Abstract
The paper deals with the problem of recovering solutions of a generalized Cauchy–Riemann system in a multidimensional spatial domain from their values on a piece of the boundary of this domain, i.e., an approximate solution of this problem based on the Carleman–Yarmukhamedov matrix method is constructed.
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É. N. Sattorov and F. É. Érmamatova, “Carleman’s formula for solutions of a generalized Cauchy–Riemann system in multidimensional spatial domain,” in Contemporary Problems in Mathematics and Physics, CMFD (Peoples’ Friendship University of Russia, Moscow, 2019), Vol. 65, pp. 95–108.
E. I. Obolashvili, “Generalized Cauchy–Riemann system in a multidimensional Euclidean space,” in Proc. International Conference on Complex Analysis and Its Applications to Partial Differential Equations, Halle, GDR, October 18–24, 1976 (Halle, 1977), pp. 36–39 [in Russian].
E. I. Obolashvili, “Generalized Cauchy–Riemann system in multidimensional space,” in Tr. Tbilisi. Matem. Inst. (Tbilisi, 1978), Vol. 58, pp. 168–173.
E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton Univ. Press, Princeton, NJ, 1971).
V. S. Vladimirov and I. V. Volovich, “Superanalysis. I. Differential calculus,” Theoret. and Math. Phys. 59 (1), 317–335 (1984).
V. S. Vladimirov and I. V. Volovich, “Superanalysis. II. Integral calculus,” Theoret. and Math. Phys. 60 (2), 743–765 (1984).
F. Brackx, K. Delanghe and F. Sommen, Clifford Analysis, in Res. Notes in Math. (Pitman, Boston, MA, 1982), Vol. 76.
J. S. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations (Yale University Press, New Haven, Conn., 1923).
M. M. Lavrent’ev, Some Ill-Posed Problems of Mathematical Physics (Izdat. Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk, 1962) [in Russian].
L. Bers, F. John and M. Schechter, Partial Differential Equations (Interscience, New York, 1964).
Sh. Yarmukhamedov, “On the Cauchy problem for Laplace’s equation,” Dokl. Akad. Nauk SSSR 235 (2), 281–283 (1977).
Sh. Yarmukhamedov, “On the continuation of the solution of the Helmholtz equation,” Dokl. AN 357 (3), 320–323 (1997).
A. N. Tikhonov, “On the solution of ill-posed problems and the method of regularization,” Dokl. Akad. Nauk SSSR 151 (3), 501–504 (1963).
V. K. Ivanov, “Cauchy problem for the Laplace equation in an infinite strip,” Differ. Uravn. 1 (1), 131–136 (1965).
M. M. Lavrent’ev, “On Cauchy problem for linear elliptical equations of the second order,” Dokl. Akad. Nauk SSSR 112 (2), 195–197 (1957).
S. N. Mergelyan, “Harmonic approximation and approximate solution of the Cauchy problem for the Laplace equation,” Uspekhi Mat. Nauk 11 (5 (71)), 3–26 (1956).
L. A. Aizenberg and N. N. Tarkhanov, “An abstract Carleman formula,” Dokl. Math. 37 (1), 235–238 (1988).
O. I. Makhmudov, “The Cauchy problem for a system of equations in the theory of elasticity and thermoelasticity in space,” Russian Math. (Iz. VUZ) 48 (2), 40–50 (2004).
O. Makhmudov, I. Niyozov and N. Tarkhanov, “The Cauchy problem of couple-stress elasticity,” in Complex Analysis and Dynamical Systems III, Contemp. Math. (Amer. Math. Soc., Providence, RI, 2008), Vol. 455, pp. 297–310.
É. N. Sattorov and D. A. Mardonov, “The Cauchy problem for the system of Maxwell equations,” Siberian Math. J. 44 (4), 671–679 (2003).
É. N. Sattorov, “Regularization of the solution of the Cauchy problem for a generalized Moisil–Teodorescu system,” Differ. Equ. 44 (8), 1136–1146 (2008).
É. N. Sattorov, “On the continuation of the solutions of a generalized Cauchy–Riemann system in space,” Math. Notes 85 (5), 733–745 (2009).
É. N. Sattorov, “Regularization of the solution of the Cauchy problem for the system of Maxwell equations in an unbounded domain,” Math. Notes 86 (3), 422–431 (2009).
É. N. Sattorov, “Reconstruction of solutions to a generalized Moisil–Teodorescu system in a spatial domain from their values on a piece of the boundary,” Russian Math. (Iz. VUZ) 55 (1), 62–73 (2011).
Sh. Yarmukhamedov, “On the analytical continuation of a holomorphic vector by its boundary-values on a piece of the boundary,” Izv. AN UzSSR. Ser. Phys.- Matem. Sciences, No. 6, 34–40 (1980).
K. O. Makhmudov, O. I. Makhmudov and N. Tarkhanov, “Equations of Maxwell Type,” J. Math. Anal. Appl. 378 (1), 64–75 (2011).
É. N. Sattorov and Z. É. Érmamatova, “Reconstruction of solutions to a generalized Moisil–Teodorescu system in a spatial domain from their values on a part of the boundary,” Russian Math. (Iz. VUZ) 55 (1), 62–73 (2011).
N. N. Tarkhanov, “The Carleman matrix for elliptic systems,” Dokl. Akad. Nauk SSSR 284 (2), 294–297 (1985).
L. A. Aizenberg, Carleman Formulas in Complex Analysis. First Applications (Nauka, Sibirsk. Otdel., Novosibirsk, 1990) [in Russian].
N. N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations (Akademie Verlag, Berlin, 1995), Vol. 7.
I. N. Vekua, Generalized analytical functions (Fizmatlit, Moscow, 1959) [in Russian].
T. I. Ishankulov, “On the possibility of extending a function from a part of a domain to a generalized analytic function on this domain,” Siberian Math. J. 41 (6), 1115–1120 (2000).
E. I. Obolashvili, “A spatial analogue of generalized analytical functions,” Soobshch. AN GSSR 73 (1), 20–24 (1974).
L. F. Nikiforov and V. B. Uvarov, Foundations of the Theory of Special The functions (Nauka, Moscow, 1974) [in Russian].
F. Tricomi, Differential Equations (Hafner Publ., New York, 1961).
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Translated from Matematicheskie Zametki, 2021, Vol. 110, pp. 405–423 https://doi.org/10.4213/mzm11103.
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Sattorov, É.N., Érmamatova, F.É. On the Recovery of Solutions of a Generalized Cauchy–Riemann System in a Multidimensional Spatial Domain from Their Values on a Piece of the Boundary of This Domain. Math Notes 110, 393–408 (2021). https://doi.org/10.1134/S000143462109008X
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DOI: https://doi.org/10.1134/S000143462109008X