Matching conditions for values of characteristic oblique derivative at the end of a string, initial data and right-hand side of the wave equation

E. V. Ustilko ; F. E. Lomovtsev

Journal of the Belarusian State University. Mathematics and Informatics, Tome 1 (2020), pp. 30-37.

Résumé

Sufficient matching conditions the time-dependent characteristic first derivatives in the boundary mode with the initial conditions and the more general vibration equation of a semi-bounded string are derived in the sets of solutions of all higher order smoothness orders. They generalize the previously found sufficient matching conditions in the case of a similar mixed problem for the simplest string vibration equation. The characteristic of non-stationary first oblique derivatives in the boundary mode means that at each moment of time they are directed along the critical characteristic.

Détail

Keywords: mixed problem; characteristic first oblique derivative; initial conditions; smoothness requirements; matching conditions.

Bibliographie

[1] A. N. Tikhonov, A. A. Samarskii, “Uravneniya matematicheskoi fiziki”, Moskva: Nauka, 2004, 798 | MR

[2] S. N. Baranovskaya, N. I. Yurchuk, “Smeshannaya zadacha dlya uravneniya kolebaniya struny s zavisyaschei ot vremeni kosoi proizvodnoi v kraevom uslovii”, Differentsialnye uravneniya, 45(8) (2009), 1188–1191 | MR | Zbl

[3] F. E. Lomovtsev, E. V. Ustilko, “Kriterii korrektnosti smeshannoi zadachi dlya obschego uravneniya kolebanii poluogranichennoi struny s nestatsionarnoi kharakteristicheskoi pervoi kosoi proizvodnoi v granichnom uslovii”, Vesnik Vitsebskaga dzyarzhaunaga universiteta, 4(101) (2018), 18–28

[4] F. E. Lomovtsev, T. S. Tochko, “Smeshannaya zadacha dlya neodnorodnogo uravneniya kolebanii ogranichennoi struny pri kharakteristicheskikh nestatsionarnykh pervykh kosykh proizvodnykh na kontsakh”, Vesnik Grodzenskaga dzyarzhaunaga universiteta imya Yanki Kupaly. Matematyka. Fizika. Infarmatyka, vylichalnaya tekhnika i kiravanne, 9(2) (2019), 56–75

[5] F. E. Lomovtsev, “Metod vspomogatelnykh smeshannykh zadach dlya poluogranichennoi struny”, Shestye Bogdanovskie chteniya po obyknovennym differentsialnym uravneniyam, Belarus, Minsk (Materialy Mezhdunarodnoi matematicheskoi konferentsii), Institut matematiki NAN Belarusi

[6] F. E. Lomovtsev, “Metod korrektirovki probnykh reshenii obschego volnovogo uravneniya v pervoi chetverti ploskosti dlya minimalnoi gladkosti ego pravoi chasti”, Zhurnal Belorusskogo gosudarstvennogo universiteta. Matematika. Informatika, 3 (2017), 38–52

[7] F. E. Lomovtsev, “Reshenie bez prodolzheniya dannykh smeshannoi zadachi dlya neodnorodnogo uravneniya kolebanii struny pri granichnykh kosykh proizvodnykh”, Differentsialnye uravneniya, 52(8) (2016), 1128–1132 | Zbl

[8] F. E. Lomovtsev, “Neobkhodimye i dostatochnye usloviya vynuzhdennykh kolebanii poluogranichennoi struny s pervoi kharakteristicheskoi kosoi proizvodnoi v nestatsionarnom granichnom uslovii”, Vestsi Natsyyanalnai akademii navuk Belarusi. Seryya fizika-matematychnykh navuk, 1 (2016), 21–27

[9] E. N. Novikov, “Smeshannye zadachi dlya uravneniya vynuzhdennykh kolebanii ogranichennoi struny pri nestatsionarnykh granichnykh usloviyakh s pervoi i vtoroi kosymi proizvodnymi”, Minsk: BGU, 2017, 258

[10] F. E. Lomovtsev, N. I. Yurchuk, “Reshenie nachalno-kraevoi zadachi dlya nestrogo giperbolicheskogo uravneniya pri smeshannykh granichnykh usloviyakh v chetverti ploskosti”, Vestsi Natsyyanalnai akademii navuk Belarusi. Seryya fizika-matematychnykh navuk, 3 (2016), 51–57

[11] F. E. Lomovtsev, “Kriterii korrektnosti smeshannoi zadachi dlya odnogo parabolicheskogo uravneniya na otrezke so smeshannymi granichnymi usloviyami na kontsakh”, Sovremennye metody teorii funktsii i smezhnye problemy; Voronezhskaya zimnyaya matematicheskaya shkola. Materialy mezhdunarodnoi konferentsii (Rossiya), 2019, 184–185, Voronezh: Izdatelskii dom VGU