We consider a new plane problem of the potential theory outside a symmetric T-shaped profile. In view of the symmetry of the profile, we reduce the analyzed problem to a mixed boundary-value problem in a half plane with perpendicular cut whose solution is constructed by the method of partial domains with the use of polar coordinates and the Mellin integral transformation. The original problem is reduced to a system of two Wiener–Hopf equations whose solution is reduced to a completely regular infinite system of linear algebraic equations. The solution of the infinite system is approximately found by the method of reduction. The obtained explicit expressions for the unknown harmonic function make it possible to efficiently determine its values for all possible values of the arguments.
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References
V. S. Vladimirov and V. V. Zharinov, Equations of Mathematical Physics [in Russian], Fizmatlit, Moscow (2004).
T. V. Klimchuk and V. I. Ostryk, “Smooth contact of a semiinfinite punch with rounded edge and an elastic strip,” Mat. Metody Fiz.-Mekh. Polya, 59, No. 2, 132–141 (2016); English translation: J. Math. Sci., 231, No. 5, 650–664 (2018); https://doi.org/10.1007/s10958-018-3842-9.
N. N. Lebedev, Special Functions and Their Applications, [in Russian], Fizmatgiz, Moscow (1963).
A. V. Loveikin, “Specific features of the stress behavior in an incompressible half space with internal V-shaped crack lying in a plane perpendicular to the surface of the half space with a tip reaching the surface,” Mat. Metody Fiz.-Mekh. Polya, 55, No. 2, 93– 106 (2012); English translation: J. Math. Sci., 192, No. 5, 593–607 (2013).
A. V. Loveikin, “Equilibrium of an elastic half plane with rigidly fastened boundary weakened by an oblique cut,” Mat. Metody Fiz.-Mekh. Polya, 62, No. 2, 146–160 (2019).
B. Noble, Methods Based on the Wiener–Hopf Technique for the Solution of Partial Differential Equations, Pergamon, London (1958).
V. I. Ostryk and A. F. Ulitko, Wiener–Hopf Technique in Contact Problems of the Theory of Elasticity [in Russian], Naukova Dumka, Kiev (2006).
Ya. S. Uflyand, Integral Transformations in Problems of the Theory of Elasticity [in Russian], Nauka, Leningrad (1968).
A. A. Khrapkov, “Certain cases of the elastic equilibrium of an infinite wedge with a nonsymmetric notch at the vertex, subjected to concentrated forces,” Prikl. Mat. Mekh., 35, No. 4, 677–689 (1971); English translation: J. Appl. Math. Mech., 35, No. 1, 625–637 (1971); https://doi.org/10.1016/0021-8928(71)90056-6.
I. D. Abrahams, “On the application of the Wiener–Hopf technique to problems in dynamic elasticity,” Wave Motion, 36, No. 4, 311–333 (2002); https://doi.org/10.1016/S0165-2125(02)00027-6.
Y. A. Antipov, “The Baker–Akhiezer function and factorization of the Chebotarev–Khrapkov matrix,” Lett. Math. Phys., 104, No. 11, 1365–1384 (2014); https://doi.org/10.1007/s11005-014-0721-2.
D. G. Crowdy and E. Luca, “Solving Wiener-Hopf problems without kernel factorization”, Proc. R. Soc. London, Ser. A, 470, 20140304 (2014); https://doi.org/10.1098/rspa.2014.0304.
D. S. Jones, “Wiener–Hopf splitting of a 2 × 2 matrix,” Proc. Roy. Soc., Ser. A, 434, No. 1891, 419–433 (1991); https://www.jstor.org/stable/51839.
A. A. Khrapkov, Wiener–Hopf Method in Mixed Elasticity Theory Problems, V. E. Vedeneev VNIIG Publ. House, St. Petersburg (2001).
A. V. Kisil, “An iterative Wiener–Hopf method for triangular matrix functions with exponential factors,” SIAM J. Appl. Math., 78, No. 1, 45–62 (2018); https://doi.org/10.1137/17M1136304.
J. Lawrie and I. D. Abrahams, “A brief historical perspective of the Wiener–Hopf technique,” J. Eng. Math., 59, No. 4, 351–358 (2007).
P. Livasov and G. Mishuris, “Numerical factorization of a matrix-function with exponential factors in an anti-plane problem for a crack with process zone,” Phil. Trans. R. Soc., Ser. A, 377, 20190109 (2019); https://doi.org/10.1098/rsta.2019.0109.
G. Mishuris and S. Rogosin, “Regular approximate factorization of a class of matrix-function with an unstable set of partial indices,” Proc. R. Soc., Ser. A, 474: 20170279 (2018); https://doi.org/10.1098/rspa.2017.0279.
B. H. Veitch and I. D. Abrahams, “On the commutative factorization of n × n matrix Wiener–Hopf kernels with distinct eigenvalues,” Proc. R. Soc., Ser. A, 463, No. 2078, 613–639 (2007); https://doi.org/10.1098/rspa.2006.1780.
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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 63, No. 2, pp. 83–97, April–June, 2020.
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Loveikin, А.V. Plane Potential Field Outside a Symmetric T-Shaped Profile. J Math Sci 272, 93–111 (2023). https://doi.org/10.1007/s10958-023-06402-4
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DOI: https://doi.org/10.1007/s10958-023-06402-4