Geodesic
To boundary-value problems for degenerating pseudoparabolic equations with Gerasimov--Caputo fractional derivative
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2018), pp. 3-16.

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In this paper we consider a boundary-value problems for degenerating pseudoparabolic equation with variable coefficients and with Gerasimov–Caputo fractional derivative. To solve the problem we obtain apriori estimates in differential and difference settings. These apriori estimates imply uniqueness and stability of the solution with respect to the initial data and the right-hand side on the layer, as well as the convergence of the solution of each of the difference problem to the solution of the corresponding differential problem.
Keywords: boundary-value problem, apriori estimate, difference scheme, the equation of pseudoparabolic type, differential equation of fractional order, Gerasimov–Caputo fractional derivative. 
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M. Kh. Beshtokov. To boundary-value problems for degenerating pseudoparabolic equations with Gerasimov--Caputo fractional derivative. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2018), pp. 3-16. http://geodesic.mathdoc.fr/item/IVM_2018_10_a0/

[1] Rubinshtein L. I., “K voprosu o protsesse rasprostraneniya tepla v geterogennykh sredakh”, Izv. AN SSSR, ser. geogr., 12:1 (1948), 27–45

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

[3] Hallaire M., “Le potentiel efficace de l'eau dans le sol en regime de dessechement”, L'Eau et la Production Vegetale, 9, Institut National de la Recherche Agronomique, Paris, 1964, 27–62

[4] Barenblat G. I., Zheltov Yu. P., Kochina I. N., “Ob osnovnykh predstavleniyakh teorii filtratsii odnorodnykh zhidkostei v treschinovatykh porodakh”, PMM, 25:5 (1960), 852–864

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

[6] Samko S. G., Kilbas A. A., Marichev O. I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, Minsk, 1987

[7] Nakhushev A. M., Elementy drobnogo ischisleniya i ikh primeneniya, NIII PMA KBNTs RAN, Nalchik, 2000

[8] Alikhanov A. A., “Apriornye otsenki reshenii kraevykh zadach dlya uravnenii drobnogo poryadka”, Differents. uravneniya, 46:5 (2010), 660–666 | Zbl

[9] Alikhanov A. A., “A new difference scheme for the time fractional diffusion equation”, J. Comput. Phys., 280 (2015), 424–438 | DOI | MR | Zbl

[10] Goloviznin V. M., Kiselev V. P., Korotkii I. A., Chislennye metody resheniya uravneniya drobnoi diffuzii s drobnoi proizvodnoi po vremeni v odnomernom sluchae, In-t probl. bezopasnogo razvitiya atomnoi energetiki RAN, M., 2003

[11] Taukenova F. I., Shkhanukov-Lafishev M. Kh., “Raznostnye metody resheniya kraevykh zadach dlya differentsialnykh uravnenii drobnogo poryadka”, Zhurn. vychisl. matem. i matem. fiz., 46:10 (2006), 1871–1881 | MR

[12] Beshtokov M. Kh., “O chislennom reshenii nelokalnoi kraevoi zadachi dlya vyrozhdayuschegosya psevdoparabolicheskogo uravneniya”, Differents. uravneniya, 52:10 (2016), 1393–1406 | DOI | Zbl

[13] Beshtokov M. Kh., “Raznostnyi metod resheniya nelokalnoi kraevoi zadachi dlya vyrozhdayuschegosya psevdoparabolicheskogo uravneniya tretego poryadka s peremennymi koeffitsientami”, Zhurn. vychisl. matem. i matem. fiz., 56:10 (2016), 1780–1794 | DOI | MR | Zbl

[14] Caputo H., “Linear model of dissipation whose $Q$ is almost frequency independent. II”, Geophys. J. Roy. Astronom. Soc., 13 (1967), 529–539 | DOI

[15] Gerasimov A. N., “Obobschenie lineinykh zakonov deformatsii i ikh prilozhenie k zadacham vnutrennego treniya”, AN SSSR. PMM, 12 (1948), 251–260 | Zbl

[16] Samarskii A. A., Teoriya raznostnykh skhem, Nauka, M., 1983

[17] Li D., Liao J.-L., Sun W., J. Wang J., Zhang J., Analysis of L1–Galerkin FEMs for time-fractional nonlinear parabolic problems, 2016, arXiv: 1612.00562v1