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 Mat. Zametki, 2015, Volume 97, Issue 2, Pages 262–276 (Mi mz9286)

The Dirichlet Problem for Higher-Order Partial Differential Equations

K. B. Sabitovab

a Novosibirsk State University
b Institute of Applied Research, Sterlitamak

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-ð_ïîâîëæüå_à This work was supported by the Russian Foundation for Basic Research (r_Povolzh'e_a) (grant no. 14-01-97003).

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

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English version:
Mathematical Notes, 2015, 97:2, 255–267

Bibliographic databases:

Document Type: Article
UDC: 517.95
Revised: 26.06.2014

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

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz9286
• https://doi.org/10.4213/mzm9286
• http://mi.mathnet.ru/eng/mz/v97/i2/p262

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2. Sadybekov M.A., Yessirkegenov N.A., “Boundary-Value Problems For Wave Equations With Data on the Whole Boundary”, Electron. J. Differ. Equ., 2016, 281
3. K. B. Sabitov, “On fixed sign solution to nonhomogeneous equation of mixed parabolic-hyperbolic type of higher order”, Russian Math. (Iz. VUZ), 61:7 (2017), 49–57
4. B. Yu. Irgashev, “On one boundary-value problem for an equation of higher even order”, Russian Math. (Iz. VUZ), 61:9 (2017), 10–26
5. Yu. K. Sabitova, “The Dirichlet problem for telegraph equation in a rectangular domain”, Russian Math. (Iz. VUZ), 61:12 (2017), 39–48
6. Sabitova Yu.K., “The Dirichlet Problem For a Hyperboli-Type Equation With Power Degeneracy in a Rectangular Domain”, Differ. Equ., 54:2 (2018), 228–238
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8. Baranetskij Ya.O. Ivasiuk I.Ya. Kalenyuk I P. Solomko V A., “the Nonlocal Boundary Problem With Perturbations of Antiperiodicity Conditions For the Eliptic Equation With Constant Coefficients”, Carpathian Math. Publ., 10:2 (2018), 215–234
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