Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Aldashev S. A. Well-posedness of the Dirichlet Problem for a Class of Multidimensional Elliptic-parabolic Equations. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2016, vol. 16, iss. 2, pp. 125-132. DOI: 10.18500/1816-9791-2016-16-2-125-132

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Well-posedness of the Dirichlet Problem for a Class of Multidimensional Elliptic-parabolic Equations

Aldashev Serik Aimurzaevich, Abai Kazakh National Pedagogical University

Correctness of boundary problems in the plane for elliptic equations is well analyzed by analitic function theory of complex variable. There appear principal difficulties in similar problems when the number of independent variables is more than two. An attractive and suitable method of singular integral equations is less strong because of lock of any complete theory of multidimensional singular integral equations. In the work, the method proposed in the author’s works, shows the unique solvability and obtained the explicit form of the Dirichlet problem in the cylindric domain for a class of multidimensional elliptic-parabolic equations.

  1. Fikera G. The unified theory of boundary value problems for elltptlc-parabolic equations of second order. Sbornik perevodov. Matematika, 1963, vol. 7, no. 6, pp. 99–121 (in Russian).
  2. Oleinik О. A., Radkevich E. V. Equations with nonnegativo characteristic form. Moscow, Moscow Univ. Press, 2010, 360 p. (in Russian).
  3. Aldashev S. A. The correctness of the Dirichlet problem in the cylindric domain for the multidimensional elliptic-parabolic equation. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2014, vol. 14, iss. 1, pp. 5–10 (in Russian).
  4. Aldashev S. A. Correctness of Dirichlet’s Problem in a Cylindric Domain for a Single Class of Manydimensional Elliptic Equations. Vestn. Novosib. Gos. Univ., Ser. Matem., Mekh., Inform., 2012, vol. 12, iss. 1, pp. 7–13 (in Russian).
  5. Mikhlin S. G. Multidimensional Singular Integrals and Integral Equations. New York, Pergamon, 1965 (Russ. ed.: Mikhlin S. G. Mnogomernye singuliarnye integraly i integral’nye uravneniia. Moscow, Physmathgiz, 1962, 254 p.).
  6. Aldashev S. A. Boundary-Value Problems for Multi-Dimensional Hyperbolic and Mixed Equations. Almaty, Kazakhstan, Gylym Press, 1994, 170 p. (in Russian).
  7. Aldashev S. A. On Darboux problems for a class of multidimensional hyperbolic equations. Differentsialnye Uravneniya, 1998, vol. 34, no. 1, pp. 64–68 (in Russian).
  8. Aldashev S. A. Degenerate multidimensional hyperbolic equation. Oral, ZKATU, 2007, 139 p. (in Russian).
  9. Каmке E. Handbook on Ordinary Differential Equations]. Moscow, Nauka, 1965, 703 p. (in Russian).
  10. Beitmen G., Erdeii A. Higher Transcendental Functions. Moscow, Nauka, 1974, vol. 2, 297 p. (in Russian).
  11. Kolmogorov A., Fomin S. Elements of the Theory of Functions and Functional Analysis. Mineola, NY, USA, Dover Publications, 1999. (Russ. ed. :Kolmogorov A. N., Fomin S. V. Elementy teorii funktsii i funktsional’nogo analiza. Moscow, Nauka, 1976, 543 p.).
  12. Tikhonov A. N., Samarskii A. A. Equations of mathematical physics. Moscow, Nauka, 1966, 724 p. (in Russian).
  13. Smirnov V. I. The higher mathematics course. Moscow, Nauka, 1981, vol. 4, pt. 2, 550 p. (in Russian).
  14. Friedman A. Uravneniia s chastnymi proizvodnymi parabolicheskogo tipa [Partial differential Equations of parabolic type]. Moscow, Mir, 1968, 527 p. (in Russian).
  15. Bers L., John F., Schechter M. Partial differential equations. Moscow, Mir, 1966, 352 p. (in Russian).