On a method of solving integral equation of Carleman type on the pair of segments.

*(English)*Zbl 07216511
Kravchenko, Vladislav V. (ed.) et al., Transmutation operators and applications. Cham: Birkhäuser (ISBN 978-3-030-35913-3/hbk; 978-3-030-35914-0/ebook). Trends in Mathematics, 431-445 (2020).

Summary: The method is considered of solving integral equations of Carleman type on the pair of adjacent and disjoint segments. The problem is reduced to boundary problem of Riemann with piecewise constant matrix and four and five singular points. The solution is expressed via the solution of a differential equation of Fuchs class in which it was possible to define all the parameters.

For the entire collection see [Zbl 1443.34001].

For the entire collection see [Zbl 1443.34001].

##### MSC:

45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |

34M99 | Ordinary differential equations in the complex domain |

##### Keywords:

integral equations of Carleman type; canonical matrix; Riemann boundary value problem; differential equation of the Fuchs class
PDF
BibTeX
XML
Cite

\textit{L. A. Khvostchinskaya}, in: Transmutation operators and applications. Cham: Birkhäuser. 431--445 (2020; Zbl 07216511)

Full Text:
DOI

##### References:

[1] | F.D. Gakhov, Boundary value problems. Russ. Mosc. Nauka 45(2), 1-47 (1990) |

[2] | S.G. Samko, A.A. Kilbas, O.I. Marichev, Integrals and Derivatives of Fractional Order and Some of Their Applications (Nauka and Technika, Minsk, 1987, Russian) · Zbl 0617.26004 |

[3] | L.A. Khvostchinskaya, Explicit solution of a single integral equation of the Carleman type on a semi-axis. Vesci AN-Belarusi 4, 32-37 (1994, Russian) |

[4] | L.A. Khvostchinskaya, Representation of the logarithm of the product of nonsingular matrices of the 2nd order, in Materials of the XVI International Scientific Conference Devoted to Acad. M. Krawtchuk, 14-15 May 2015, vol. 2 (NTUU “KPI”, Kiev, 2015), pp. 192-195 |

[5] | L.A. Khvostchinskaya, To the Riemann problem in the case of arbitrary number of points, in Proceedings of International Conference “Boundary Value Problems, Special Functions and Fractional Calculus”, pp. 377-382 (1996) |

[6] | L.A. Khvostchinskaya, Solution of the Carleman type integral equation on a pair of intervals, in Mathematical Methods in Technics and Technologies. Proceedings of International Scientific Conference MMTT-30 in 12 volumes, vol. 3, SpB: Politechn. University, pp. 9-13 (2017) |

[7] | L.A. Khvostchinskaya, T.N. Zhorovina, Construction of differential equation of a hydromechanical problem, in Mathematical Methods in Technics and Technologies. Proceedings of International Scientific Conference MMTT-30 in 12 volumes, vol. 3, SpB: Politechn. University, pp. 14-18 (2017) |

[8] | N.I. Muskhelishvili, N.P. Vekua, The Riemann boundary problem for several unknown functions and its application to systems of singular integral equations. Trudy Tbil. Mat. Inst. XII, 1-46 (1943, Russian) |

[9] | F.D. Gakhov, One case of the Riemann boundary value problem for a system of n pairs of functions. Izv. AN SSSR Ser. Mat. 14, 549-568 (1950, Russian) · Zbl 0041.04801 |

[10] | F.D. Gakhov, Riemann’s boundary value problem for a system of n pairs of functions. Uspekhi Mat. Nauk VII (4(50)), 3-54 (1952, Russian) · Zbl 0049.05703 |

[11] | G.S. Litvinchuk, Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift. Mathematics and Its Applications, vol. 523 (Kluwer Academic Publishers, Dordrecht, 2000) · Zbl 0980.45001 |

[12] | N.P. Erugin, Riemann Problem (Nauka i Tekhnika, Minsk, 1982, Russian) · Zbl 0659.01009 |

[13] | J. Plemelj, Riemannsche Funktionenscharen mit gegebener Monodromiegruppe. Monat. Math. Phys. 19, 211-245 (1908) · JFM 39.0461.01 |

[14] | A.A. Bolibruch, The Riemann-Hilbert problem. Russ. Math. Surv. 45(2), 1-47 (1990) · Zbl 0706.34005 |

[15] | A. |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.