M. Radok, P. Noordhoff, Ltd., Groningen, 1963. MR 0176648
- G.Ia. Popov, Plane contact problem of the theory of elasticity with bonding or frictional forces, J. Appl. Math. Mech., 30 (1967), 653–667.
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- Gabor Szegö, Orthogonal polynomials, American Mathematical Society Colloquium Publications, Vol. 23. Revised ed, American Mathematical Society, Providence, R.I., 1959. MR 0106295
- E. C. Titchmarsh, Introduction to the theory of Fourier integrals, 3rd ed., Chelsea Publishing Co., New York, 1986. MR 942661
- F. G. Tricomi, On the finite Hilbert transformation, Quart. J. Math., Oxford Ser. (2) 2 (1951), 199–211. MR 0043258, DOI https://doi.org/10.1093/qmath/2.1.199
- F. G. Tricomi, Integral equations, Pure and Applied Mathematics. Vol. V, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1957. MR 0094665
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Additional Information
Y. A. Antipov
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
MR Author ID:
245270
Email:
yantipov@lsu.edu
S. M. Mkhitaryan
Affiliation:
Department of Mechanics of Elastic and Viscoelastic Bodies, National Academy of Sciences, Yerevan 0019, Armenia
MR Author ID:
194038
Email:
smkhitaryan39@rambler.ru
Keywords:
Hilbert transform,
orthogonal polynomials,
singular integral equations,
quadrature formulas
Received by editor(s):
February 13, 2018
Published electronically:
May 16, 2018
Additional Notes:
The research of the first author was sponsored by the Army Research Office and was accomplished under Grant Number W911NF-17-1-0157. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.
Article copyright:
© Copyright 2018
Brown University
