Geodesic
Classical solution of one problem of a perfectly inelastic impact on a long elastic semi-infinite bar with a linear elastic element at the end
Journal of the Belarusian State University. Mathematics and Informatics, Tome 2 (2022), pp. 34-46.

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

In this article, we study the classical solution of the mixed problem in a quarter of a plane for a one-dimensional wave equation. This mixed problem models the propagation of displacement waves during a longitudinal impact on a bar, when the load remains in contact with the bar and the bar has a linear elastic element at the end. On the lower boundary, the Cauchy conditions are specified, and the second of them has a discontinuity of the first kind at one point. The boundary condition, including the unknown function and its first and second order partial derivatives, is set at the side boundary. The solution is built using the method of characteristics in an explicit analytical form. The uniqueness is proven and the conditions are established under which a piecewise-smooth solution exists. The problem with matching conditions is considered.
Keywords: One-dimensional wave equation; inhomogeneous equation; mixed problem; non-smooth boundary conditions; longitudinal impact; method of characteristics. 
@article{BGUMI_2022_2_a3,
     author = {V. I. Korzyuk and J. V. Rudzko},
     title = {Classical solution of one problem of a perfectly inelastic impact on a long elastic semi-infinite bar with a linear elastic element at the end},
     journal = {Journal of the Belarusian State University. Mathematics and Informatics},
     pages = {34--46},
     publisher = {mathdoc},
     volume = {2},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/BGUMI_2022_2_a3/}
}
TY  - JOUR
AU  - V. I. Korzyuk
AU  - J. V. Rudzko
TI  - Classical solution of one problem of a perfectly inelastic impact on a long elastic semi-infinite bar with a linear elastic element at the end
JO  - Journal of the Belarusian State University. Mathematics and Informatics
PY  - 2022
SP  - 34
EP  - 46
VL  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BGUMI_2022_2_a3/
LA  - ru
ID  - BGUMI_2022_2_a3
ER  - 
%0 Journal Article
%A V. I. Korzyuk
%A J. V. Rudzko
%T Classical solution of one problem of a perfectly inelastic impact on a long elastic semi-infinite bar with a linear elastic element at the end
%J Journal of the Belarusian State University. Mathematics and Informatics
%D 2022
%P 34-46
%V 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BGUMI_2022_2_a3/
%G ru
%F BGUMI_2022_2_a3
V. I. Korzyuk; J. V. Rudzko. Classical solution of one problem of a perfectly inelastic impact on a long elastic semi-infinite bar with a linear elastic element at the end. Journal of the Belarusian State University. Mathematics and Informatics, Tome 2 (2022), pp. 34-46. http://geodesic.mathdoc.fr/item/BGUMI_2022_2_a3/

[1] V. A. Lazaryan, “O dinamicheskikh usiliyakh v upryazhnykh priborakh odnorodnykh poezdov pri soprotivleniyakh otnositelnym peremescheniyam ekipazhei”, Trudy Dnepropetrovskogo instituta inzhenerov zheleznodorozhnogo transporta, 20 (1950), 3–32

[2] A. I. Mavrin, “K teorii udarnogo pogruzheniya svai”, Izvestiya vuzov. Stroitelstvo i arkhitektura, 8 (1967), 24–28

[3] J. Boussinesq, “Du choc longitudinal d’une barre prismatique, fixee a un bout et heurtee a l’autre”, Comptes Rendus de l’Académie des Sciences, 97:2 (1883), 154–157

[4] S. I. Gaiduk, “O nekotorykh zadachakh, svyazannykh s teoriei poperechnogo udara po sterzhnyam”, Differentsialnye uravneniya, 13:7 (1977), 1233–1243 | Zbl

[5] S. I. Gaiduk, “Matematicheskoe rassmotrenie nekotorykh zadach, svyazannykh s teoriei prodolnogo udara po konechnym sterzhnyam”, Differentsialnye uravneniya, 13:11 (1977), 2009–2025 | Zbl

[6] V. I. Korzyuk, Ya. V. Rudko, “Klassicheskoe reshenie odnoi zadachi ob absolyutno neuprugom udare po dlinnomu uprugomu polubeskonechnomu sterzhnyu”, Vestsi Natsyyanalnai akademii navuk Belarusi. Seryya fizika-matematychnykh navuk, 57:4 (2021), 417–427 | DOI | MR

[7] V. I. Korzyuk, Uravneniya matematicheskoi fiziki, URSS, Moskva, 2021, 480 pp.

[8] V. I. Korzyuk, I. S. Kozlovskaya, Klassicheskie resheniya zadach dlya giperbolicheskikh uravnenii, BGU, Minsk, 2017, 48 pp.

[9] V. I. Korzyuk, Ya. V. Rudko, “Klassicheskoe reshenie smeshannoi zadachi dlya odnomernogo volnovogo uravneniya s negladkim vtorym usloviem Koshi”, Vestsi Natsyyanalnai akademii navuk Belarusi. Seryya fizika-matematychnykh navuk, 57:1 (2021), 23–32 | DOI | MR

[10] V. I. Korzyuk, Ya. V. Rudko, “Klassicheskoe reshenie smeshannoi zadachi dlya odnomernogo volnovogo uravneniya s negladkim vtorym usloviem Koshi”, Doklady Natsionalnoi akademii nauk Belarusi, 64:6 (2020), 657–662 | DOI | MR

[11] F. E. Lomovtsev, E. N. Novikov, “Metod Dyuamelya resheniya neodnorodnogo uravneniya kolebanii poluogranichennoi struny s kosoi proizvodnoi v nestatsionarnom granichnom uslovii”, Vestnik BGU. Fizika. Matematika. Informatika, 1 (2012), 83–86 | Zbl

[12] T. S. Shlapakova, N. I. Yurchuk, “Smeshannaya zadacha dlya uravneniya kolebaniya ogranichennoi struny s zavisyaschei ot vremeni proizvodnoi v kraevom uslovii, napravlennoi po kharakteristike”, Vestnik BGU. Fizika. Matematika. Informatika, 2 (2013), 84–90 | Zbl

[13] T. S. Shlapakova, N. I. Yurchuk, “Smeshannaya zadacha dlya uravneniya kolebaniya ogranichennoi struny s proizvodnoi v kraevom uslovii, napravlennoi ne po kharakteristike”, Vestnik BGU. Fizika. Matematika. Informatika, 1 (2013), 64–69 | Zbl

[14] C. I. Gaiduk, “Matematicheskoe rassmotrenie odnoi zadachi o prodolnom udare po relaksiruyuschemu sterzhnyu”, Differentsialnye uravneniya, 12:4 (1976), 668–685 | MR | Zbl

[15] N. I. Yurchuk, E. N. Novikov, “Neobkhodimye usloviya dlya suschestvovaniya klassicheskikh reshenii uravneniya kolebanii poluogranichennoi struny”, Vestsi Natsyyanalnai akademii navuk Belarusi. Seryya fizika-matematychnykh navuk, 4 (2016), 116–120 | MR