Geodesic
Solution of a semi-boundary-value problem for a first-order degenerate partial differential equation
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh Winter Mathematical School "Modern Methods of Function Theory and Related Problems." January 28 – February 2, 2019. Part 4, Tome 173 (2019), pp. 48-57.

Voir la notice de l'article provenant de la source Math-Net.Ru

A first-order partial differential equation with constant irreversible coefficients in a Banach space is considered. In the particular case of a finite-dimensional space, the initial-boundary-value problem with irreversible matrix coefficients has no solution; hence, we pose Showalter-type conditions. Due to the regularity of the operator pencil, the equation splits into differential equations in subspaces and given conditions lead to initial conditions in subspaces. A solution to the problem is constructed and an example is provided.
Keywords: Banach space, degenerate partial differential equation, $0$-normal eigenvalue, Showalter-type conditions. 
@article{INTO_2019_173_a3,
     author = {S. P. Zubova and A. H. Mohamad and V. I. Uskov},
     title = {Solution of a semi-boundary-value problem for a first-order degenerate partial differential equation},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {48--57},
     publisher = {mathdoc},
     volume = {173},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2019_173_a3/}
}
TY  - JOUR
AU  - S. P. Zubova
AU  - A. H. Mohamad
AU  - V. I. Uskov
TI  - Solution of a semi-boundary-value problem for a first-order degenerate partial differential equation
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2019
SP  - 48
EP  - 57
VL  - 173
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2019_173_a3/
LA  - ru
ID  - INTO_2019_173_a3
ER  - 
%0 Journal Article
%A S. P. Zubova
%A A. H. Mohamad
%A V. I. Uskov
%T Solution of a semi-boundary-value problem for a first-order degenerate partial differential equation
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2019
%P 48-57
%V 173
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2019_173_a3/
%G ru
%F INTO_2019_173_a3
S. P. Zubova; A. H. Mohamad; V. I. Uskov. Solution of a semi-boundary-value problem for a first-order degenerate partial differential equation. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh Winter Mathematical School "Modern Methods of Function Theory and Related Problems." January 28 – February 2, 2019. Part 4, Tome 173 (2019), pp. 48-57. http://geodesic.mathdoc.fr/item/INTO_2019_173_a3/

[1] Boyarintsev Yu. E., Regulyarnye i singulyarnye sistemy lineinykh obyknovennykh differentsialnykh uravnenii, Nauka, Novosibirsk, 1980

[2] Vainberg M. M., Trenogin V. A., Teoriya vetvleniya reshenii nelineinykh uravnenii, Nauka, M., 1969

[3] Gelfand I. M., Lektsii po lineinoi algebre, MTsNMO, M., 1998 | MR

[4] Gokhberg I. Ts., Krein M. G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov, Nauka, M., 1965

[5] Demidenko G. V., Uspenskii S. V., Uravneniya i sistemy, ne razreshennye otnositelno starshei proizvodnoi, Nauchnaya kniga, Novosibirsk, 1998

[6] Zubova S. P., Singulyarnoe vozmuschenie lineinykh differentsialnykh uravnenii, nerazreshennykh otnositelno proizvodnoi, Diss na soisk. uch. step. kand. fiz.-mat. nauk, Voronezh, 1973 | Zbl

[7] Zubova S. P., “Svoistva vozmuschennogo fredgolmovskogo operatora”, Reshenie differentsialnogo uravneniya s fredgolmovskim operatorom pri proizvodnoi, VGU, Voronezh, 1991

[8] Zubova S. P., “Reshenie zadachi Koshi dlya dvukh differentsialno-algebraicheskikh uravnenii s fredgolmovym operatorom”, Differ. uravn., 41:10 (2005), 1410–1412 | MR | Zbl

[9] Zubova S. P., “Reshenie odnorodnoi zadachi Koshi dlya uravneniya s neterovym operatorom pri proizvodnoi”, Dokl. RAN., 428:4 (2009), 444–446 | Zbl

[10] Zubova S. P. Chernyshov K. I., “O lineinom differentsialnom uravnenii s fredgolmovym operatorom pri proizvodnoi”, Differentsialnye uravneniya i ikh primenenie, 14, In-t fiz. mat. AN Lit. SSR, Vilnyus, 1976, 21–39

[11] Krein S. G., Lineinye differentsialnye uravneniya v banakhovom prostranstve, Nauka, M., 1967

[12] Nguen Khak Diep, Chistyakov V. F., “O modelirovanii s ispolzovaniem differentsialno-algebraicheskikh uravnenii v chastnykh proizvodnykh”, Vestn. YuUrGU. Ser. Mat. model. program., 6:1 (2013), 98–111

[13] Sobolev S. L., “Ob odnoi novoi zadache matematicheskoi fiziki”, Izv. AN SSSR. Ser. mat., 18:1 (1954), 3–50 | MR | Zbl

[14] Chistyakov V. F., Scheglova A. A., Izbrannye glavy teorii algebro-differentsialnykh sistem, Nauka, Novosibirsk, 2003 | MR

[15] Campbell S. L., Singular Systems of Differential Equations, Pitman, London, 1980 | MR | Zbl

[16] Kunkel P., Mehrmann V., Differential-Algebraic Equations: Analysis and Numerical Solutions, EMA, Zürich, 2006 | MR

[17] Poincaré H., “Sur l'equilibre d'une masse fluide animee d'un movement de rotation”, Acta Math., 7 (1885), 259–380 | DOI | MR | Zbl

[18] Wade S. M., Paul I. B., “A differentiation index for partial differential algebraic equations”, SIAM J. Sci. Comp., 21:6 (2000), 2295–2316 | DOI