Geodesic
Generalized solution of the initial-boundary-value problem for the wave equation with a mixed derivative and a general potential
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 3, Tome 232 (2024), pp. 99-121.

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We study the initial-boundary-value problem in a half-strip for a second-order inhomogeneous hyperbolic equation with constant coefficients and a nonzero potential containing a mixed derivative. The equation considered is the equation of transverse vibrations of a moving finite string. The problems with general initial conditions (nonzero string profile and nonzero initial velocity of string points) and fixed ends (Dirichlet conditions) are examined. Theorems on the existence and uniqueness of a solution are formulated and formulas for the solution are obtained.
Mots-clés : partial differential equation, nonzero potential, wave equation, hyperbolic equation, mixed derivative, generalized solution 
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V. S. Rykhlov. Generalized solution of the initial-boundary-value problem for the wave equation with a mixed derivative and a general potential. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 3, Tome 232 (2024), pp. 99-121. https://geodesic-test.mathdoc.fr/item/INTO_2024_232_a8/

[1] Burlutskaya M. Sh., Khromov A. P., “Rezolventnyi podkhod v metode Fure”, Dokl. RAN., 458:2 (2014), 138–140 | DOI | MR | Zbl

[2] Burlutskaya M. Sh., Khromov A. P., “Rezolventnyi podkhod dlya volnovogo uravneniya”, Zh. vychisl. mat. mat. fiz., 55:2 (2015), 229–241 | DOI | MR | Zbl

[3] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1976 | MR

[4] Kornev V. V., “O primenenii raskhodyaschikhsya ryadov v smeshannykh zadachakh, ne imeyuschikh klassicheskogo resheniya”, Mat. Mezhdunar. konf. «Sovremennye metody teorii kraevykh zadach». Voronezhskaya vesennyaya matematicheskaya shkola «Pontryaginskie chteniya–XXXIII» (Voronezh, 3-9 maya 2022 g.), VGU, Voronezh, 2022, 132–137

[5] Kornev V. V., Khromov A. P., “O klassicheskom i obobschennom reshenii smeshannoi zadachi dlya volnovogo uravneniya”, Mat. Mezhdunar. konf. «Sovremennye metody teorii kraevykh zadach», posv. 90-letiyu akad. V. A. Ilina. Voronezhskaya vesennyaya matematicheskaya shkola «Pontryaginskie chteniya–XXIX» (Moskva, 3-5 maya 2018 g.), M., 2018, 132–133

[6] Kornev V. V., Khromov A. P., “Klassicheskoe reshenie smeshannoi zadachi dlya odnorodnogo volnovogo uravneniya s zakreplennymi kontsami”, Itogi nauki i tekhn. Ser. Sovr. mat. prilozh. Temat. obz., 172 (2019), 119–133 | DOI

[7] Kornev V. V., Khromov A. P., “Ispolzovanie rezolventnogo podkhoda i raskhodyaschikhsya ryadov pri reshenii smeshannykh zadach”, Matematika. Mekhanika. T. 23, Izd-vo Saratov. un-ta, 2021, 18–24

[8] Krylov A. N., O nekotorykh differentsialnykh uravneniyakh matematicheskoi fiziki, imeyuschikh prilozheniya v tekhnicheskikh voprosakh, GITTL, M.-L., 1950

[9] Kurdyumov V. P., Khromov A. P., Khalova V. A., “Smeshannaya zadacha dlya odnorodnogo volnovogo uravneniya s nenulevoi nachalnoi skorostyu s summiruemym potentsialom”, Izv. Saratov. un-ta. Nov. ser. Mat. Mekh. Inform., 20:4 (2020), 444–456 | MR | Zbl

[10] Lomov I. S., “Effektivnoe primenenie metoda Fure dlya postroeniya resheniya smeshannoi zadachi dlya telegrafnogo uravneniya”, Vestn. Mosk. un-ta. Ser. 15. Vychisl. mat. kibern., 2021, no. 4, 37–42

[11] Lomov I. S., “Obobschennaya formula Dalambera dlya telegrafnogo uravneniya”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obz., 199 (2021), 66–79 | DOI

[12] Lomov I. S., “Effektivnoe primenenie metoda Fure k resheniyu smeshannoi zadachi dlya telegrafnogo uravneniya”, Mat. 21 Mezhdunar. Saratov. zimnei shkoly «Sovremennye problemy teorii funktsii i ikh prilozheniya» (Saratov, 31 yanvarya — 4 fevralya 2022 g.), Izd-vo Saratov. un-ta, Saratov, 2022, 178–180

[13] Lomov I. S., “Postroenie obobschennogo resheniya smeshannoi zadachi dlya telegrafnogo uravneniya: sekventsialnyi i aksiomaticheskii podkhody”, Differ. uravn., 58:11 (2022), 1471–1483 | Zbl

[14] Lomov I. S., Vestn. Mosk.un-ta. Ser. 15. Vychisl. mat. kibern., 2022, no. 3, 33–40

[15] Lomovtsev F. E., “Metod korrektirovki probnykh reshenii volnovogo uravneniya v krivolineinoi pervoi chetverti ploskosti dlya minimalnoi gladkosti pravoi chasti”, Zh. Belorus. gos. un-ta. Mat. Inf., 2017, no. 3, 38–52

[16] Lomovtsev F. E., “Globalnaya teorema korrektnosti po Adamaru pervoi smeshannoi zadachi dlya volnovogo uravneniya v polupolose ploskosti”, Vesn. GrDU imya Yanki Kupaly. Ser. 2. Mat. Fiz. Іnfarm. Vylich. tekhn. kiravanne., 11:1 (2021), 68–82

[17] Lomovtsev F. E., “Pervaya smeshannaya zadacha dlya obschego telegrafnogo uravneniya s peremennymi koeffitsientami na polupryamoi”, Zh. Belorus. gos. un-ta. Mat. Inf., 1 (2021), 18–38

[18] Lomovtsev F. E., “Globalnaya teorema korrektnosti pervoi smeshannoi zadachi dlya obschego telegrafnogo uravneniya s peremennymi koeffitsientami na otrezke”, Probl. fiz. mat. tekhn., 1:50 (2022), 62–73

[19] Lomovtsev F. E., Lysenko V. N., Vesn. MDU imya Kulyashova A. A. Ser. B. Pryrod. navuki: Mat. Fiz. Biyalogiya., 2:58 (Smeshannaya zadacha dlya obschego odnomernogo volnovogo uravneniya v polupolose ploskosti pri nestatsionarny), 28–55

[20] Moiseev E. I., Lomovtsev F. E., Novikov E. N., “Neodnorodnoe faktorizovannoe giperbolicheskoe uravnenie vtorogo poryadka v chetverti ploskosti pri polunestatsionarnoi faktorizovannoi vtoroi kosoi proizvodnoi v granichnom uslovii”, Dokl. RAN., 459:5 (2014), 544–549 | DOI | Zbl

[21] Muravei L. A., Petrov V. M., Romanenkov A. M., “O zadache gasheniya poperechnykh kolebanii prodolno dvizhuscheisya struny”, Vestn. Mordov. un-ta., 28:4 (2018), 472–485

[22] Muravei L. A., Romanenkov A. M., “Chislennye metody gasheniya kolebanii dvizhuschegosya bumazhnogo polotna”, Sb. mat. Mezhdunar. konf. «Differentsialnye uravneniya, matematicheskoe modelirovanie i vychislitelnye algoritmy» (Belgorod, 25–29 oktyabrya 2021 g.), Belgorod, 2021, 194–196

[23] Naimark M. A., Lineinye differentsialnye operatory, Nauka, M., 1969

[24] Natanson I. P., Teoriya funktsii veschestvennoi peremennoi, Nauka, M., 1974 | MR

[25] Rykhlov V. S., “Razreshimost smeshannoi zadachi dlya giperbolicheskogo uravneniya pri otsutstvii polnoty sobstvennykh funktsii”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obz., 200 (2021), 95–104 | DOI

[26] Rykhlov V. S., “Razreshimost smeshannoi zadachi dlya giperbolicheskogo uravneniya s raspadayuschimisya kraevymi usloviyami pri otsutstvii polnoty sobstvennykh funktsii”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obz., 204 (2022), 124–134 | DOI

[27] Rykhlov V. S., “Edinstvennost resheniya nachalno-granichnoi zadachi dlya giperbolicheskogo uravneniya so smeshannoi proizvodnoi i formula dlya resheniya”, Izv. Saratov. un-ta. Nov. ser. Mat. Mekh. Inform., 23:2 (2023), 183–194 | MR | Zbl

[28] Rykhlov V. S., “Reshenie nachalno-granichnoi zadachi dlya uravneniya giperbolicheskogo tipa so smeshannoi proizvodnoi”, Mat. 21 Mezhdunar. Saratovskoi zimnei shkoly «Sovremennye problemy teorii funktsii i ikh prilozheniya» (Saratov, 31 yanvarya — 4 fevralya 2022 g.), Izd-vo Saratoa. un-ta, Saratov, 2022, 252–255

[29] Rykhlov V. S., “O reshenii nachalno-granichnoi zadachi dlya giperbolicheskogo uravneniya so smeshannoi proizvodnoi”, Mat. Mezhdunar. konf. «Sovremennye metody teorii kraevykh zadach». Voronezhskaya vesennyaya matematicheskaya shkola «Pontryaginskie chteniya–XXXIII» (Voronezh, 3-9 maya 2022 g.), VGU, Voronezh, 2022

[30] Rykhlov V. S., “Reshenie nachalno-granichnoi zadachi v polupolose dlya giperbolicheskogo uravneniya so smeshannoi proizvodnoi”, Matematika. Mekhanika. T. 24, Izd-vo Saratov. un-ta, Saratov, 2022, 53–58

[31] Rykhlov V. S., “Obobschennoe reshenie nachalno-granichnoi zadachi dlya volnovogo uravneniya so smeshannoi proizvodnoi i nenulevym potentsialom”, Mat. Mezhdunar. konf. «Sovremennye metody teorii kraevykh zadach». Voronezhskaya vesennyaya matematicheskaya shkola «Pontryaginskie chteniya–XXXIV» (Voronezh, 3-9 maya 2023 g.), VGU, Voronezh, 2023, 343–345

[32] Rykhlov V. S., “O reshenii nachalno-granichnoi zadachi v polupolose dlya giperbolicheskogo uravneniya so smeshannoi proizvodnoi”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obz., 226 (2023), 89–107

[33] Rykhlov V. S., “Obobschennaya nachalno-granichnaya zadacha dlya volnovogo uravneniya so smeshannoi proizvodnoi”, Sovr. mat. Fundam. napr., 69:2 (2023), 342–363 | MR

[34] Rykhlov V. S., “Obobschennoe reshenie nachalno-granichnoi zadachi dlya uravneniya giperbolicheskogo tipa so smeshannoi proizvodnoi”, Tez. dokl. XVII Mezhdunar. nauch. konf. «Poryadkovyi analiz i smezhnye voprosy matematicheskogo modelirovaniya. Teoriya operatorov i differentsialnye uravneniya» (Vladikavkaz, 14–20 iyulya 2013 g.), YuMI VNTs RAN, Vladikavkaz, 2023, 207–208

[35] Rykhlov V. S., “Reshenie nachalno-granichnoi zadachi dlya volnovogo uravneniya so smeshannoi proizvodnoi i potentsalom obschego vida”, Sb. trudov XVI Mezhdunar. Kazanskoi shkoly-konf. «Teoriya funktsii, ee prilozheniya i smezhnye voprosy» (Kazan, 22–27 avgusta 2023 g.), Kazan, 2023, 205–207

[36] Tolstov G. P., “O vtoroi smeshannoi proizvodnoi”, Mat. sb., 24 (66):1 (1949), 27–51

[37] Khardi G., Raskhodyaschiesya ryady., Izd-vo inostrannoi literatury, M., 1951

[38] Khromov A. P., “Raskhodyaschiesya ryady i obobschennaya smeshannaya zadacha dlya volnovogo uravneniya”, Mat. 21 Mezhdunar. Saratovskoi zimnei shkoly «Sovremennye problemy teorii funktsii i ikh prilozheniya» (Saratov, 31 yanvarya — 4 fevralya 2022 g.), SGU, Saratov, 2022, 319–324

[39] Khromov A. P., “Raskhodyaschiesya ryady i obobschennaya smeshannaya zadacha dlya volnovogo uravneniya prosteishego vida”, Izv. Saratov. un-ta. Nov. ser. Mat. Mekh. Inform., 22:3 (2022), 322–331 | MR | Zbl

[40] Khromov A. P., “Povedenie formalnogo resheniya smeshannoi zadachi dlya volnovogo uravneniya”, Zh. vychisl. mat. mat. fiz., 56:2 (2016), 239–251 | DOI | MR | Zbl

[41] Khromov A. P., “Raskhodyaschiesya ryady i smeshannaya zadacha dlya volnovogo uravneniya”, V sb.: Matematika. Mekhanika., 21 (2019), 62–67

[42] Khromov A. P., “O klassicheskom reshenii smeshannoi zadachi dlya odnorodnogo volnovogo uravneniya s zakreplennymi kontsami i nulevoi nachalnoi skorostyu”, Izv. Saratov. un-ta. Nov. ser. Mat. Mekh. Inform., 19:3 (2019), 280–288 | MR | Zbl

[43] Khromov A. P., “Raskhodyaschiesya ryady i funktsionalnye uravneniya, svyazannye s analogami geometricheskoi progressii”, Mat. Mezhdunar. konf. «Sovremennye metody teorii kraevykh zadach». Voronezhskaya vesennyaya matematicheskaya shkola «Pontryaginskie chteniya–XXX» (Voronezh, 3-9 maya 2019 g.), VGU, Voronezh, 2019, 291–300

[44] Khromov A. P., “Raskhodyaschiesya ryady i metod Fure dlya volnovogo uravneniya”, Mat. 20 Mezhdunar. Saratovskoi zimnei shkoly «Sovremennye problemy teorii funktsii i ikh prilozheniya» (Saratov, 28 yanvarya — 1 fevralya 2020 g.), Nauchnaya kniga, Saratov, 2020, 433–439

[45] Khromov A. P., “Raskhodyaschiesya ryady i obobschennaya smeshannaya zadacha”, Matematika. Mekhanika. T. 23, Izd-vo Saratov. un-ta., Saratov, 2021, 63–67

[46] Khromov A. P., Kornev V. V., “Klassicheskoe i obobschennoe resheniya smeshannoi zadachi dlya neodnorodnogo volnovogo uravneniya”, Zh. vychisl. mat. mat. fiz., 59:2 (2019), 286–300 | DOI | Zbl

[47] Khromov A. P., Kornev V. V., “Raskhodyaschiesya ryady v metode Fure dlya volnovogo uravneniya”, Tr. In-ta mat. mekh. UrO RAN., 27:4 (2021), 215–238 | MR

[48] Khromov A. P., Kornev V. V., “Raskhodyaschiesya ryady i obobschennaya smeshannaya zadacha, ne dopuskayuschaya razdeleniya peremennykh”, Tr. Mat. tsentra im. N. I. Lobachevskogo., 60 (2021), 325–328

[49] Eiler L., Differentsialnoe ischislenie, GITTL, M., L., 1949

[50] Archibald F. R., Emslie A. G., “The vibration of a string having a uniform motion along its length”, J. Appl. Mech., 25:1 (1958), 347–348 | DOI | Zbl

[51] Mahalingam S., “Transverse vibrations of power transmission chains”, British J. Appl. Phys., 8:4 (1957), 145–148 | DOI

[52] Sack R. A., “Transverse oscillations in traveling strings”, British J. Appl. Phys., 5:6 (1954), 224–226 | DOI