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Russian Mathematics

, Volume 63, Issue 9, pp 43–54 | Cite as

Integral Equations of Curvilinear Convolution Type with Hypergeometric Function in a Kernel

  • A. I. PeschanskiiEmail author
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Abstract

We study the class of integral equations of first kind over the circumference in the complex plane. The kernels of the equations contain a Gaussian hypergeometric function and depend on the arguments ratio. This class includes such specific cases as the equations with power and logarithmic kernels. To set the Noetherian property of equations correctly the method of operator normalization with a non-closed image is applied. The space of the right-hand sides of equations is described as the space of fractional integrals of curvilinear convolution type. The solutions of equations in explicit form are obtained as a result of consequent solving characteristic singular equations with a Cauchy kernel and inversion of a curvilinear convolution operator by means of Laurent transform of the functions defined on the circumference.

Key words

operator of curvilinear convolution with a Gaussian function in a kernel inversion of a curvilinear convolution operator Laurent transform Noetherian property of an integral equation 

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References

  1. 1.
    Gahov, F.D. Boundary value problems (Nauka, Moscow, 1977).Google Scholar
  2. 2.
    Muskhelishvili, N.I. Singular integral equations (Nauka, Moscow, 1968).zbMATHGoogle Scholar
  3. 3.
    Gahov, F.D., Cherskii, Yu.I. Equations of convolution type (Nauka, Moscow, 1978).Google Scholar
  4. 4.
    Polyanin, A.D., Manzhirov, A.V. Singular integral equations, v. 1–2 (Yurait, Moscow, 2018).Google Scholar
  5. 5.
    Prössdorf, S. Some classes of singular equations (Mir, Moscow, 1979).zbMATHGoogle Scholar
  6. 6.
    Samko, S.G., Kilbas, A.A., Marichev, O.I. Fractional integrals and derivatives: theory and applications (Gordon and Breach Sci., 1993).Google Scholar
  7. 7.
    Pleschinskii, N.B. Singular integral equations with complex feature of kernel (Kazan univ., Kazan, 2018).Google Scholar
  8. 8.
    Peters, A.S. “Some integral equations related to Abel’s equation and the Hilbert transform”, Comm. Pure Appl. Math. 22 (4), 539–560 (1969).MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chumakov, V.F., Vasiliev, I.L. “Integral equations of Abel type on closed curve”, Vestn. Belorussk. univ. Ser. 1 2, 40–44 (1980).Google Scholar
  10. 10.
    Saks, R.S. Boundary value problems for elliptic systems of differential equations (NGU publishers, Novosibirsk, 1975).zbMATHGoogle Scholar
  11. 11.
    Tovmasyan, N.E. “On theory of singular integral equations”, Differential equations 3 (1), 69–80 (1967).MathSciNetGoogle Scholar
  12. 12.
    Haikin, M.I. “On regularization of operators with non-closed range”, Izv. vuz. Mat., 8, 118–123 (1970).MathSciNetGoogle Scholar
  13. 13.
    Peschanskii, A.I., Cherskii, Yu.I. “Integral equation with curvilinear convolutions on closed curve”, Ukr. mathem. journ. 36 (3), 335–340 (1984).Google Scholar
  14. 14.
    Biberbach, L. Analytical continuation (Mir, Moscow, 1967).Google Scholar
  15. 15.
    Smirnov, V.I., Lebedev, A.N. Constructive function of complex variable (Nauka, Moscow, 1964).Google Scholar
  16. 16.
    Zygmund, A. Trigonometric series, v. 2 (Mir, Moscow, 1967).zbMATHGoogle Scholar
  17. 17.
    Peschanskii, A.I. “Description of a space of fractional integrals of curvilinear convolution type”, Soviet Math. (Iz. VUZ) 33 (7), 37–50 (1989).MathSciNetGoogle Scholar
  18. 18.
    Peschanskii, A.I. “Description of a space of fractional integrals of curvilinear convolution type”(in: Matherials of XXY International scientific-techn. conf. “Prikl. zadachi matem.”, Sevastopol, 18–22 Sent. 2017, SevGU publishers, Sevastopol, 104–108 (2017)).Google Scholar
  19. 19.
    Isakhanov, R.S. “Differential boundary problem of linear conjugation with applications in theory of integral-differential equations”, Soobsch. AN GruzSSR 20 (6), 659–666 (1958).Google Scholar
  20. 20.
    Cherskii, Yu.I. “Integral equations reducible to two Riemann problems”, DAN SSSR 248 (4), 802–805 (1979).MathSciNetGoogle Scholar

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© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.Sevastopol State UniversitySevastopolRussia

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