On a Digital Version of Pseudo-Differential Operators and Its Applications

  • Vladimir B. VasilyevEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)


We develop discrete variant of a theory of pseudo-differential operators and equations. For some canonical domains we obtain solvability results for such equations and use these results to construct approximate solutions.


Digital pseudo-differential operator Periodic factorization Approximate solution 



The work is supported by the State contract of the Russian Ministry of Education and Science (contract No 1.7311.2017/8.9).


  1. 1.
    Eskin, G.: Boundary Value Problems for Elliptic Pseudodifferential Equations. AMS, Providence (1981)zbMATHGoogle Scholar
  2. 2.
    Taylor, M.E.: Pseudodifferential Operators. Princeton Univ. Press, Princeton (1981)CrossRefGoogle Scholar
  3. 3.
    Trevés, F.: Introduction to Pseudodifferential Operators and Fourier Integral Operators. Springer, New York (1980). Scholar
  4. 4.
    Samarskii, A.A.: The Theory of Difference Schemes. CRC Press, Boca Raton (2001)CrossRefGoogle Scholar
  5. 5.
    Ryaben’kii, V.S.: Method of Difference Potentials and its Applications. Springer, Heidelberg (2002). Scholar
  6. 6.
    Vasilyev, V.B.: The periodic Cauchy kernel, the periodic Bochner kernel, and discrete pseudo-differential operators. In: AIP Conference Proceedings, vol. 1863, p. 14014 (2017).
  7. 7.
    Vasilyev, V.B.: Discreteness, periodicity, holomorphy, and factorization. In: Constanda, C., Dalla Riva, M., Lamberti, P.D., Musolino, P. (eds.) Integral Methods in Science and Engineering, Theoretical Technique, vol. 1, pp. 315–324. Birkhäuser, Cham (2017). Scholar
  8. 8.
    Vasilyev, A.V., Vasilyev, V.B.: Discrete approximations for multidimensional singular integral operators. In: Dimov, I., Faragó, I., Vulkov, L. (eds.) NAA 2016. LNCS, vol. 10187, pp. 706–712. Springer, Heidelberg (2017). Scholar
  9. 9.
    Vasilyev, A.V., Vasilyev, V.B.: Two-scale estimates for special finite discrete operators. Math. Model. Anal. 22(3), 300–310 (2017). Scholar
  10. 10.
    Vasilyev, A.V., Vasilyev, V.B.: On a digital approximation for pseudo-differential operators. Proc. Appl. Math. Mech. 17(1), 763–764 (2017). Scholar
  11. 11.
    Vasilyev, V.B.: On some equations on non-smooth manifolds: canonical domains and model operators. In: Kalmenov, T., Nursultanov, E., Ruzhansky, M., Sadybekov, M. (eds.) FAIA 2017. PROMS, vol. 216, pp. 363–375. Springer, Heidelberg (2017). Scholar
  12. 12.
    Vasilyev, V.B.: Discrete pseudo-differential operators and boundary value problems in a half-space and a cone. Lobachevskii J. Math. 39(2), 289–296 (2018). Scholar
  13. 13.
    Vasilev, A.V., Vasilev, V.B.: Periodic Riemann problem and discrete convolution equations. Differ. Equ. 51(5), 652–660 (2015). Scholar
  14. 14.
    Vasilyev, A.V., Vasilyev, V.B.: Discrete singular operators and equations in a half-space. Azerb. J. Math. 3(1), 84–93 (2013)MathSciNetzbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Belgorod National Research UniversityBelgorodRussia

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