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A method for solving ill-posed nonlocal problem for the elliptic equation with data on the whole boundary

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Abstract
In this paper a nonlocal problem for the elliptic equation in a cylindrical domain is considered. It is shown that this problem is ill-posed as well as the Cauchy problem for the Laplace equation. The method of spectral expansion in eigenfunctions of the nonlocal problem for equations with involution establishes a criterion of the strong solvability of the considered nonlocal problem. It is shown that the ill-posedness of the nonlocal problem is equivalent to the existence of an isolated point of the continuous spectrum for a nonself-adjoint operator with involution.
Keywords
Elliptic operator, Nonlocal boundary conditions, Operator with involution, Criterion of well-posedness, Riesz basis, Primary 35J25, 35C10, Secondary 35P10, MIXED CAUCHY-PROBLEM, STRONG SOLVABILITY, LAPLACE, REGULARIZATION, CRITERION

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MLA
Kal’menov, Tynysbek Sh., and Berikbol Torebek. “A Method for Solving Ill-Posed Nonlocal Problem for the Elliptic Equation with Data on the Whole Boundary.” JOURNAL OF PSEUDO-DIFFERENTIAL OPERATORS AND APPLICATIONS, vol. 10, no. 1, 2019, pp. 177–85, doi:10.1007/s11868-017-0231-y.
APA
Kal’menov, T. Sh., & Torebek, B. (2019). A method for solving ill-posed nonlocal problem for the elliptic equation with data on the whole boundary. JOURNAL OF PSEUDO-DIFFERENTIAL OPERATORS AND APPLICATIONS, 10(1), 177–185. https://doi.org/10.1007/s11868-017-0231-y
Chicago author-date
Kal’menov, Tynysbek Sh., and Berikbol Torebek. 2019. “A Method for Solving Ill-Posed Nonlocal Problem for the Elliptic Equation with Data on the Whole Boundary.” JOURNAL OF PSEUDO-DIFFERENTIAL OPERATORS AND APPLICATIONS 10 (1): 177–85. https://doi.org/10.1007/s11868-017-0231-y.
Chicago author-date (all authors)
Kal’menov, Tynysbek Sh., and Berikbol Torebek. 2019. “A Method for Solving Ill-Posed Nonlocal Problem for the Elliptic Equation with Data on the Whole Boundary.” JOURNAL OF PSEUDO-DIFFERENTIAL OPERATORS AND APPLICATIONS 10 (1): 177–185. doi:10.1007/s11868-017-0231-y.
Vancouver
1.
Kal’menov TSh, Torebek B. A method for solving ill-posed nonlocal problem for the elliptic equation with data on the whole boundary. JOURNAL OF PSEUDO-DIFFERENTIAL OPERATORS AND APPLICATIONS. 2019;10(1):177–85.
IEEE
[1]
T. Sh. Kal’menov and B. Torebek, “A method for solving ill-posed nonlocal problem for the elliptic equation with data on the whole boundary,” JOURNAL OF PSEUDO-DIFFERENTIAL OPERATORS AND APPLICATIONS, vol. 10, no. 1, pp. 177–185, 2019.
@article{8663367,
  abstract     = {{In this paper a nonlocal problem for the elliptic equation in a cylindrical domain is considered. It is shown that this problem is ill-posed as well as the Cauchy problem for the Laplace equation. The method of spectral expansion in eigenfunctions of the nonlocal problem for equations with involution establishes a criterion of the strong solvability of the considered nonlocal problem. It is shown that the ill-posedness of the nonlocal problem is equivalent to the existence of an isolated point of the continuous spectrum for a nonself-adjoint operator with involution.}},
  author       = {{Kal’menov, Tynysbek Sh. and Torebek, Berikbol}},
  issn         = {{1662-9981}},
  journal      = {{JOURNAL OF PSEUDO-DIFFERENTIAL OPERATORS AND APPLICATIONS}},
  keywords     = {{Elliptic operator,Nonlocal boundary conditions,Operator with involution,Criterion of well-posedness,Riesz basis,Primary 35J25,35C10,Secondary 35P10,MIXED CAUCHY-PROBLEM,STRONG SOLVABILITY,LAPLACE,REGULARIZATION,CRITERION}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{177--185}},
  title        = {{A method for solving ill-posed nonlocal problem for the elliptic equation with data on the whole boundary}},
  url          = {{http://doi.org/10.1007/s11868-017-0231-y}},
  volume       = {{10}},
  year         = {{2019}},
}

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