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A new and self-contained presentation of the theory of boundary operators for slit diffraction and their logarithmic approximations. (English) Zbl 1230.78008

Summary: We present a new and self-contained theory for mapping properties of the boundary operators for slit diffraction occurring in Sommerfeld’s diffraction theory, covering two different cases of the polarisation of the light. This theory is entirely developed in the context of the boundary operators with a Hankel kernel and not based on the corresponding mixed boundary value problem for the Helmholtz equation. For a logarithmic approximation of the Hankel kernel, we also study the corresponding mapping properties and derive explicit solutions together with certain regularity results.

MSC:

78A45 Diffraction, scattering
42A50 Conjugate functions, conjugate series, singular integrals
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
45H05 Integral equations with miscellaneous special kernels
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[1] Abramowitz, Handbook of Mathematical Functions (1968)
[2] Bastos, Convolution equations of the first kind on a finite interval in Sobolev spaces, Integral Equations Oper. Theory 13 pp 638– (1990) · Zbl 0715.47016 · doi:10.1007/BF01732316
[3] Belward, The solution of an integral equation of the first kind on a finite interval, Q. Appl. Math. 27 pp 313– (1969) · Zbl 0187.04601 · doi:10.1090/qam/410301
[4] Born, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Seventh ed (1999) · doi:10.1017/CBO9781139644181
[5] Bouwkamp, Diffraction theory, Reports on Progress in Physics XVII pp 35– (1954) · doi:10.1088/0034-4885/17/1/302
[6] Castro, Pseudo differential operators in a wave diffraction problem with impedance conditions, Fract. Calc. Appl. Anal. 11 (1) pp 15– (2008)
[7] Castro, The impedance boundary-value problem of diffraction by a strip, J. Math. Anal. Appl. 337 pp 1031– (2008) · Zbl 1151.35015 · doi:10.1016/j.jmaa.2007.04.037
[8] L. Castro D. Kapanadze Dirichlet-Neumann-impedance boundary-value problems arising in rectangular wedge diffraction problems 2008 2113 2123
[9] Dörr, Zwei Integralgleichungen erster Art, die sich mit Hilfe Mathieuscher Funktionen lösen lassen, ZAMP III pp 427– (1952)
[10] Eskin, Boundary Value Problems for Elliptic Pseudodifferential Operators (1981)
[11] Gorenflo, Null space distributions - a new approach to finite convolution equations with a Hankel kernel, Integral Equations Oper. Theory 35 pp 366– (1999) · Zbl 0944.45001 · doi:10.1007/BF01193904
[12] Gorenflo, A characterization of the range of a finite convolution operator with a Hankel kernel, Integral Transforms Spec. Funct. 12 pp 27– (2001) · Zbl 1035.47023 · doi:10.1080/10652460108819331
[13] Gorenflo, A new explicit solution method for the diffraction through a slit, ZAMP 53 pp 877– (2002) · Zbl 1016.45001 · doi:10.1007/s00033-002-8187-y
[14] Gorenflo, A new explicit solution method for the diffraction through a slit - Part 2, ZAMP 58 pp 16– (2007) · Zbl 1118.45002 · doi:10.1007/s00033-006-5106-7
[15] Gorenflo, Solution of a finite convolution equation with a Hankel kernel by matrix factorization, SIAM J. Math. Anal. 28 pp 434– (1997) · Zbl 0912.45001 · doi:10.1137/S0036141095289154
[16] Gradshteyn, Table of Integrals, Series, and Products, Corrected and Enlarged ed. (1980) · Zbl 0521.33001
[17] Kunik, Diffraction of light revisited, Math. Methods Appl. Sci. 31 (7) pp 793– (2008) · Zbl 1138.78006 · doi:10.1002/mma.945
[18] Latta, The solution of a class of integral equations, J. Ration. Mech. Anal. 5 pp 821– (1956) · Zbl 0070.32901
[19] Mandrik, Application of Laplace and Hankel transforms to solution of mixed nonstationary boundary value problems, Integral Transforms Spec. Funct. 13 pp 277– (2002) · Zbl 1034.35061 · doi:10.1080/10652460213519
[20] McLean, Strongly Elliptic Systems and Boundary Integral Equations (2000) · Zbl 0948.35001
[21] Meister, Einige gelöste und ungelöste kanonische Probleme der mathematischen Beugungstheorie, Expo. Math. 5 pp 193– (1987) · Zbl 0647.35019
[22] Meister, A Sommerfeld-type diffraction problem with second-order boundary conditions, Z. Angew. Math. Mech. 72 (12) pp 621– (1992) · doi:10.1002/zamm.19920721206
[23] Meixner, Die Kantenbedingung in der Theorie der Beugung elektromagnetischer Wellen an vollkommen leitenden ebenen Schirmen, Annalen der Physik 441 (1) pp 1– (1949) · Zbl 0034.12501 · doi:10.1002/andp.19494410103
[24] Morse, Methods of Theoretical Physics, Part I (1953) · Zbl 0051.40603
[25] Santos, Sommerfeld diffraction problems with oblique derivatives, Math. Methods Appl. Sci. 20 (7) pp 635– (1997) · Zbl 0888.35028 · doi:10.1002/(SICI)1099-1476(19970510)20:7<635::AID-MMA875>3.0.CO;2-O
[26] Noble, Methods based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations (1958) · Zbl 0082.32101
[27] Okikiolu, Aspects of the Theory of Bounded Integral Operators in Lp-spaces (1971) · Zbl 0219.44002
[28] Pal’cev, A generalization of the Wiener-Hopf method for convolution equations on a finite interval with symbols having power-like asymptotics at infinity, Math. USSR Sb. 41 (3) pp 289– (1982) · Zbl 0477.45002 · doi:10.1070/SM1982v041n03ABEH002235
[29] Penzel, Sobolev space methods for dual integral equations in axialsymmetric screen problems, SIAM J. Math. Anal. 23 pp 1167– (1992) · Zbl 0756.45007 · doi:10.1137/0523065
[30] Polyanin, Handbook of Integral Equations (1998) · doi:10.1201/9781420050066
[31] Rudin, Functional Analysis (1979)
[32] Santos, The Sommerfeld problem revisited: Solution spaces and the edge conditions, J. Math. Anal. Appl. 143 (2) pp 341– (1989) · Zbl 0713.35023 · doi:10.1016/0022-247X(89)90045-0
[33] Schmeidler, Integralgleichungen mit Anwendungen in Physik und Technik, Band I, Lineare Integralgleichungen (1950) · Zbl 0035.34901
[34] A. V. Shanin To the problem of diffraction on a slit: Some properties of Schwarzschild’s series 2000 143 155
[35] Shanin, Three theorems concerning diffraction by a strip or a slit, Q. J. Mech. Appl. Math. 54 pp 107– (2001) · Zbl 1033.78008 · doi:10.1093/qjmam/54.1.107
[36] Sommerfeld, Vorlesungen über Theoretische Physik, Band IV, Optik (1989)
[37] Sommerfeld, Mathematische Theorie der Diffraction, Math. Ann. 47 (2,3) pp 317– (1896) · JFM 27.0706.03 · doi:10.1007/BF01447273
[38] Spahn, The diffraction of a plane wave by an infinite slit. I, Q. Appl. Math. 40 (1) pp 105– (1982/83) · doi:10.1090/qam/652055
[39] Williams, Diffraction by a finite strip, Q. J. Mech. Appl. Math. 35 (1) pp 103– (1982) · Zbl 0482.73024 · doi:10.1093/qjmam/35.1.103
[40] P. Wolfe On the inverse of an integral operator 1970 443 448 · Zbl 0194.45003
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