Correction method of test solutions of the general wave equation in the first quarter of the plane for minimal smoothness of its right-hand side

F. E. Lomovtsev

Journal of the Belarusian State University. Mathematics and Informatics, Tome 3 (2017), pp. 38-52.

Résumé

A method is proposed for correcting of test classical solutions of the general inhomogeneous factorized oscillation equation for a semibounded string in order that they have minimal (necessary) smoothness requirements on its righthand side. The idea of the method is to calculate the correction to some of its trial (test) classical solutions, which may require an overestimated smoothness from the right-hand side of the equation. To this end, the correcting Goursat problem for the canonical form of this oscillation equation of a string is formulated and solved. Then, in the resulting solution, the smoothness of the test solution is analyzed and, if necessary, it is corrected by the corresponding solution of the homogeneous oscillation equation of the string. We find new classical solutions and the previously unknown necessary smoothness of the right-hand side.

Détail

Keywords: correction method of solutions; necessary smoothness; correcting Goursat problem; test solution; correction of the solution; corrective solution; corrected solution.

Bibliographie

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

[2] F. E. Lomovtsev, “Metod vspomogatelnykh smeshannykh zadach dlya poluogranichennoi struny”, Shestye Bogdanovskie chteniya po obyknovennym differentsialnym uravneniyam: materialy Mezhdunar. mat. konf. (Minsk), 2 (2015), 74–75

[3] F. E. Lomovtsev, E. N. Novikov, “Neobkhodimye i dostatochnye usloviya kolebanii ogranichennoi struny pri kosykh proizvodnykh v granichnykh usloviyakh”, Diff. uravneniya, 50, # 1 (2014), 126–129 | Zbl

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

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

[6] F. E. Lomovtsev, E. N. Novikov, “Klassicheskie resheniya neodnorodnogo faktorizovannogo giperbolicheskogo uravneniya vtorogo poryadka v chetverti ploskosti pri polunestatsionarnoi vtoroi kosoi proizvodnoi v granichnom uslovii”, Vesn. Vitseb. dzyarzh. un-ta imya PM Masherava, 4(88) (2015), 5–11

[7] F. E. Lomovtsev, “O razryvakh pervykh i vtorykh chastnykh proizvodnykh reshenii obschego odnomernogo faktorizovannogo volnovogo uravneniya v chetverti ploskosti”, Vestn. Polots. gos. un-ta, 12 (2016), 117–124

[8] V. S. Vladimirov, “Uravneniya matematicheskoi fiziki”, Nauka, 2003 | MR

[9] N. I. Brish, N. I. Yurchuk, “Zadacha Gursa dlya abstraktnykh lineinykh differentsialnykh uravnenii vtorogo poryadka”, Diff. uravneniya, 7, # 6 (1971), 1020 | MR

[10] S. L. Sobolev, “Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike”, 1988 | MR

[11] N. I. Yurchuk, E. N. Novikov, “Neobkhodimye usloviya dlya suschestvovaniya klassicheskikh reshenii kolebanii poluogranichennoi struny”, Vestsi NAN Belarusi. Ser. fiz.-mat. navuk, 4 (2016), 116–120 | MR

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

[13] F. E. Lomovtsev, Yu. F. Novik, “Nachalno-kraevaya zadacha dlya neodnorodnogo uravneniya kolebanii ogranichennoi struny obschego vida s pervymi nekharakteristicheskimi kosymi proizvodnymi v nestatsionarnykh granichnykh usloviyakh”, Vestn. BGU. Fizika. Matematika. Informatika, 2016, 129–135

[14] F. E. Lomovtsev, E. N. Novikov, “Reshenie smeshannoi zadachi dlya faktorizovannogo uravneniya kolebanii ogranichennoi struny pri polunestatsionarnykh faktorizovannykh vtorykh kosykh proizvodnykh v granichnykh usloviyakh”, Vesn. Vitseb. dzyarzh. un-ta imya PM Masherava, 2015, 15–21

[15] V. I. Korzyuk, “Uravneniya matematicheskoi fiziki”, Minsk, 2011

[16] A. I. Kozhanov, “Zadachi s usloviyami integralnogo vida dlya nekotorykh klassov nestatsionarnykh uravnenii”, Dokl. Akad. nauk, 457, # 2 (2014), 152–156 | Zbl

[17] V. I. Korzyuk, I. S. Kozlovskaya, S. N. Naumovets, “Klassicheskoe reshenie pervoi smeshannoi zadachi odnomernogo volnovogo uravneniya s usloviyami tipa Koshi”, Vestsi NAN Belarusi, fiz (Ser), -mat

[18] V. V. Kornev, A. P. Khromov, “Rezolventnyi podkhod k metodu Fure v smeshannoi zadache dlya neodnorodnogo volnovogo uravneniya”, Izv. Sarat. un-ta. Ser.: Matematika. Mekhanika. Informatika, 16 (2016), 403–413

[19] V. I. Korzyuk, A. A. Mandrik, “Granichnye zadachi dlya nestrogo giperbolicheskogo uravneniya tretego poryadka”, Diff. uravneniya, 52, # 2 (2016), 209–219 | Zbl

[20] V. I. Korzyuk, A. A. Mandrik, “Pervaya smeshannaya zadacha dlya nestrogo giperbolicheskogo uravneniya tretego poryadka v ogranichennoi oblasti”, Diff. uravneniya, 52, # 6 (2016), 788–802 | Zbl

[21] V. S. Gavrilov, “Suschestvovanie i edinstvennost reshenii giperbolicheskikh uravnenii divergentnogo vida s raznymi kraevymi usloviyami na raznykh chastyakh granitsy”, Diff. uravneniya, 52, # 8 (2016), 1050–1061 | Zbl

[22] V. I. Korzyuk, I. I. Stolyarchuk, “Klassicheskoe reshenie pervoi smeshannoi zadachi dlya giperbolicheskogo uravneniya vtorogo poryadka v krivolineinoi polupolose s peremennymi koeffitsientami”, Diff. uravneniya, 53, # 1 (2017), 77–88 | MR | Zbl