Geodesic
Solution of the Mixed Problem for a Third-Order Partial Differential Equation
Matematičeskie zametki, Tome 111 (2022) no. 6, pp. 895-903.

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We consider the initial-boundary value problem for a third-order partial differential equation with highest mixed derivative. An abstract Cauchy problem for a first-order algebraic-differential equation in a Banach space with a distinguished time variable is solved. It is proved that, by equivalent replacements, the original problem is reduced to the Cauchy problem for an algebraic-differential equations. To solve the stated problem, the Fredholm property of the operator before the highest derivative is used. Conditions under which the solution of the problem exists and is unique are determined, and this solution is found in analytical form.
Keywords: partial differential equation of third order, mixed derivative, Cauchy problem, algebraic-differential equation, Banach space, Fredholm operator. 
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V. I. Uskov. Solution of the Mixed Problem for a Third-Order Partial Differential Equation. Matematičeskie zametki, Tome 111 (2022) no. 6, pp. 895-903. http://geodesic.mathdoc.fr/item/MZM_2022_111_6_a8/

[1] G. I. Barenblat, Yu. P. Zheltov, I. N. Kochina, “Ob osnovnykh predstavleniyakh teorii filtratsii v treschinovatykh sredakh”, Prikl. matem. mekh., 24:5 (1960), 58–73

[2] A. F. Chudnovskii, Teplofizika pochv, Nauka, M., 1976

[3] V. L. Ginzburg, A. A. Rukhadze, Volny v magnitoaktivnoi plazme, Nauka, M., 1975

[4] A. G. Sveshnikov, A. B. Alshin, M. O. Korpusov, Yu. D. Pletner, Lineinye i nelineinye uravneniya sobolevskogo tipa, Fizmatlit, M., 2007 | Zbl

[5] A. F. Chudnovskii, “Vliyanie obrabotki pochvy na ee teplovoi rezhim”, Sb. trudov po agrofizike, 1969, no. 23, 55–63

[6] M. Kh. Shkhanukov, “O nekotorykh kraevykh zadachakh dlya uravneniya tretego poryadka, voznikayuschikh pri modelirovanii filtratsii zhidkosti v poristykh sredakh”, Differents. uravneniya, 18:4 (1982), 689–699 | MR | Zbl

[7] M. Kh. Beshtokov, “Metod funktsii Rimana i raznostnyi metod resheniya odnoi nelokalnoi kraevoi zadachi dlya uravneniya tretego poryadka giperbolicheskogo tipa”, Izv. Vuzov. Severo-Kavkazskii region, 2007, no. 5, 6–9

[8] S. P. Zubova, V. I. Uskov, “Reshenie zadachi dlya uravneniya tretego poryadka sobolevskogo tipa metodom kaskadnoi dekompozitsii”, Vestn. PMM, 11 (2015), 70–83

[9] A. A. Scheglova, “Issledovanie i reshenie vyrozhdennykh sistem obyknovennykh differentsialnykh uravnenii s pomoschyu zamen peremennykh”, Sib. matem. zhurn., 36:6 (1995), 1435–1445 | MR | Zbl

[10] G. A. Sviridyuk, V. E. Fedorov, “Polugruppy operatorov s yadrami”, Vestn. Chelyabinskogo gos. un-ta, 2002, no. 6, 42–70 | Zbl

[11] S. P. Zubova, “O razreshimosti zadachi Koshi dlya deskriptornogo psevdoregulyarnogo uravneniya v banakhovom prostranstve”, Vestn. Voronezhskogo gos. un-ta. Ser. Fiz. Matem., 2013, no. 2, 192–198 | Zbl

[12] S. P. Zubova, E. V. Raetskaya, V. I. Uskov, “Degeneracy property of a matrix-differential operator and applications”, J. Math. Sci. (N.Y.), 255:5 (2021), 640–652 | DOI | MR | Zbl

[13] V. I. Uskov, “Initial-boundary value problem for perturbed third order partial differential equations”, J. Math. Sci. (N.Y.), 255:6 (2021), 779–789 | DOI | MR | Zbl

[14] S. G. Krein, Lineinye differentsialnye uravneniya v banakhovom prostranstve, Nauka, M., 1967 | MR | Zbl