Geodesic
A boundary value problem for the third-order equation not solvable with respect to the highest-order derivative
Matematičeskie zametki SVFU, Tome 24 (2017), pp. 28-36.

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We consider a boundary value problem for the third-order equation not solvable with respect to the highest-order derivative. Equations of this type, often called Sobolev type equations, occur in many applied problems. The nonstationary Galerkin method and regularization method are applied to prove the existence and uniqueness theorem for a regular solution of the boundary value problem. Also we obtain an error estimate via regularization parameter and in terms of eigenvalues of the spectral problem for the Laplace operator.
Mots-clés : pseudoparabolic equation
Keywords: boundary value problem, solvability, a priori estimate, approximate solution, error estimate. 
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     title = {A boundary value problem for the third-order equation not solvable with respect to the highest-order derivative},
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I. E. Egorov; E. S. Efimova. A boundary value problem for the third-order equation not solvable with respect to the highest-order derivative. Matematičeskie zametki SVFU, Tome 24 (2017), pp. 28-36. http://geodesic.mathdoc.fr/item/SVFU_2017_24_a2/

[1] Yakubov S. Y., Linear Differential-Operator Equations and Their Applications, Izdat. ELM, Baku, 1985

[2] Egorov I. E., Fedorov V. E., Nonclassical Higher-Order Equations in Mathematical Physics, Izdat. Vychisl. Tsentra SO RAN, Novosibirsk, 1995 | MR

[3] Demidenko G. V., Uspenskiy S. V., Equations and Systems not Solvable with Respect to the Highest-Order Derivative, Nauchn. Kniga, Novosibirsk, 1998 | MR

[4] Egorov I. E., Pyatkov S. G., Popov S. V., Nonclassical Differential-Operator Equations, Nauka, Novosibirsk, 2000 | MR

[5] Kozhanov A. I., “Boundary value problems for some classes of higher-order equations not solvable with respect to the highest-order derivative”, Sib. Math. J., 35:2 (1994), 361–376

[6] Sveshnikov A. G., Alshin A. B., Korpusov M. O., Pletner Yu. D., Linear and Nonlinear Equations of Sobolev Type, Fizmatlit, Moscow, 2007

[7] Lions J.-L., Some Methods of Solving Nonlinear Boundary Value Problems, Mir, Moscow, 1973 | MR

[8] Ficera G., “The unified theory of boundary value problems for elliptic-parabolic equations”, Math., 7:6 (1963), 99–121

[9] Nakhushev A. M., Problems with Shift for the Equations in Partial Derivatives, Nauka, Moscow, 2006

[10] Beals R., Protopescu V., “Half-range completeness for the Fokker-Planck equation”, J. Statist. Phys., 32:3 (1983), 565–584 | DOI | MR | Zbl

[11] Kozhanov A. I., “The existence of regular solutions of the first boundary value Problem for one class of Sobolev type equations with alternating direction”, Mat. Zamet. YaGU, 4:2 (1997), 39–48 | Zbl

[12] Kozhanov A. I. Potapova S. V., “The Dirichlet problem for a class of composite type equations with a discontinuous coefficient of the highest derivative”, Dal'nevost. Mat. Zhurn., 14:1 (2014), 48–65 | MR | Zbl

[13] Ladyzhenskaya O. A., Boundary Value Problems of Mathematical Physics, Nauka, Moscow, 1973 | MR

[14] Vinogradova P. V., Zarubin A. G., “Error estimation of Galerkin method for non-stationary equations”, J. Comput. Math. Math. Phys., 49:9 (2009), 1643–1651 | MR | Zbl

[15] Egorov I. E., “On a modified Galerkin method for a parabolic equation with alternating time direction”, Uzbek Math. J., 2013, no. 3, 33–40