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THE CAUCHY PROBLEM FOR MATRIX FACTORIZATIONS OF THE HELMHOLTZ EQUATION IN R^3

Year 2018, Volume: 1 Issue: 3 - To memory of Prof. RNDr. Beloslav Rieˇcan, DrSc., 312 - 319, 24.10.2018

Abstract

In the paper it is considered the regularization of the Cauchy problem for systems of elliptic type equations of the first order with constant coefficients factorisable Helmholtz operator in three-dimensional bounded domain.
Using the results of [1-6; 19,20,21,22], we construct in explicit form Carleman
matrix and, based on the regularized solution of the Cauchy problem.

References

  • \begin{thebibliography}{}\bibitem {1} N.N. Tarkhanov, {\it On the integral representation of solutions of systems offirst-order linear differential equations in partial derivatives andsome of its applications,} Some questions of multidimensionalcomplex analysis. Institute of Physics, USSR Academy of Sciences,Krasnoyarsk, (1980) 147--160.\bibitem {2} N.N. Tarkhanov, {\it On the Carleman matrix for ellipticsystems,} Dokl. Akad. Nauk SSSR. 284 (1985), no.~2., 294--297.\bibitem {3} N.N. Tarkhanov, {\it The Cauchy problem for solutions of ellipticequations,} Akad. Verl., Berlin, V. 7. 1995 (in English).\bibitem {4} T. Carleman, {\it Les fonctions quasi analytiques,} Gautier-Villars et Cie., Paris, 1926.\bibitem {5} M.M. Lavrent'ev, {\it On the Cauchy problem for second-order linear elliptic equations,} Dokl. Akad. Nauk SSSR. 112 (1957), no.~2., 195--197.\bibitem {6} M.M.~Lavrent'ev, {\it On some ill-posed problems ofmathematical physics,} Nauka, Novosibirsk, 1962 (in Russian).\bibitem {7} Sh. Yarmukhamedov, {\it On the Cauchy problem for the Laplaceequation,} Dokl. Akad. Nauk SSSR. 235 (1977), no.~2., 281--283.\bibitem {8} Sh. Yarmukhamedov, {\it On the extension of the solution of the Helmholtzequation,} Dokl. Ross. Akad. Nauk. 357 (1997), no.~3., 320--323.\bibitem {9} Sh. Yarmukhamedov, {\it The Carleman function and the Cauchy problem for the Laplaceequation,} Sibirsk. Math. Journal. 45 (2004), no.~3., 702--719.\bibitem {10} J. Adamar, {\it The Cauchy problem for linear partial differential equations of hyperbolictype,} Nauka, Moscow, 1978 (in Russian).\bibitem{11} L.A. Aizenberg, {\it Carleman's formulas in complex analysis,} Nauka, Novosibirsk, 1990 (in Russian).\bibitem{12} G.M. Goluzin, V.M. Krylov, {\it The generalized Carleman formula and its application to the analytic continuation of functions,} Math. Sbornik. 40 (1993), no.~2., 144--149.\bibitem{13} A.N. Tikhonov, {\it On the solution of ill-posed problems and the regularizationmethod,} Dokl. Akad. Nauk SSSR. 151 (1963), no.~3., 501--504.\bibitem{14} A. Bers, F. John, M. Shekhter {\it Partial DifferentialEquations,} Mir, Moscow, 1966 (in Russian).\bibitem{15} M.A. Aleksidze, {\it Fundamental functions in approximate solutions of boundary valueproblems,} Nauka, Moscow, 1991 (in Russian).\bibitem{16} E.V. Arbuzov, A.L. Bukhgeim, {\it The Carleman formula for the Helmholtzequation,} Sib. Math. Journal. 47 (1979), no.~3., 518--526.\bibitem{17} O.I. Makhmudov {\it On the Cauchy problem for elliptic systems in the space${\mathbb R}^{m}$,} Mat. Zametki. 75 (2004), no.~3., 849--860.\bibitem{18} I.E. Niyozov, O.I. Makhmudov {\it TheCauchy problem of the moment elasticity theory in ${\mathbbR}^{m}$,} Izv. Vuz. Mat. 58 (2014), no.~2., 24--30.\bibitem{19} D.A. Juraev, {\it The construction of the fundamental solution of the Helmholtzequation,} Reports of the Academy of Sciences of the Republic ofUzbekistan. (2012), no.~2., 14--17.\bibitem{20} D.A. Juraev, {\it The Cauchy problem for matrix factorizations of the Helmholtz equation in an unbounded domain,} Sib. Electron. Mat. Izv. 14 (2017),752--764. \\ doi:10.17377/semi.2017.14.064, Zbl 1375.35153.\bibitem{21} D.A. Zhuraev, {\it Cauchy problemfor matrix factorizations of the Helmholtz equation,} UkrainianMathematical Journal. 69 (2018), no.~10., 1583-1592. (UkrainianOriginal Vol. 69, No. 10, October, 2017). \\doi:10.1007/s11253-018-1456-5.\bibitem{22} D.A. Juraev, {\it On the Cauchy problem for matrix factorizations of the Helmholtzequation in a bounded domain,} Sib. Electron. Mat. Izv. 15 (2018),11--20. \\ doi:10.17377/semi.2018.15.002, Zbl 1387.35176.\end{thebibliography}
Year 2018, Volume: 1 Issue: 3 - To memory of Prof. RNDr. Beloslav Rieˇcan, DrSc., 312 - 319, 24.10.2018

Abstract

References

  • \begin{thebibliography}{}\bibitem {1} N.N. Tarkhanov, {\it On the integral representation of solutions of systems offirst-order linear differential equations in partial derivatives andsome of its applications,} Some questions of multidimensionalcomplex analysis. Institute of Physics, USSR Academy of Sciences,Krasnoyarsk, (1980) 147--160.\bibitem {2} N.N. Tarkhanov, {\it On the Carleman matrix for ellipticsystems,} Dokl. Akad. Nauk SSSR. 284 (1985), no.~2., 294--297.\bibitem {3} N.N. Tarkhanov, {\it The Cauchy problem for solutions of ellipticequations,} Akad. Verl., Berlin, V. 7. 1995 (in English).\bibitem {4} T. Carleman, {\it Les fonctions quasi analytiques,} Gautier-Villars et Cie., Paris, 1926.\bibitem {5} M.M. Lavrent'ev, {\it On the Cauchy problem for second-order linear elliptic equations,} Dokl. Akad. Nauk SSSR. 112 (1957), no.~2., 195--197.\bibitem {6} M.M.~Lavrent'ev, {\it On some ill-posed problems ofmathematical physics,} Nauka, Novosibirsk, 1962 (in Russian).\bibitem {7} Sh. Yarmukhamedov, {\it On the Cauchy problem for the Laplaceequation,} Dokl. Akad. Nauk SSSR. 235 (1977), no.~2., 281--283.\bibitem {8} Sh. Yarmukhamedov, {\it On the extension of the solution of the Helmholtzequation,} Dokl. Ross. Akad. Nauk. 357 (1997), no.~3., 320--323.\bibitem {9} Sh. Yarmukhamedov, {\it The Carleman function and the Cauchy problem for the Laplaceequation,} Sibirsk. Math. Journal. 45 (2004), no.~3., 702--719.\bibitem {10} J. Adamar, {\it The Cauchy problem for linear partial differential equations of hyperbolictype,} Nauka, Moscow, 1978 (in Russian).\bibitem{11} L.A. Aizenberg, {\it Carleman's formulas in complex analysis,} Nauka, Novosibirsk, 1990 (in Russian).\bibitem{12} G.M. Goluzin, V.M. Krylov, {\it The generalized Carleman formula and its application to the analytic continuation of functions,} Math. Sbornik. 40 (1993), no.~2., 144--149.\bibitem{13} A.N. Tikhonov, {\it On the solution of ill-posed problems and the regularizationmethod,} Dokl. Akad. Nauk SSSR. 151 (1963), no.~3., 501--504.\bibitem{14} A. Bers, F. John, M. Shekhter {\it Partial DifferentialEquations,} Mir, Moscow, 1966 (in Russian).\bibitem{15} M.A. Aleksidze, {\it Fundamental functions in approximate solutions of boundary valueproblems,} Nauka, Moscow, 1991 (in Russian).\bibitem{16} E.V. Arbuzov, A.L. Bukhgeim, {\it The Carleman formula for the Helmholtzequation,} Sib. Math. Journal. 47 (1979), no.~3., 518--526.\bibitem{17} O.I. Makhmudov {\it On the Cauchy problem for elliptic systems in the space${\mathbb R}^{m}$,} Mat. Zametki. 75 (2004), no.~3., 849--860.\bibitem{18} I.E. Niyozov, O.I. Makhmudov {\it TheCauchy problem of the moment elasticity theory in ${\mathbbR}^{m}$,} Izv. Vuz. Mat. 58 (2014), no.~2., 24--30.\bibitem{19} D.A. Juraev, {\it The construction of the fundamental solution of the Helmholtzequation,} Reports of the Academy of Sciences of the Republic ofUzbekistan. (2012), no.~2., 14--17.\bibitem{20} D.A. Juraev, {\it The Cauchy problem for matrix factorizations of the Helmholtz equation in an unbounded domain,} Sib. Electron. Mat. Izv. 14 (2017),752--764. \\ doi:10.17377/semi.2017.14.064, Zbl 1375.35153.\bibitem{21} D.A. Zhuraev, {\it Cauchy problemfor matrix factorizations of the Helmholtz equation,} UkrainianMathematical Journal. 69 (2018), no.~10., 1583-1592. (UkrainianOriginal Vol. 69, No. 10, October, 2017). \\doi:10.1007/s11253-018-1456-5.\bibitem{22} D.A. Juraev, {\it On the Cauchy problem for matrix factorizations of the Helmholtzequation in a bounded domain,} Sib. Electron. Mat. Izv. 15 (2018),11--20. \\ doi:10.17377/semi.2018.15.002, Zbl 1387.35176.\end{thebibliography}
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Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Davron Juraev (zhuraev) 0000-0003-1224-6764

Publication Date October 24, 2018
Submission Date August 28, 2018
Acceptance Date October 23, 2018
Published in Issue Year 2018 Volume: 1 Issue: 3 - To memory of Prof. RNDr. Beloslav Rieˇcan, DrSc.

Cite

APA Juraev (zhuraev), D. (2018). THE CAUCHY PROBLEM FOR MATRIX FACTORIZATIONS OF THE HELMHOLTZ EQUATION IN R^3. Journal of Universal Mathematics, 1(3), 312-319.
AMA Juraev (zhuraev) D. THE CAUCHY PROBLEM FOR MATRIX FACTORIZATIONS OF THE HELMHOLTZ EQUATION IN R^3. JUM. October 2018;1(3):312-319.
Chicago Juraev (zhuraev), Davron. “THE CAUCHY PROBLEM FOR MATRIX FACTORIZATIONS OF THE HELMHOLTZ EQUATION IN R^3”. Journal of Universal Mathematics 1, no. 3 (October 2018): 312-19.
EndNote Juraev (zhuraev) D (October 1, 2018) THE CAUCHY PROBLEM FOR MATRIX FACTORIZATIONS OF THE HELMHOLTZ EQUATION IN R^3. Journal of Universal Mathematics 1 3 312–319.
IEEE D. Juraev (zhuraev), “THE CAUCHY PROBLEM FOR MATRIX FACTORIZATIONS OF THE HELMHOLTZ EQUATION IN R^3”, JUM, vol. 1, no. 3, pp. 312–319, 2018.
ISNAD Juraev (zhuraev), Davron. “THE CAUCHY PROBLEM FOR MATRIX FACTORIZATIONS OF THE HELMHOLTZ EQUATION IN R^3”. Journal of Universal Mathematics 1/3 (October 2018), 312-319.
JAMA Juraev (zhuraev) D. THE CAUCHY PROBLEM FOR MATRIX FACTORIZATIONS OF THE HELMHOLTZ EQUATION IN R^3. JUM. 2018;1:312–319.
MLA Juraev (zhuraev), Davron. “THE CAUCHY PROBLEM FOR MATRIX FACTORIZATIONS OF THE HELMHOLTZ EQUATION IN R^3”. Journal of Universal Mathematics, vol. 1, no. 3, 2018, pp. 312-9.
Vancouver Juraev (zhuraev) D. THE CAUCHY PROBLEM FOR MATRIX FACTORIZATIONS OF THE HELMHOLTZ EQUATION IN R^3. JUM. 2018;1(3):312-9.