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Uspekhi Mat. Nauk, 2012, Volume 67, Issue 3(405), Pages 147–172 (Mi umn9483)  

This article is cited in 12 scientific papers (total in 12 papers)

Difference potentials analogous to Cauchy integrals

V. S. Ryaben'kii

M. V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences

Abstract: This work presents the state of the art in the theory of potentials for the solutions of systems of linear difference equations, which was proposed by the author in 1969. The role played by difference potentials in the solution of linear difference schemes of general form is for the first time compared in detail to the role played by Cauchy-type integrals in the theory of analytic functions. New vistas are exposed, which are opened up by the theory of difference potentials and arise through combining the universality and algorithmicity of difference schemes with certain properties of Cauchy-type integrals. A brief bibliographical review covers some of the fundamental applications of the theory which have already been implemented.
Bibliography: 61 titles.

Keywords: difference potentials, Cauchy integrals, numerical solution of boundary value problems, artificial boundary conditions, mathematical theory of noise control.


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English version:
Russian Mathematical Surveys, 2012, 67:3, 541–567

Bibliographic databases:

UDC: 512.64+517.5+519.6+577.9
MSC: 31C20, 65N06
Received: 05.12.2011

Citation: V. S. Ryaben'kii, “Difference potentials analogous to Cauchy integrals”, Uspekhi Mat. Nauk, 67:3(405) (2012), 147–172; Russian Math. Surveys, 67:3 (2012), 541–567

Citation in format AMSBIB
\by V.~S.~Ryaben'kii
\paper Difference potentials analogous to Cauchy integrals
\jour Uspekhi Mat. Nauk
\yr 2012
\vol 67
\issue 3(405)
\pages 147--172
\jour Russian Math. Surveys
\yr 2012
\vol 67
\issue 3
\pages 541--567

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    This publication is cited in the following articles:
    1. M. Medvinsky, S. Tsynkov, E. Turkel, “High order numerical simulation of the transmission and scattering of waves using the method of difference potentials”, J. Comput. Phys., 243 (2013), 305–322  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. D. S. Britt, S. V. Tsynkov, E. Turkel, “A high-order numerical method for the Helmholtz equation with nonstandard boundary conditions”, SIAM J. Sci. Comput., 35:5 (2013), A2255–A2292  crossref  mathscinet  zmath  isi  scopus
    3. V. S. Ryaben'kii, “Boundary equations of the difference potential theory”, Dokl. Math., 90:1 (2014), 469–471  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    4. Y. Epshteyn, “Algorithms composition approach based on difference potentials method for parabolic problems”, Commun. Math. Sci., 12:4 (2014), 723–755  crossref  mathscinet  zmath  isi  elib  scopus
    5. I. L. Sofronov, L. Dovgilovich, N. Krasnov, “Application of transparent boundary conditions to high-order finite-difference schemes for the wave equation in waveguides”, Appl. Numer. Math., 93 (2015), 195–205  crossref  mathscinet  zmath  isi  elib  scopus
    6. W. H. Woodward, S. Utyuzhnikov, P. Massin, “On the application of the method of difference potentials to linear elastic fracture mechanics”, Int. J. Numer. Meth. Eng., 103:10 (2015), 703–736  crossref  mathscinet  zmath  isi  elib  scopus
    7. Y. Epshteyn, S. Phippen, “High-order difference potentials methods for 1D elliptic type models”, Appl. Numer. Math., 93 (2015), 69–86  crossref  mathscinet  zmath  isi  scopus
    8. J. Albright, Y. Epshteyn, K. R. Steffen, “High-order accurate difference potentials methods for parabolic problems”, Appl. Numer. Math., 93 (2015), 87–106  crossref  mathscinet  zmath  isi  scopus
    9. Y. Epshteyn, M. Medvinsky, “On the solution of the elliptic interface problems by difference potentials method”, Spectral and High Order Methods For Partial Differential Equations ICOSAHOM 2014, Lecture Notes in Computational Science and Engineering, 106, eds. Kirby R., Berzins M., Hesthaven J., Springer-Verlag, Berlin, 2015, 197–205  crossref  mathscinet  zmath  isi  scopus
    10. J. Albright, Y. Epshteyn, M. Medvinsky, Q. Xia, “High-order numerical schemes based on difference potentials for 2D elliptic problems with material interfaces”, Appl. Numer. Math., 111 (2017), 64–91  crossref  mathscinet  zmath  isi  elib  scopus
    11. J. Albright, Y. Epshteyn, Q. Xia, “High-order accurate methods based on difference potentials for 2D parabolic interface models”, Commun. Math. Sci., 15:4 (2017), 985–1019  crossref  mathscinet  zmath  isi
    12. G. Ludvigsson, K. R. Steffen, S. Sticko, S. Wang, Q. Xia, Y. Epshteyn, G. Kreiss, “High-order numerical methods for 2D parabolic problems in single and composite domains”, J. Sci. Comput., 76:2 (2018), 812–847  crossref  mathscinet  isi  scopus
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