K. B. Sabitov, “The Dirichlet Problem for Higher-Order Partial Differential Equations”, Mat. Zametki, 97:2 (2015), 262–276; Math. Notes, 97:2 (2015), 255

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Matematicheskie Zametki, 2015, Volume 97, Issue 2, Pages 262–276
DOI: https://doi.org/10.4213/mzm9286 (Mi mzm9286)

This article is cited in 27 scientific papers (total in 27 papers)

The Dirichlet Problem for Higher-Order Partial Differential Equations

K. B. Sabitov^ab

^a Novosibirsk State University
^b Institute of Applied Research, Sterlitamak

Full-text PDF (556 kB) Citations (27)

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DOI: https://doi.org/10.4213/mzm9286

Abstract: For higher-order partial differential equations in two or three variables, the Dirichlet problem in rectangular domains is studied. Small denominators hampering the convergence of series appear in the process of constructing the solution of the problem by the spectral decomposition method. A uniqueness criterion for the solution is established. In the two-dimensional case, estimates justifying the existence of a solution of the Dirichlet problem are obtained. In the three-dimensional case where the domain is a cube, it is shown that the uniqueness of the solution of the Dirichlet problem is equivalent to the great Fermat problem.

Keywords: higher-order partial differential equation, Dirichlet problem, spectral decomposition method, Fourier series, Fermat problem.

Funding agency	Grant number
Russian Foundation for Basic Research	14-01-97003-р_поволжье_а
This work was supported by the Russian Foundation for Basic Research (r_Povolzh'e_a) (grant no. 14-01-97003).

Received: 30.11.2011
Revised: 26.06.2014

English version:
Mathematical Notes, 2015, Volume 97, Issue 2, Pages 255–267
DOI: https://doi.org/10.1134/S0001434615010277

Bibliographic databases:

Document Type: Article

UDC: 517.95

Language: Russian

Citation: K. B. Sabitov, “The Dirichlet Problem for Higher-Order Partial Differential Equations”, Mat. Zametki, 97:2 (2015), 262–276; Math. Notes, 97:2 (2015), 255–267

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Linking options:

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https://doi.org/10.4213/mzm9286

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This publication is cited in the following 27 articles:

A. K. Urinov, M. S. Azizov, “Initial-Boundary Value Problem for a Degenerate High Even-Order Partial Differential Equation with the Bessel Operator”, Lobachevskii J Math, 45:2 (2024), 864
Muataz Almohamed, 2024 Intelligent Technologies and Electronic Devices in Vehicle and Road Transport Complex (TIRVED), 2024, 1
A. K. Urinov, M. S. Azizov, “About an initial boundary problem for a degenerate higher even order partial differential equation”, J. Appl. Industr. Math., 17:2 (2023), 414–426
K. B. Sabitov, “Forward and inverse source reconstruction problems for the equations of vibrations of a rectangular plate”, Comput. Math. Math. Phys., 63:4 (2023), 582–595
B. Yu. Irgashev, “On the Conditions for the Solvability of Boundary-Value Problems for Higher-Order Equations with Discontinuous Coefficients”, Math. Notes, 111:2 (2022), 217–229
T. K. Yuldashev, “On a nonlocal boundary value problem for a partial integro-differential equations with degenerate kernel”, Vladikavk. matem. zhurn., 24:2 (2022), 130–141
A. K. Urinov, M. S. Azizov, “O razreshimosti nelokalnykh nachalno-granichnykh zadach dlya odnogo differentsialnogo uravneniya v chastnykh proizvodnykh vysokogo chetnogo poryadka”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 32:2 (2022), 240–255
D. Amanov, O. Sh. Kilichov, “Nonlocal Boundary Value Problem for a Fourth Order Differential Equation”, Lobachevskii J Math, 43:2 (2022), 293
A. V. Glushak, “Uniqueness Criterion for the Solution of Boundary-Value Problems for the Abstract Euler–Poisson–Darboux Equation on a Finite Interval”, Math. Notes, 109:6 (2021), 867–875
N. V. Zaitseva, “Keldysh type problem with an integral condition for two-dimensional $B$ -hyperbolic equation”, Moscow University Mathematics Bulletin, 76:5 (2021), 221–229
Ya. O. Baranetskij, P. I. Kalenyuk, “Nonlocal Problem with Multipoint Perturbations of Dirichlet Conditions for Even-Order Partial Differential Equations with Constant Coefficients”, J Math Sci, 256:4 (2021), 375
T. K. Yuldashev, F. D. Rakhmonov, “On a Benney–Luke Type Differential Equation with Nonlinear Boundary Value Conditions”, Lobachevskii J Math, 42:15 (2021), 3761
A. H. Babayan, Springer Proceedings in Mathematics & Statistics, 357, Operator Theory and Harmonic Analysis, 2021, 55
E. Providas, I. N. Parasidis, Springer Optimization and Its Applications, 179, Mathematical Analysis in Interdisciplinary Research, 2021, 641
B. Yu. Irgashev, “On a boundary value problem for a high order mixed type equation”, Sib. elektron. matem. izv., 17 (2020), 899–912
K. B. Sabitov, “Inverse problems of determining the right-hand side and the initial conditions for the beam vibration equation”, Differ. Equ., 56:6 (2020), 761–774
A. Gimaltdinova, “The Dirichlet problem for an equation of mixed type with two internal lines of type change”, Lobachevskii J. Math., 41:11, SI (2020), 2155–2167
Ya. O. Baranetskij, P. І. Kalenyuk, M. І. Kopach, “Nonlocal Multipoint Problem for Partial Differential Equations of Even Order with Constant Coefficients”, J Math Sci, 249:3 (2020), 307
Yu. K. Sabitova, “The Dirichlet problem for a hyperboli-type equation with power degeneracy in a rectangular domain”, Differ. Equ., 54:2 (2018), 228–238
Yu. K. Sabitova, “Dirichlet problem for Lavrent'ev–Bitsadze equation with loaded summands”, Russian Math. (Iz. VUZ), 62:9 (2018), 35–51

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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