Обобщенная смешанная задача для волнового уравнения простейшего вида и ее приложения

Обложка
  • Авторы: Хромов А.П.1
  • Учреждения:
    1. Саратовский национальный исследовательский государственный университет имени Н. Г. Чернышевского
  • Выпуск: Том 229 (2023)
  • Страницы: 83-89
  • Раздел: Статьи
  • URL: https://journal-vniispk.ru/2782-4438/article/view/261893
  • ID: 261893

Цитировать

Полный текст

Аннотация

Приведены результаты по обобщенной смешанной задаче (однородной и неоднородной) для волнового уравнения, основанные на операции интегрирования расходящегося ряда формального решения по методу разделения переменных. Найдено решение обобщенной смешанной задачи для неоднородного уравнения в предположении, что функция, характеризующая неоднородность, локально суммируема. В качестве приложения рассмотрена смешанная задача с ненулевым потенциалом.

Полный текст

Введение

Обобщенная смешанная задача для волнового уравнения является одним из наиболее сильных обобщений смешанной задачи. Она впервые появилась в [6]. Внешний вид ее такой же, как и у исходной смешанной задачи и характеризуется тем, что в формальном решении ее по методу Фурье потенциал и начальные данные считаются произвольными суммируемыми функциями, а возмущение в случае неоднородной задачи - произвольной локально суммируемой функцией. Ряд формального решения может быть и расходящимся. Расходящийся ряд рассматривается в понимании Л. Эйлера (см. [7, с. 100-101]), основоположника теории суммирования расходящихся рядов. Найти решение обобщенной смешанной задачи - значит найти сумму ряда формального решения.

В настоящей статье основное внимание уделяется следующей обобщенной смешанной задаче простейшего вида:

                                  2 u(x,t) t 2 = 2 u(x,t) x 2 ,(x,t)[0,1]×[0,), MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaadaWcaaqaaiabgkGi2oaaCaaaleqaba GaaGOmaaaakiaadwhacaaIOaGaamiEaiaaiYcacaWG0bGaaGykaaqa aiabgkGi2kaadshadaahaaWcbeqaaiaaikdaaaaaaOGaaGypamaala aabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaamyDaiaaiIcacaWG 4bGaaGilaiaadshacaaIPaaabaGaeyOaIyRaamiEamaaCaaaleqaba GaaGOmaaaaaaGccaaISaGaaGzbVlaaiIcacaWG4bGaaGilaiaadsha caaIPaGaeyicI4SaaG4waiaaicdacaaISaGaaGymaiaai2facqGHxd aTcaaIBbGaaGimaiaaiYcacqGHEisPcaaIPaGaaGilaaaa@5B2E@                                         (1)

                                                     u(0,t)=u(1,t)=0, MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaaicdacaaISaGaam iDaiaaiMcacaaI9aGaamyDaiaaiIcacaaIXaGaaGilaiaadshacaaI PaGaaGypaiaaicdacaaISaaaaa@3E50@                                                            (2)

                                              u(x,0)=φ(x), u t (x,0)=0 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaadIhacaaISaGaaG imaiaaiMcacaaI9aGaeqOXdOMaaGikaiaadIhacaaIPaGaaGilaiaa ywW7caWG1bWaaSbaaSqaaiqadshagaqbaaqabaGccaaIOaGaamiEai aaiYcacaaIWaGaaGykaiaai2dacaaIWaaaaa@453F@                                                     (3)

в случае φ(x)L[0,1] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqaHgpGAcaaIOaGaamiEaiaaiMcacq GHiiIZcaWGmbGaaG4waiaaicdacaaISaGaaGymaiaai2faaaa@3C2C@ . Ее удается решить, привлекая аксиомы о расходящихся рядах из [3, с. 19], используя следующее правило интегрирования расходящегося ряда:

                                                            = , MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaadaWdbaqabSqabeqaniabgUIiYdGcda aeabqabSqabeqaniabggHiLdGccaaI9aWaaabqaeqaleqabeqdcqGH ris5aOWaa8qaaeqaleqabeqdcqGHRiI8aOGaaGilaaaa@3B88@                                                                  (4)

где MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaadaWdbaqabSqabeqaniabgUIiYdaaaa@33BD@   MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefmuySLMyYL gaiuaajugGbabaaaaaaaaapeGaa8hfGaaa@3A95@  определенный интеграл, и опираясь на известные результаты, относящиеся к почленному интегрированию тригонометрического ряда Фурье по синусам.

Затем показано, как полученный результат помогает дать решение и обобщенной смешанной задачи для неоднородного уравнения. Наконец, в качестве приложения к вышеприведенным результатам рассмотрена смешанная задача для волнового уравнения с ненулевым потенциалом. Показано, что эта задача приводится к интегральному уравнению, решение которого получается по методу последовательных подстановок.

Кратко содержание статьи представлено в [5].

1 Простейшая однородная обобщенная смешанная задача

Рассмотрим обобщенную смешанную задачу (1) MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefmuySLMyYL gaiuaajugGbabaaaaaaaaapeGaa83eGaaa@3A94@ (3) в случае φ(x)L[0,1] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqaHgpGAcaaIOaGaamiEaiaaiMcacq GHiiIZcaWGmbGaaG4waiaaicdacaaISaGaaGymaiaai2faaaa@3C2C@ . Формальное решение ее по методу Фурье имеет вид

                                  u(x,t)=2 n=1 (φ(ξ),sinnπξ)sinnπxcosnπt, MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaadIhacaaISaGaam iDaiaaiMcacaaI9aGaaGOmamaaqahabeWcbaGaamOBaiaai2dacaaI XaaabaGaeyOhIukaniabggHiLdGccaaIOaGaeqOXdOMaaGikaiabe6 7a4jaaiMcacaaISaGaci4CaiaacMgacaGGUbGaamOBaiabec8aWjab e67a4jaaiMcaciGGZbGaaiyAaiaac6gacaWGUbGaeqiWdaNaamiEai GacogacaGGVbGaai4Caiaad6gacqaHapaCcaWG0bGaaGilaaaa@5A84@                                        (5)

где

                                                   (f,g)= 0 1 f(x)g(x)dx. MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaaIOaGaamOzaiaaiYcacaWGNbGaaG ykaiaai2dadaWdXbqabSqaaiaaicdaaeaacaaIXaaaniabgUIiYdGc caWGMbGaaGikaiaadIhacaaIPaGaam4zaiaaiIcacaWG4bGaaGykai aayIW7caWGKbGaamiEaiaai6caaaa@452D@

Имеем

                    u(x,t)= Σ + + Σ ,где Σ ± = n=1 (φ(ξ),sinnπξ)sinnπ(x±t). MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaadIhacaaISaGaam iDaiaaiMcacaaI9aGaeu4Odm1aaSbaaSqaaiabgUcaRaqabaGccqGH RaWkcqqHJoWudaWgaaWcbaGaeyOeI0cabeaakiaaiYcacaaMf8Uaae 4meiaabsdbcaqG1qGaaGzbVlabfo6atnaaBaaaleaacqGHXcqSaeqa aOGaaGypamaaqahabeWcbaGaamOBaiaai2dacaaIXaaabaGaeyOhIu kaniabggHiLdGccaaIOaGaeqOXdOMaaGikaiabe67a4jaaiMcacaaI SaGaci4CaiaacMgacaGGUbGaamOBaiabec8aWjabe67a4jaaiMcaci GGZbGaaiyAaiaac6gacaWGUbGaeqiWdaNaaGikaiaadIhacqGHXcqS caWG0bGaaGykaiaai6caaaa@6831@                           (6)

Отсюда следует, что для вычисления суммы ряда (6) требуется найти сумму тригонометрического ряда Фурье функции φ(x) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqaHgpGAcaaIOaGaamiEaiaaiMcaaa a@35E0@ , т.е. ряда

                                               2 n=1 (φ(ξ),sinnπξ)sinnπx. MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaaIYaWaaabCaeqaleaacaWGUbGaaG ypaiaaigdaaeaacqGHEisPa0GaeyyeIuoakiaaiIcacqaHgpGAcaaI OaGaeqOVdGNaaGykaiaaiYcaciGGZbGaaiyAaiaac6gacaWGUbGaeq iWdaNaeqOVdGNaaGykaiGacohacaGGPbGaaiOBaiaad6gacqaHapaC caWG4bGaaGOlaaaa@4E38@                                                     (7)

Пусть сумма ряда (7) при x[0,1] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG4bGaeyicI4SaaG4waiaaicdaca aISaGaaGymaiaai2faaaa@3839@  есть какая-либо функция g(x)L[0,1] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGNbGaaGikaiaadIhacaaIPaGaey icI4SaamitaiaaiUfacaaIWaGaaGilaiaaigdacaaIDbaaaa@3B5B@  (в запасе имеются только функции из L[0,1] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGmbGaaG4waiaaicdacaaISaGaaG ymaiaai2faaaa@3689@  ). Тогда в соответствии с правилом (4) имеем

                                  0 x g(η)dη=2 n=1 (φ(ξ),sinnπξ) 0 x sinnπηdη. MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaadaWdXbqabSqaaiaaicdaaeaacaWG4b aaniabgUIiYdGccaWGNbGaaGikaiabeE7aOjaaiMcacaaMi8Uaamiz aiabeE7aOjaai2dacaaIYaWaaabCaeqaleaacaWGUbGaaGypaiaaig daaeaacqGHEisPa0GaeyyeIuoakiaaiIcacqaHgpGAcaaIOaGaeqOV dGNaaGykaiaaiYcaciGGZbGaaiyAaiaac6gacaWGUbGaeqiWdaNaeq OVdGNaaGykamaapehabeWcbaGaaGimaaqaaiaadIhaa0Gaey4kIipa kiGacohacaGGPbGaaiOBaiaad6gacqaHapaCcqaH3oaAcaaMi8Uaam izaiabeE7aOjaai6caaaa@644D@                                         (8)

По теореме 3 из [2, с. 320] ряд в (8) сходится при любом x[0,1] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG4bGaeyicI4SaaG4waiaaicdaca aISaGaaGymaiaai2faaaa@3839@ , а его сумма равна

                                  2 n=1 (φ(ξ),sinnπξ) 0 x sinnπηdη= 0 x φ(η)dη. MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaaIYaWaaabCaeqaleaacaWGUbGaaG ypaiaaigdaaeaacqGHEisPa0GaeyyeIuoakiaaiIcacqaHgpGAcaaI OaGaeqOVdGNaaGykaiaaiYcaciGGZbGaaiyAaiaac6gacaWGUbGaeq iWdaNaeqOVdGNaaGykamaapehabeWcbaGaaGimaaqaaiaadIhaa0Ga ey4kIipakiGacohacaGGPbGaaiOBaiaad6gacqaHapaCcqaH3oaAca aMi8UaamizaiabeE7aOjaai2dadaWdXbqabSqaaiaaicdaaeaacaWG 4baaniabgUIiYdGccqaHgpGAcaaIOaGaeq4TdGMaaGykaiaayIW7ca WGKbGaeq4TdGMaaGOlaaaa@651E@

Таким образом, получили, что

                                                    0 x g(η)dη= 0 x φ(η)dη. MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaadaWdXbqabSqaaiaaicdaaeaacaWG4b aaniabgUIiYdGccaWGNbGaaGikaiabeE7aOjaaiMcacaaMi8Uaamiz aiabeE7aOjaai2dadaWdXbqabSqaaiaaicdaaeaacaWG4baaniabgU IiYdGccqaHgpGAcaaIOaGaeq4TdGMaaGykaiaayIW7caWGKbGaeq4T dGMaaGOlaaaa@4CAD@

Отсюда g(x)=φ(x) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGNbGaaGikaiaadIhacaaIPaGaaG ypaiabeA8aQjaaiIcacaWG4bGaaGykaaaa@39F5@  почти всюду, т.е. найдена сумма g(x) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGNbGaaGikaiaadIhacaaIPaaaaa@350F@  расходящегося ряда (7). Далее, sinnπx MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaaciGGZbGaaiyAaiaac6gacaWGUbGaeq iWdaNaamiEaaaa@3846@  нечетна и 2-периодична. Тогда получаем, что сумма ряда (7) при x(,) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG4bGaeyicI4SaaGikaiabgkHiTi abg6HiLkaaiYcacqGHEisPcaaIPaaaaa@3A2C@  равна φ ˜ (x) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacuaHgpGAgaacaiaaiIcacaWG4bGaaG ykaaaa@35EF@ , где φ ˜ (x) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacuaHgpGAgaacaiaaiIcacaWG4bGaaG ykaaaa@35EF@   MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefmuySLMyYL gaiuaajugGbabaaaaaaaaapeGaa8hfGaaa@3A95@ нечетное, 2-периодическое продолжение φ(x) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqaHgpGAcaaIOaGaamiEaiaaiMcaaa a@35E0@  с отрезка [0,1] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaaIBbGaaGimaiaaiYcacaaIXaGaaG yxaaaa@35B8@  на всю ось. В силу (6) получаем, что сумма u(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaadIhacaaISaGaam iDaiaaiMcaaaa@36CC@  ряда (5) есть

                                            u(x,t)= 1 2 [ φ ˜ (x+t)+ φ ˜ (xt)]. MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaadIhacaaISaGaam iDaiaaiMcacaaI9aWaaSaaaeaacaaIXaaabaGaaGOmaaaacaaIBbGa fqOXdOMbaGaacaaIOaGaamiEaiabgUcaRiaadshacaaIPaGaey4kaS IafqOXdOMbaGaacaaIOaGaamiEaiabgkHiTiaadshacaaIPaGaaGyx aiaai6caaaa@489D@                                                   (9)

Таким образом, получено следующее утверждение.

Теорема 1 Решением обобщенной смешанной задачи (1) MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefmuySLMyYL gaiuGajugGbabaaaaaaaaapeGaa83eGaaa@3A96@ (3) является функция u(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaadIhacaaISaGaam iDaiaaiMcaaaa@36CC@  класса Q MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGrbaaaa@3297@ , определенная по формуле (9).

Функция u(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaadIhacaaISaGaam iDaiaaiMcaaaa@36CC@  класса Q MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGrbaaaa@3297@  означает, что u(x,t)L[ Q T ] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaadIhacaaISaGaam iDaiaaiMcacqGHiiIZcaWGmbGaaG4waiaadgfadaWgaaWcbaGaamiv aaqabaGccaaIDbaaaa@3CD2@  при любом T>0 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGubGaaGOpaiaaicdaaaa@341C@ , где Q T MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGrbWaaSbaaSqaaiaadsfaaeqaaa aa@339C@   MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefmuySLMyYL gaiuaajugGbabaaaaaaaaapeGaa8hfGaaa@3A95@ множество [0,1]×[0,T] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaaIBbGaaGimaiaaiYcacaaIXaGaaG yxaiabgEna0kaaiUfacaaIWaGaaGilaiaadsfacaaIDbaaaa@3BE4@ .

2 Приложение. Простейшая неоднородная смешанная задача

Рассмотрим следующую простейшую неоднородную смешанную задачу:

                           2 u(x,t) t 2 = 2 u(x,t) x 2 +f(x,t),(x,t)[0,1]×[0,), MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaadaWcaaqaaiabgkGi2oaaCaaaleqaba GaaGOmaaaakiaadwhacaaIOaGaamiEaiaaiYcacaWG0bGaaGykaaqa aiabgkGi2kaadshadaahaaWcbeqaaiaaikdaaaaaaOGaaGypamaala aabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaamyDaiaaiIcacaWG 4bGaaGilaiaadshacaaIPaaabaGaeyOaIyRaamiEamaaCaaaleqaba GaaGOmaaaaaaGccqGHRaWkcaWGMbGaaGikaiaadIhacaaISaGaamiD aiaaiMcacaaISaGaaGzbVlaaiIcacaWG4bGaaGilaiaadshacaaIPa GaeyicI4SaaG4waiaaicdacaaISaGaaGymaiaai2facqGHxdaTcaaI BbGaaGimaiaaiYcacqGHEisPcaaIPaGaaGilaaaa@610C@                                (10)

                                                     u(0,t)=u(1,t)=0, MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaaicdacaaISaGaam iDaiaaiMcacaaI9aGaamyDaiaaiIcacaaIXaGaaGilaiaadshacaaI PaGaaGypaiaaicdacaaISaaaaa@3E50@                                                          (11)

                                                   u(x,0)= u t (x,0)=0, MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaadIhacaaISaGaaG imaiaaiMcacaaI9aGaamyDamaaBaaaleaaceWG0bGbauaaaeqaaOGa aGikaiaadIhacaaISaGaaGimaiaaiMcacaaI9aGaaGimaiaaiYcaaa a@3F92@                                                        (12)

где f(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGMbGaaGikaiaadIhacaaISaGaam iDaiaaiMcaaaa@36BD@  есть функция класса Q MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGrbaaaa@3297@ . Формальное решение ее по методу Фурье есть

                      u(x,t)=2 n=1 0 t (f(ξ,τ),sinnπξ) 1 nπ sinnπxsinnπ(tτ)dτ. MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaadIhacaaISaGaam iDaiaaiMcacaaI9aGaaGOmamaaqahabeWcbaGaamOBaiaai2dacaaI XaaabaGaeyOhIukaniabggHiLdGcdaWdXbqabSqaaiaaicdaaeaaca WG0baaniabgUIiYdGccaaIOaGaamOzaiaaiIcacqaH+oaEcaaISaGa eqiXdqNaaGykaiaaiYcaciGGZbGaaiyAaiaac6gacaWGUbGaeqiWda NaeqOVdGNaaGykamaalaaabaGaaGymaaqaaiaad6gacqaHapaCaaGa ci4CaiaacMgacaGGUbGaamOBaiabec8aWjaadIhaciGGZbGaaiyAai aac6gacaWGUbGaeqiWdaNaaGikaiaadshacqGHsislcqaHepaDcaaI PaGaaGjcVlaadsgacqaHepaDcaaIUaaaaa@6C2C@                           (13)

Так как

                                   2 nπ sinnπxsinnπ(tτ)= xt+τ x+tτ sinnπηdη, MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaadaWcaaqaaiaaikdaaeaacaWGUbGaeq iWdahaaiGacohacaGGPbGaaiOBaiaad6gacqaHapaCcaWG4bGaci4C aiaacMgacaGGUbGaamOBaiabec8aWjaaiIcacaWG0bGaeyOeI0Iaeq iXdqNaaGykaiaai2dadaWdXbqabSqaaiaadIhacqGHsislcaWG0bGa ey4kaSIaeqiXdqhabaGaamiEaiabgUcaRiaadshacqGHsislcqaHep aDa0Gaey4kIipakiGacohacaGGPbGaaiOBaiaad6gacqaHapaCcqaH 3oaAcaaMi8UaamizaiabeE7aOjaaiYcaaaa@60B9@

то (13) переходит в

                              u(x,t)= n=1 0 t (f(ξ,τ),sinnπξ)dτ xt+τ x+tτ sinnπηdη. MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaadIhacaaISaGaam iDaiaaiMcacaaI9aWaaabCaeqaleaacaWGUbGaaGypaiaaigdaaeaa cqGHEisPa0GaeyyeIuoakmaapehabeWcbaGaaGimaaqaaiaadshaa0 Gaey4kIipakiaaiIcacaWGMbGaaGikaiabe67a4jaaiYcacqaHepaD caaIPaGaaGilaiGacohacaGGPbGaaiOBaiaad6gacqaHapaCcqaH+o aEcaaIPaGaaGjcVlaadsgacqaHepaDdaWdXbqabSqaaiaadIhacqGH sislcaWG0bGaey4kaSIaeqiXdqhabaGaamiEaiabgUcaRiaadshacq GHsislcqaHepaDa0Gaey4kIipakiGacohacaGGPbGaaiOBaiaad6ga cqaHapaCcqaH3oaAcaaMi8UaamizaiabeE7aOjaai6caaaa@6FBA@                                   (14)

Из (14) в силу правила (4) получим

             u(x,t)= 0 t dτ xt+τ x+tτ n=1 (f(ξ,τ),sinnπξ)sinnπηdη= 1 2 0 t dτ xt+τ x+tτ f ˜ (η,τ)dη, MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaadIhacaaISaGaam iDaiaaiMcacaaI9aWaa8qCaeqaleaacaaIWaaabaGaamiDaaqdcqGH RiI8aOGaaGjcVlaadsgacqaHepaDdaWdXbqabSqaaiaadIhacqGHsi slcaWG0bGaey4kaSIaeqiXdqhabaGaamiEaiabgUcaRiaadshacqGH sislcqaHepaDa0Gaey4kIipakmaaqahabeWcbaGaamOBaiaai2daca aIXaaabaGaeyOhIukaniabggHiLdGccaaIOaGaamOzaiaaiIcacqaH +oaEcaaISaGaeqiXdqNaaGykaiaaiYcaciGGZbGaaiyAaiaac6gaca WGUbGaeqiWdaNaeqOVdGNaaGykaiGacohacaGGPbGaaiOBaiaad6ga cqaHapaCcqaH3oaAcaaMi8UaamizaiabeE7aOjaai2dadaWcaaqaai aaigdaaeaacaaIYaaaamaapehabeWcbaGaaGimaaqaaiaadshaa0Ga ey4kIipakiaayIW7caWGKbGaeqiXdq3aa8qCaeqaleaacaWG4bGaey OeI0IaamiDaiabgUcaRiabes8a0bqaaiaadIhacqGHRaWkcaWG0bGa eyOeI0IaeqiXdqhaniabgUIiYdGcceWGMbGbaGaacaaIOaGaeq4TdG MaaGilaiabes8a0jaaiMcacaaMi8UaamizaiabeE7aOjaaiYcaaaa@92A0@                 (15)

поскольку ряд в (15), как это следует из п. 1, имеет сумму 1 2 f ˜ (η,τ) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaadaWcaaqaaiaaigdaaeaacaaIYaaaai qadAgagaacaiaaiIcacqaH3oaAcaaISaGaeqiXdqNaaGykaaaa@39CE@ , где f ˜ (η,τ) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaaceWGMbGbaGaacaaIOaGaeq4TdGMaaG ilaiabes8a0jaaiMcaaaa@3847@   MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefmuySLMyYL gaiuaajugGbabaaaaaaaaapeGaa8hfGaaa@3A95@ нечетное, 2-периодическое продолжение по η MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqaH3oaAaaa@336D@  на всю ось функции f(η,τ) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGMbGaaGikaiabeE7aOjaaiYcacq aHepaDcaaIPaaaaa@3838@  с отрезка [0,1] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaaIBbGaaGimaiaaiYcacaaIXaGaaG yxaaaa@35B8@ . Таким образом, справедливо следующее утверждение.

Теорема 2 Решение u(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaadIhacaaISaGaam iDaiaaiMcaaaa@36CC@  обобщенной смешанной задачи (10) MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefmuySLMyYL gaiuGajugGbabaaaaaaaaapeGaa83eGaaa@3A96@ (12) есть функция класса Q MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGrbaaaa@3297@ , определяемая по формуле

                                             u(x,t)= 1 2 0 t dτ xt+τ x+tτ f ˜ (η,τ)dη. MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaadIhacaaISaGaam iDaiaaiMcacaaI9aWaaSaaaeaacaaIXaaabaGaaGOmaaaadaWdXbqa bSqaaiaaicdaaeaacaWG0baaniabgUIiYdGccaaMi8Uaamizaiabes 8a0naapehabeWcbaGaamiEaiabgkHiTiaadshacqGHRaWkcqaHepaD aeaacaWG4bGaey4kaSIaamiDaiabgkHiTiabes8a0bqdcqGHRiI8aO GabmOzayaaiaGaaGikaiabeE7aOjaaiYcacqaHepaDcaaIPaGaaGjc VlaadsgacqaH3oaAcaaIUaaaaa@5A6C@                                                  (16)

Отметим, что без привлечения операции интегрирования расходящегося ряда формула (16) приводится в [1].

Тот факт, что u(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaadIhacaaISaGaam iDaiaaiMcaaaa@36CC@  есть функция класса Q MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGrbaaaa@3297@ , дается следующей леммой.

Лемма 1 Имеет место оценка

                                      u(x,t) L[ Q T ] T(T+2)f(x,t) L[ Q T ] . MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaarqqr1ngBPrgifHhDYfgaiqaacqWFLi cucaWG1bGaaGikaiaadIhacaaISaGaamiDaiaaiMcacqWFLicudaWg aaWcbaGaamitaiaaiUfacaWGrbWaaSbaaeaacaWGubaabeaacaaIDb aabeaakiabgsMiJkaadsfacaaIOaGaamivaiabgUcaRiaaikdacaaI PaGae8xjIaLaamOzaiaaiIcacaWG4bGaaGilaiaadshacaaIPaGae8 xjIa1aaSbaaSqaaiaadYeacaaIBbGaamyuamaaBaaabaGaamivaaqa baGaaGyxaaqabaGccaaIUaaaaa@553B@

Proof. Из (16) имеем

                                            |u(x,t)| 1 2 0 T dτ T T+1 | f ˜ (η,τ)|dη. MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaaI8bGaamyDaiaaiIcacaWG4bGaaG ilaiaadshacaaIPaGaaGiFaiabgsMiJoaalaaabaGaaGymaaqaaiaa ikdaaaWaa8qCaeqaleaacaaIWaaabaGaamivaaqdcqGHRiI8aOGaam izaiabes8a0naapehabeWcbaGaeyOeI0IaamivaaqaaiaadsfacqGH RaWkcaaIXaaaniabgUIiYdGccaaI8bGabmOzayaaiaGaaGikaiabeE 7aOjaaiYcacqaHepaDcaaIPaGaaGiFaiaayIW7caWGKbGaeq4TdGMa aGOlaaaa@56E9@

Пусть m MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGTbaaaa@32B3@  - наименьшее натуральное число, для которого Tm MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGubGaeyizImQaamyBaaaa@3541@ . Тогда в силу нечетности f ˜ (η,τ) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaaceWGMbGbaGaacaaIOaGaeq4TdGMaaG ilaiabes8a0jaaiMcaaaa@3847@  по η MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqaH3oaAaaa@336D@  имеем

             T T+1 | f ˜ (η,τ)|dη m m+1 | f ˜ (η,τ)|dη= m 0 | f ˜ (η,τ)|dη+ 0 m+1 | f ˜ (η,τ)|dη= MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaadaWdXbqabSqaaiabgkHiTiaadsfaae aacaWGubGaey4kaSIaaGymaaqdcqGHRiI8aOGaaGiFaiqadAgagaac aiaaiIcacqaH3oaAcaaISaGaeqiXdqNaaGykaiaaiYhacaaMi8Uaam izaiabeE7aOjabgsMiJoaapehabeWcbaGaeyOeI0IaamyBaaqaaiaa d2gacqGHRaWkcaaIXaaaniabgUIiYdGccaaI8bGabmOzayaaiaGaaG ikaiabeE7aOjaaiYcacqaHepaDcaaIPaGaaGiFaiaayIW7caWGKbGa eq4TdGMaaGypamaapehabeWcbaGaeyOeI0IaamyBaaqaaiaaicdaa0 Gaey4kIipakiaaiYhaceWGMbGbaGaacaaIOaGaeq4TdGMaaGilaiab es8a0jaaiMcacaaI8bGaaGjcVlaadsgacqaH3oaAcqGHRaWkdaWdXb qabSqaaiaaicdaaeaacaWGTbGaey4kaSIaaGymaaqdcqGHRiI8aOGa aGiFaiqadAgagaacaiaaiIcacqaH3oaAcaaISaGaeqiXdqNaaGykai aaiYhacaaMi8UaamizaiabeE7aOjaai2daaaa@8122@

             = m 0 | f ˜ (η,τ)|dη+ 0 m+1 | f ˜ (η,τ)|dη= 0 m | f ˜ (η,τ)|dη+ 0 m+1 | f ˜ (η,τ)|dη MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaaI9aWaa8qCaeqaleaacqGHsislca WGTbaabaGaaGimaaqdcqGHRiI8aOGaaGiFaiqadAgagaacaiaaiIca cqGHsislcqaH3oaAcaaISaGaeqiXdqNaaGykaiaaiYhacaaMi8Uaam izaiabeE7aOjabgUcaRmaapehabeWcbaGaaGimaaqaaiaad2gacqGH RaWkcaaIXaaaniabgUIiYdGccaaI8bGabmOzayaaiaGaaGikaiabeE 7aOjaaiYcacqaHepaDcaaIPaGaaGiFaiaayIW7caWGKbGaeq4TdGMa aGypamaapehabeWcbaGaaGimaaqaaiaad2gaa0Gaey4kIipakiaaiY haceWGMbGbaGaacaaIOaGaeq4TdGMaaGilaiabes8a0jaaiMcacaaI 8bGaaGjcVlaadsgacqaH3oaAcqGHRaWkdaWdXbqabSqaaiaaicdaae aacaWGTbGaey4kaSIaaGymaaqdcqGHRiI8aOGaaGiFaiqadAgagaac aiaaiIcacqaH3oaAcaaISaGaeqiXdqNaaGykaiaaiYhacaaMi8Uaam izaiabeE7aOjabgsMiJcaa@7F3C@

                                     2 0 m+1 | f ˜ (η,τ)|dη=2 k=0 m k k+1 | f ˜ (η,τ)|dη. MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqGHKjYOcaaIYaWaa8qCaeqaleaaca aIWaaabaGaamyBaiabgUcaRiaaigdaa0Gaey4kIipakiaaiYhaceWG MbGbaGaacaaIOaGaeq4TdGMaaGilaiabes8a0jaaiMcacaaI8bGaaG jcVlaadsgacqaH3oaAcaaI9aGaaGOmamaaqahabeWcbaGaam4Aaiaa i2dacaaIWaaabaGaamyBaaqdcqGHris5aOWaa8qCaeqaleaacaWGRb aabaGaam4AaiabgUcaRiaaigdaa0Gaey4kIipakiaaiYhaceWGMbGb aGaacaaIOaGaeq4TdGMaaGilaiabes8a0jaaiMcacaaI8bGaaGjcVl aadsgacqaH3oaAcaaIUaaaaa@613B@

Пусть k MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGRbaaaa@32B1@  четно, т.е. k=2v MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGRbGaaGypaiaaikdacaWG2baaaa@352F@ . Рассмотрим следующий интеграл:

             k k+1 | f ˜ (η,τ)|dη= 2v 2v+1 | f ˜ (η,τ)|dη= 0 1 | f ˜ (2v+ξ,τ)|dξ= 0 1 |f(ξ,τ)|dξ. MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaadaWdXbqabSqaaiaadUgaaeaacaWGRb Gaey4kaSIaaGymaaqdcqGHRiI8aOGaaGiFaiqadAgagaacaiaaiIca cqaH3oaAcaaISaGaeqiXdqNaaGykaiaaiYhacaaMi8UaamizaiabeE 7aOjaai2dadaWdXbqabSqaaiaaikdacaWG2baabaGaaGOmaiaadAha cqGHRaWkcaaIXaaaniabgUIiYdGccaaI8bGabmOzayaaiaGaaGikai abeE7aOjaaiYcacqaHepaDcaaIPaGaaGiFaiaayIW7caWGKbGaeq4T dGMaaGypamaapehabeWcbaGaaGimaaqaaiaaigdaa0Gaey4kIipaki aaiYhaceWGMbGbaGaacaaIOaGaaGOmaiaadAhacqGHRaWkcqaH+oaE caaISaGaeqiXdqNaaGykaiaaiYhacaaMi8Uaamizaiabe67a4jaai2 dadaWdXbqabSqaaiaaicdaaeaacaaIXaaaniabgUIiYdGccaaI8bGa amOzaiaaiIcacqaH+oaEcaaISaGaeqiXdqNaaGykaiaaiYhacaaMi8 Uaamizaiabe67a4jaai6caaaa@7FD6@

Если k MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGRbaaaa@32B1@  нечетно, т.е. k=2v+1 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGRbGaaGypaiaaikdacaWG2bGaey 4kaSIaaGymaaaa@36CC@ , то

             k k+1 | f ˜ (η,τ)|dη= 2v+1 2v+2 | f ˜ (η,τ)|dη= 0 1 | f ˜ (2v+1+ξ,τ)|dξ= 0 1 | f ˜ (1+ξ,τ)|dξ= MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaadaWdXbqabSqaaiaadUgaaeaacaWGRb Gaey4kaSIaaGymaaqdcqGHRiI8aOGaaGiFaiqadAgagaacaiaaiIca cqaH3oaAcaaISaGaeqiXdqNaaGykaiaaiYhacaaMi8UaamizaiabeE 7aOjaai2dadaWdXbqabSqaaiaaikdacaWG2bGaey4kaSIaaGymaaqa aiaaikdacaWG2bGaey4kaSIaaGOmaaqdcqGHRiI8aOGaaGiFaiqadA gagaacaiaaiIcacqaH3oaAcaaISaGaeqiXdqNaaGykaiaaiYhacaaM i8UaamizaiabeE7aOjaai2dadaWdXbqabSqaaiaaicdaaeaacaaIXa aaniabgUIiYdGccaaI8bGabmOzayaaiaGaaGikaiaaikdacaWG2bGa ey4kaSIaaGymaiabgUcaRiabe67a4jaaiYcacqaHepaDcaaIPaGaaG iFaiaayIW7caWGKbGaeqOVdGNaaGypamaapehabeWcbaGaaGimaaqa aiaaigdaa0Gaey4kIipakiaaiYhaceWGMbGbaGaacaaIOaGaaGymai abgUcaRiabe67a4jaaiYcacqaHepaDcaaIPaGaaGiFaiaayIW7caWG KbGaeqOVdGNaaGypaaaa@84CC@

                        = 0 1 | f ˜ (1ξ,τ)|dξ= 0 1 |f(1ξ,τ)|dξ= 0 1 |f(ξ,τ)|dξ. MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaaI9aWaa8qCaeqaleaacaaIWaaaba GaaGymaaqdcqGHRiI8aOGaaGiFaiqadAgagaacaiaaiIcacqGHsisl caaIXaGaeyOeI0IaeqOVdGNaaGilaiabes8a0jaaiMcacaaI8bGaaG jcVlaadsgacqaH+oaEcaaI9aWaa8qCaeqaleaacaaIWaaabaGaaGym aaqdcqGHRiI8aOGaaGiFaiaadAgacaaIOaGaaGymaiabgkHiTiabe6 7a4jaaiYcacqaHepaDcaaIPaGaaGiFaiaayIW7caWGKbGaeqOVdGNa aGypamaapehabeWcbaGaaGimaaqaaiaaigdaa0Gaey4kIipakiaaiY hacaWGMbGaaGikaiabe67a4jaaiYcacqaHepaDcaaIPaGaaGiFaiaa yIW7caWGKbGaeqOVdGNaaGOlaaaa@6B5A@

Таким образом, при всех k MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGRbaaaa@32B1@  (четных и нечетных) получаем один и тот же результат:

                                            k k+1 | f ˜ (η,τ)|dη= 0 1 |f(ξ,τ)|dξ. MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaadaWdXbqabSqaaiaadUgaaeaacaWGRb Gaey4kaSIaaGymaaqdcqGHRiI8aOGaaGiFaiqadAgagaacaiaaiIca cqaH3oaAcaaISaGaeqiXdqNaaGykaiaaiYhacaaMi8UaamizaiabeE 7aOjaai2dadaWdXbqabSqaaiaaicdaaeaacaaIXaaaniabgUIiYdGc caaI8bGaamOzaiaaiIcacqaH+oaEcaaISaGaeqiXdqNaaGykaiaaiY hacaaMi8Uaamizaiabe67a4jaai6caaaa@56A9@

Отсюда

                   |u(x,t)| 1 2 0 T dτ2(m+1) 0 1 |f(η,τ)|dη=(m+1)f(x,t) L[ Q T ] . MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaaI8bGaamyDaiaaiIcacaWG4bGaaG ilaiaadshacaaIPaGaaGiFaiabgsMiJoaalaaabaGaaGymaaqaaiaa ikdaaaWaa8qCaeqaleaacaaIWaaabaGaamivaaqdcqGHRiI8aOGaam izaiabes8a0jaaikdacaaIOaGaamyBaiabgUcaRiaaigdacaaIPaWa a8qCaeqaleaacaaIWaaabaGaaGymaaqdcqGHRiI8aOGaaGiFaiaadA gacaaIOaGaeq4TdGMaaGilaiabes8a0jaaiMcacaaI8bGaamizaiab eE7aOjaai2dacaaIOaGaamyBaiabgUcaRiaaigdacaaIPaqeeuuDJX wAKbsr4rNCHbaceaGae8xjIaLaamOzaiaaiIcacaWG4bGaaGilaiaa dshacaaIPaGae8xjIa1aaSbaaSqaaiaadYeacaaIBbGaamyuamaaBa aabaGaamivaaqabaGaaGyxaaqabaGccaaIUaaaaa@6C5F@

Значит,

                Q T |u(x,t)|dxdtT(m+1)f(x,t) L[ Q T ] T(T+2)f(x,t) L[ Q T ] . MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaadaWdrbqabSqaaiaadgfadaWgaaqaai aadsfaaeqaaaqab0Gaey4kIipakiaaiYhacaWG1bGaaGikaiaadIha caaISaGaamiDaiaaiMcacaaI8bGaamizaiaadIhacaaMi8Uaamizai aadshacqGHKjYOcaWGubGaaGikaiaad2gacqGHRaWkcaaIXaGaaGyk aebbfv3ySLgzGueE0jxyaGabaiab=vIiqjaadAgacaaIOaGaamiEai aaiYcacaWG0bGaaGykaiab=vIiqnaaBaaaleaacaWGmbGaaG4waiaa dgfadaWgaaqaaiaadsfaaeqaaiaai2faaeqaaOGaeyizImQaamivai aaiIcacaWGubGaey4kaSIaaGOmaiaaiMcacqWFLicucaWGMbGaaGik aiaadIhacaaISaGaamiDaiaaiMcacqWFLicudaWgaaWcbaGaamitai aaiUfacaWGrbWaaSbaaeaacaWGubaabeaacaaIDbaabeaakiaai6ca aaa@6C43@

3 Приложение. Смешанная задача с ненулевым потенциалом

Сначала рассмотрим следующую обобщенную задачу:

                           2 u(x,t) t 2 = 2 u(x,t) x 2 +f(x,t),(x,t)[0,1]×[0,), MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaadaWcaaqaaiabgkGi2oaaCaaaleqaba GaaGOmaaaakiaadwhacaaIOaGaamiEaiaaiYcacaWG0bGaaGykaaqa aiabgkGi2kaadshadaahaaWcbeqaaiaaikdaaaaaaOGaaGypamaala aabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaamyDaiaaiIcacaWG 4bGaaGilaiaadshacaaIPaaabaGaeyOaIyRaamiEamaaCaaaleqaba GaaGOmaaaaaaGccqGHRaWkcaWGMbGaaGikaiaadIhacaaISaGaamiD aiaaiMcacaaISaGaaGzbVlaaiIcacaWG4bGaaGilaiaadshacaaIPa GaeyicI4SaaG4waiaaicdacaaISaGaaGymaiaai2facqGHxdaTcaaI BbGaaGimaiaaiYcacqGHEisPcaaIPaGaaGilaaaa@610C@                                (17)

                                                     u(0,t)=u(1,t)=0, MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaaicdacaaISaGaam iDaiaaiMcacaaI9aGaamyDaiaaiIcacaaIXaGaaGilaiaadshacaaI PaGaaGypaiaaicdacaaISaaaaa@3E50@                                                          (18)

                                              u(x,0)=φ(x), u t (x,0)=0. MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaadIhacaaISaGaaG imaiaaiMcacaaI9aGaeqOXdOMaaGikaiaadIhacaaIPaGaaGilaiaa ywW7caWG1bWaaSbaaSqaaiqadshagaqbaaqabaGccaaIOaGaamiEai aaiYcacaaIWaGaaGykaiaai2dacaaIWaGaaGOlaaaa@45F7@                                                  (19)

Здесь f(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGMbGaaGikaiaadIhacaaISaGaam iDaiaaiMcaaaa@36BD@   MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefmuySLMyYL gaiuaajugGbabaaaaaaaaapeGaa8hfGaaa@3A95@ функция класса Q MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGrbaaaa@3297@  и φ(x)L[0,1] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqaHgpGAcaaIOaGaamiEaiaaiMcacq GHiiIZcaWGmbGaaG4waiaaicdacaaISaGaaGymaiaai2faaaa@3C2C@ . Формальное решение ее по методу Фурье есть

                                                u(x,t)= u 0 (x,t)+ u 1 (x,t), MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaadIhacaaISaGaam iDaiaaiMcacaaI9aGaamyDamaaBaaaleaacaaIWaaabeaakiaaiIca caWG4bGaaGilaiaadshacaaIPaGaey4kaSIaamyDamaaBaaaleaaca aIXaaabeaakiaaiIcacaWG4bGaaGilaiaadshacaaIPaGaaGilaaaa @4522@

где функция u 0 (x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bWaaSbaaSqaaiaaicdaaeqaaO GaaGikaiaadIhacaaISaGaamiDaiaaiMcaaaa@37BC@  определена формулой (5), а u 1 (x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bWaaSbaaSqaaiaaigdaaeqaaO GaaGikaiaadIhacaaISaGaamiDaiaaiMcaaaa@37BD@   MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefmuySLMyYL gaiuaajugGbabaaaaaaaaapeGaa8hfGaaa@3A95@ ряд (13). Поэтому, исходя из пп. 0.1, 0.2, получаем следующее утверждение.

Теорема 3  Обобщенная смешанная задача (17) MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefmuySLMyYL gaiuGajugGbabaaaaaaaaapeGaa83eGaaa@3A96@ (19) имеет решение класса Q MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGrbaaaa@3297@ , определяемое по формуле

                           u(x,t)= 1 2 [ φ ˜ (x+t)+ φ ˜ (xt)]+ 1 2 0 t dτ xt+τ x+tτ f ˜ (η,τ)dη. MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaadIhacaaISaGaam iDaiaaiMcacaaI9aWaaSaaaeaacaaIXaaabaGaaGOmaaaacaaIBbGa fqOXdOMbaGaacaaIOaGaamiEaiabgUcaRiaadshacaaIPaGaey4kaS IafqOXdOMbaGaacaaIOaGaamiEaiabgkHiTiaadshacaaIPaGaaGyx aiabgUcaRmaalaaabaGaaGymaaqaaiaaikdaaaWaa8qCaeqaleaaca aIWaaabaGaamiDaaqdcqGHRiI8aOGaamizaiabes8a0naapehabeWc baGaamiEaiabgkHiTiaadshacqGHRaWkcqaHepaDaeaacaWG4bGaey 4kaSIaamiDaiabgkHiTiabes8a0bqdcqGHRiI8aOGabmOzayaaiaGa aGikaiabeE7aOjaaiYcacqaHepaDcaaIPaGaaGjcVlaadsgacqaH3o aAcaaIUaaaaa@6A0F@                                (20)

Теперь приступаем к смешанной задаче с ненулевым потенциалом:

                        2 u(x,t) t 2 = 2 u(x,t) x 2 q(x)u(x,t),(x,t)[0,1]×[0,), MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaadaWcaaqaaiabgkGi2oaaCaaaleqaba GaaGOmaaaakiaadwhacaaIOaGaamiEaiaaiYcacaWG0bGaaGykaaqa aiabgkGi2kaadshadaahaaWcbeqaaiaaikdaaaaaaOGaaGypamaala aabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaamyDaiaaiIcacaWG 4bGaaGilaiaadshacaaIPaaabaGaeyOaIyRaamiEamaaCaaaleqaba GaaGOmaaaaaaGccqGHsislcaWGXbGaaGikaiaadIhacaaIPaGaamyD aiaaiIcacaWG4bGaaGilaiaadshacaaIPaGaaGilaiaaywW7caaIOa GaamiEaiaaiYcacaWG0bGaaGykaiabgIGiolaaiUfacaaIWaGaaGil aiaaigdacaaIDbGaey41aqRaaG4waiaaicdacaaISaGaeyOhIuQaaG ykaiaaiYcaaaa@647E@                             (21)

                                                     u(0,t)=u(1,t)=0, MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaaicdacaaISaGaam iDaiaaiMcacaaI9aGaamyDaiaaiIcacaaIXaGaaGilaiaadshacaaI PaGaaGypaiaaicdacaaISaaaaa@3E50@                                                          (22)

                                             u(x,0)=φ(x), u t (x,0)=0, MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaadIhacaaISaGaaG imaiaaiMcacaaI9aGaeqOXdOMaaGikaiaadIhacaaIPaGaaGilaiaa ywW7caWG1bWaaSbaaSqaaiqadshagaqbaaqabaGccaaIOaGaamiEai aaiYcacaaIWaGaaGykaiaai2dacaaIWaGaaGilaaaa@45F5@                                                  (23)

где φ(x)L[0,1] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqaHgpGAcaaIOaGaamiEaiaaiMcacq GHiiIZcaWGmbGaaG4waiaaicdacaaISaGaaGymaiaai2faaaa@3C2C@ , q(x)L[0,1] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGXbGaaGikaiaadIhacaaIPaGaey icI4SaamitaiaaiUfacaaIWaGaaGilaiaaigdacaaIDbaaaa@3B65@ , q(x)u(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGXbGaaGikaiaadIhacaaIPaGaam yDaiaaiIcacaWG4bGaaGilaiaadshacaaIPaaaaa@3A24@   MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefmuySLMyYL gaiuaajugGbabaaaaaaaaapeGaa8hfGaaa@3A95@ функция класса Q MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGrbaaaa@3297@ .

В этой задаче будем рассматривать q(x)u(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqGHsislcaWGXbGaaGikaiaadIhaca aIPaGaamyDaiaaiIcacaWG4bGaaGilaiaadshacaaIPaaaaa@3B11@  как возмущение в задаче (17) MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefmuySLMyYL gaiuaajugGbabaaaaaaaaapeGaa83eGaaa@3A94@ (19). Тогда по теореме 3 перейдем от задачи (21) MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefmuySLMyYL gaiuaajugGbabaaaaaaaaapeGaa83eGaaa@3A94@ (23) к интегральному уравнению:

                        u(x,t)= 1 2 [ φ ˜ (x+t)+ φ ˜ (xt)] 1 2 0 t dτ xt+τ x+tτ q(η)u(η ˜ ,τ)dη, MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaadIhacaaISaGaam iDaiaaiMcacaaI9aWaaSaaaeaacaaIXaaabaGaaGOmaaaacaaIBbGa fqOXdOMbaGaacaaIOaGaamiEaiabgUcaRiaadshacaaIPaGaey4kaS IafqOXdOMbaGaacaaIOaGaamiEaiabgkHiTiaadshacaaIPaGaaGyx aiabgkHiTmaalaaabaGaaGymaaqaaiaaikdaaaWaa8qCaeqaleaaca aIWaaabaGaamiDaaqdcqGHRiI8aOGaamizaiabes8a0naapehabeWc baGaamiEaiabgkHiTiaadshacqGHRaWkcqaHepaDaeaacaWG4bGaey 4kaSIaamiDaiabgkHiTiabes8a0bqdcqGHRiI8aOWaaacaaeaacaWG XbGaaGikaiabeE7aOjaaiMcacaWG1bGaaGikaiabeE7aObGaay5ada GaaGilaiabes8a0jaaiMcacaaMi8UaamizaiabeE7aOjaaiYcaaaa@6EE1@                            (24)

где q(η)u(η ˜ ,τ) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaadaaiaaqaaiaadghacaaIOaGaeq4TdG MaaGykaiaadwhacaaIOaGaeq4TdGgacaGLdmaacaaISaGaeqiXdqNa aGykaaaa@3D10@   MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefmuySLMyYL gaiuaajugGbabaaaaaaaaapeGaa8hfGaaa@3A95@ нечетное, 2-периодическое продолжение q(η)u(η,τ) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGXbGaaGikaiabeE7aOjaaiMcaca WG1bGaaGikaiabeE7aOjaaiYcacqaHepaDcaaIPaaaaa@3C4E@  на всю ось.

Приступаем к решению уравнения (24). Тот факт, что φ ˜ (x) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacuaHgpGAgaacaiaaiIcacaWG4bGaaG ykaaaa@35EF@  есть нечетное, 2-периодическое продолжение φ(x) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqaHgpGAcaaIOaGaamiEaiaaiMcaaa a@35E0@  с [0,1] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaaIBbGaaGimaiaaiYcacaaIXaGaaG yxaaaa@35B8@  на всю ось, трактуется следующим образом: сначала нечетно находится φ ˜ (x) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacuaHgpGAgaacaiaaiIcacaWG4bGaaG ykaaaa@35EF@  при x[1,0] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG4bGaeyicI4SaaG4waiabgkHiTi aaigdacaaISaGaaGimaiaai2faaaa@3926@ , т.е. φ ˜ (x)=φ(x) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacuaHgpGAgaacaiaaiIcacaWG4bGaaG ykaiaai2dacqGHsislcqaHgpGAcaaIOaGaeyOeI0IaamiEaiaaiMca aaa@3CAF@  при x0 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG4bGaeyizImQaaGimaaaa@352D@ ; затем полученная φ ˜ (x) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacuaHgpGAgaacaiaaiIcacaWG4bGaaG ykaaaa@35EF@  на [1,1] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaaIBbGaeyOeI0IaaGymaiaaiYcaca aIXaGaaGyxaaaa@36A6@  продолжается 2-периодически на всю ось. Отсюда получаются следующие утверждения.

Лемма 2  Функция φ ˜ (x) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacuaHgpGAgaacaiaaiIcacaWG4bGaaG ykaaaa@35EF@  определяется однозначно по φ(x) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqaHgpGAcaaIOaGaamiEaiaaiMcaaa a@35E0@ .

Лемма 3  Операция φ ˜ (x) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacuaHgpGAgaacaiaaiIcacaWG4bGaaG ykaaaa@35EF@  линейна, т.е.

                                        α φ 1 (x)+β φ 2 (x) ˜ =α φ ˜ 1 (x)+β φ ˜ 2 (x). MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaadaaiaaqaaiabeg7aHjabeA8aQnaaBa aaleaacaaIXaaabeaakiaaiIcacaWG4bGaaGykaiabgUcaRiabek7a IjabeA8aQnaaBaaaleaacaaIYaaabeaakiaaiIcacaWG4bGaaGykaa Gaay5adaGaaGypaiabeg7aHjqbeA8aQzaaiaWaaSbaaSqaaiaaigda aeqaaOGaaGikaiaadIhacaaIPaGaey4kaSIaeqOSdiMafqOXdOMbaG aadaWgaaWcbaGaaGOmaaqabaGccaaIOaGaamiEaiaaiMcacaaIUaaa aa@50A6@

Proof. Обе операции в формулировке леммы нечетны и 2-периодичны. Но на [0,1] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaaIBbGaaGimaiaaiYcacaaIXaGaaG yxaaaa@35B8@  они обе равны α φ 1 (x)+β φ 2 (x) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqaHXoqycqaHgpGAdaWgaaWcbaGaaG ymaaqabaGccaaIOaGaamiEaiaaiMcacqGHRaWkcqaHYoGycqaHgpGA daWgaaWcbaGaaGOmaaqabaGccaaIOaGaamiEaiaaiMcaaaa@4004@ . Поэтому из леммы 2 следует лемма 3.

Введем оператор

                                           Bf= 1 2 0 t dτ xt+τ x+tτ q(η)f(η ˜ ,τ)dη, MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGcbGaamOzaiaai2dacqGHsislda WcaaqaaiaaigdaaeaacaaIYaaaamaapehabeWcbaGaaGimaaqaaiaa dshaa0Gaey4kIipakiaadsgacqaHepaDdaWdXbqabSqaaiaadIhacq GHsislcaWG0bGaey4kaSIaeqiXdqhabaGaamiEaiabgUcaRiaadsha cqGHsislcqaHepaDa0Gaey4kIipakmaaGaaabaGaamyCaiaaiIcacq aH3oaAcaaIPaGaamOzaiaaiIcacqaH3oaAaiaawoWaaiaaiYcacqaH epaDcaaIPaGaaGjcVlaadsgacqaH3oaAcaaISaaaaa@5B27@                                               (25)

где f(x,t)C[ Q T ] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGMbGaaGikaiaadIhacaaISaGaam iDaiaaiMcacqGHiiIZcaWGdbGaaG4waiaadgfadaWgaaWcbaGaamiv aaqabaGccaaIDbaaaa@3CBA@ .

Лемма 4 Оператор B MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGcbaaaa@3288@  является линейным и ограниченным в C[ Q T ] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGdbGaaG4waiaadgfadaWgaaWcba GaamivaaqabaGccaaIDbaaaa@363A@ , причем

                                   Bf C[ Q T ] T(T+2)q 1 f(x,t) C[ Q T ] , MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaarqqr1ngBPrgifHhDYfgaiqaacqWFLi cucaWGcbGaamOzaiab=vIiqnaaBaaaleaacaWGdbGaaG4waiaadgfa daWgaaqaaiaadsfaaeqaaiaai2faaeqaaOGaeyizImQaamivaiaaiI cacaWGubGaey4kaSIaaGOmaiaaiMcacqWFLicucaWGXbGae8xjIa1a aSbaaSqaaiaaykW7caaIXaaabeaakiabgwSixlab=vIiqjaadAgaca aIOaGaamiEaiaaiYcacaWG0bGaaGykaiab=vIiqnaaBaaaleaacaWG dbGaaG4waiaadgfadaWgaaqaaiaadsfaaeqaaiaai2faaeqaaOGaaG ilaaaa@59C2@

где 1 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaarqqr1ngBPrgifHhDYfgaiqaacqWFLi cucqGHflY1cqWFLicudaWgaaWcbaGaaGymaaqabaaaaa@3BC5@   MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefmuySLMyYL gaiuaajugGbabaaaaaaaaapeGaa8hfGaaa@3A95@ норма в L[0,1] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGmbGaaG4waiaaicdacaaISaGaaG ymaiaai2faaaa@3689@ .

Proof. Линейность B MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGcbaaaa@3288@  следует из леммы 3. Докажем ограниченность. Как и в лемме 1, имеем

             |Bf| 1 2 0 T dτ T T+1 | q(η)f(η ˜ ,τ)|dη(m+1)q(x)f(x,t) L[ Q T ] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaaI8bGaamOqaiaadAgacaaI8bGaey izIm6aaSaaaeaacaaIXaaabaGaaGOmaaaadaWdXbqabSqaaiaaicda aeaacaWGubaaniabgUIiYdGccaWGKbGaeqiXdq3aa8qCaeqaleaacq GHsislcaWGubaabaGaamivaiabgUcaRiaaigdaa0Gaey4kIipakiaa iYhadaaiaaqaaiaadghacaaIOaGaeq4TdGMaaGykaiaadAgacaaIOa Gaeq4TdGgacaGLdmaacaaISaGaeqiXdqNaaGykaiaaiYhacaaMi8Ua amizaiabeE7aOjabgsMiJkaaiIcacaWGTbGaey4kaSIaaGymaiaaiM carqqr1ngBPrgifHhDYfgaiqaacqWFLicucaWGXbGaaGikaiaadIha caaIPaGaamOzaiaaiIcacaWG4bGaaGilaiaadshacaaIPaGae8xjIa 1aaSbaaSqaaiaadYeacaaIBbGaamyuamaaBaaabaGaamivaaqabaGa aGyxaaqabaGccqGHKjYOaaa@72BA@

                     (m+1)Tq 1 f(x,t) C[ Q T ] T(T+2)q 1 f(x,t) C[ Q T ] . MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqGHKjYOcaaIOaGaamyBaiabgUcaRi aaigdacaaIPaGaamivaebbfv3ySLgzGueE0jxyaGabaiab=vIiqjaa dghacqWFLicudaWgaaWcbaGaaGPaVlaaigdaaeqaaOGae8xjIaLaam OzaiaaiIcacaWG4bGaaGilaiaadshacaaIPaGae8xjIa1aaSbaaSqa aiaadoeacaaIBbGaamyuamaaBaaabaGaamivaaqabaGaaGyxaaqaba GccqGHKjYOcaWGubGaaGikaiaadsfacqGHRaWkcaaIYaGaaGykaiab =vIiqjaadghacqWFLicudaWgaaWcbaGaaGPaVlaaigdaaeqaaOGae8 xjIaLaamOzaiaaiIcacaWG4bGaaGilaiaadshacaaIPaGae8xjIa1a aSbaaSqaaiaadoeacaaIBbGaamyuamaaBaaabaGaamivaaqabaGaaG yxaaqabaGccaaIUaaaaa@66F0@

Образуем ряд

                                                    A 1 (x,t)= n=1 a n (x,t), MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGbbWaaSbaaSqaaiaaigdaaeqaaO GaaGikaiaadIhacaaISaGaamiDaiaaiMcacaaI9aWaaabCaeqaleaa caWGUbGaaGypaiaaigdaaeaacqGHEisPa0GaeyyeIuoakiaadggada WgaaWcbaGaamOBaaqabaGccaaIOaGaamiEaiaaiYcacaWG0bGaaGyk aiaaiYcaaaa@4559@

где a n (x,t)=B a n1 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaO GaaGikaiaadIhacaaISaGaamiDaiaaiMcacaaI9aGaamOqaiaadgga daWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaaaa@3D1C@  ( n1 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGUbGaeyyzImRaaGymaaaa@3535@  ) и a 0 (x,t)= 1 2 [ φ ˜ (x+t)+ φ ˜ (xt)] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGHbWaaSbaaSqaaiaaicdaaeqaaO GaaGikaiaadIhacaaISaGaamiDaiaaiMcacaaI9aWaaSaaaeaacaaI XaaabaGaaGOmaaaacaaIBbGafqOXdOMbaGaacaaIOaGaamiEaiabgU caRiaadshacaaIPaGaey4kaSIafqOXdOMbaGaacaaIOaGaamiEaiab gkHiTiaadshacaaIPaGaaGyxaaaa@48C1@ .

Лемма 5 (см. 6 [с. 220-221]). Если m MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGTbaaaa@32B3@  - наименьшее натуральное число, для которого Tm MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGubGaeyizImQaamyBaaaa@3541@ , то

                                 a n (x,t) C[ Q T ] M 1 M 2 2 n1 T n1 (n1)! ,n1, MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaarqqr1ngBPrgifHhDYfgaiqaacqWFLi cucaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaaGikaiaadIhacaaISaGa amiDaiaaiMcacqWFLicudaWgaaWcbaGaam4qaiaaiUfacaWGrbWaaS baaeaacaWGubaabeaacaaIDbaabeaakiabgsMiJkaad2eadaWgaaWc baGaaGymaaqabaGcdaqadaqaamaalaaabaGaamytamaaBaaaleaaca aIYaaabeaaaOqaaiaaikdaaaaacaGLOaGaayzkaaWaaWbaaSqabeaa caWGUbGaeyOeI0IaaGymaaaakmaalaaabaGaamivamaaCaaaleqaba GaamOBaiabgkHiTiaaigdaaaaakeaacaaIOaGaamOBaiabgkHiTiaa igdacaaIPaGaaGyiaaaacaaISaGaaGzbVlaad6gacqGHLjYScaaIXa GaaGilaaaa@5C85@                                      (26)

где M 1 = a 1 (x,t) C[ Q T ] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGnbWaaSbaaSqaaiaaigdaaeqaaO GaaGypaebbfv3ySLgzGueE0jxyaGabaiab=vIiqjaadggadaWgaaWc baGaaGymaaqabaGccaaIOaGaamiEaiaaiYcacaWG0bGaaGykaiab=v IiqnaaBaaaleaacaWGdbGaaG4waiaadgfadaWgaaqaaiaadsfaaeqa aiaai2faaeqaaaaa@4596@ , M 2 =(2m+1)q 1 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGnbWaaSbaaSqaaiaaikdaaeqaaO GaaGypaiaaiIcacaaIYaGaamyBaiabgUcaRiaaigdacaaIPaqeeuuD JXwAKbsr4rNCHbaceaGae8xjIaLaamyCaiab=vIiqnaaBaaaleaaca aIXaaabeaaaaa@41AC@ . Кроме того, M 1 C T φ 1 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGnbWaaSbaaSqaaiaaigdaaeqaaO GaeyizImQaam4qamaaBaaaleaacaWGubaabeaarqqr1ngBPrgifHhD Yfgaiqaakiab=vIiqjabeA8aQjab=vIiqnaaBaaaleaacaaIXaaabe aaaaa@4087@  и постоянная C T MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGdbWaaSbaaSqaaiaadsfaaeqaaa aa@338E@  не зависит от φ(x) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqaHgpGAcaaIOaGaamiEaiaaiMcaaa a@35E0@ .

Приведем необходимое для дальнейшего доказательство этой леммы.

Proof. Положим f n (x,t)=q(x) a n (x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGMbWaaSbaaSqaaiaad6gaaeqaaO GaaGikaiaadIhacaaISaGaamiDaiaaiMcacaaI9aGaeyOeI0IaamyC aiaaiIcacaWG4bGaaGykaiaadggadaWgaaWcbaGaamOBaaqabaGcca aIOaGaamiEaiaaiYcacaWG0bGaaGykaaaa@4312@ . Очевидно, f n (x,t)L[ Q T ] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGMbWaaSbaaSqaaiaad6gaaeqaaO GaaGikaiaadIhacaaISaGaamiDaiaaiMcacqGHiiIZcaWGmbGaaG4w aiaadgfadaWgaaWcbaGaamivaaqabaGccaaIDbaaaa@3DEC@ , a n (x,t)C[ Q T ] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaO GaaGikaiaadIhacaaISaGaamiDaiaaiMcacqGHiiIZcaWGdbGaaG4w aiaadgfadaWgaaWcbaGaamivaaqabaGccaaIDbaaaa@3DDE@  при n1 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGUbGaeyyzImRaaGymaaaa@3535@ . При n=1 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGUbGaaGypaiaaigdaaaa@3436@  оценка (26) справедлива. Предположим, что она выполняется и при некотором n MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGUbaaaa@32B4@ , и докажем ее справедливость при n+1 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGUbGaey4kaSIaaGymaaaa@3451@ . Имеем

             | a n+1 (x,t)| 1 2 0 t dτ xt+τ x+tτ | f ˜ n (η,τ)|dη MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaaI8bGaamyyamaaBaaaleaacaWGUb Gaey4kaSIaaGymaaqabaGccaaIOaGaamiEaiaaiYcacaWG0bGaaGyk aiaaiYhacqGHKjYOdaWcaaqaaiaaigdaaeaacaaIYaaaamaapehabe WcbaGaaGimaaqaaiaadshaa0Gaey4kIipakiaadsgacqaHepaDdaWd XbqabSqaaiaadIhacqGHsislcaWG0bGaey4kaSIaeqiXdqhabaGaam iEaiabgUcaRiaadshacqGHsislcqaHepaDa0Gaey4kIipakiaaiYha ceWGMbGbaGaadaWgaaWcbaGaamOBaaqabaGccaaIOaGaeq4TdGMaaG ilaiabes8a0jaaiMcacaaI8bGaaGjcVlaadsgacqaH3oaAcqGHKjYO aaa@62B9@

                      1 2 0 t dτ m m+1 | f ˜ n (η,τ)|dη 2m+1 2 0 t dτ 0 1 |q(η)|| a n (η,τ)|dη MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqGHKjYOdaWcaaqaaiaaigdaaeaaca aIYaaaamaapehabeWcbaGaaGimaaqaaiaadshaa0Gaey4kIipakiaa dsgacqaHepaDdaWdXbqabSqaaiabgkHiTiaad2gaaeaacaWGTbGaey 4kaSIaaGymaaqdcqGHRiI8aOGaaGiFaiqadAgagaacamaaBaaaleaa caWGUbaabeaakiaaiIcacqaH3oaAcaaISaGaeqiXdqNaaGykaiaaiY hacaaMi8UaamizaiabeE7aOjabgsMiJoaalaaabaGaaGOmaiaad2ga cqGHRaWkcaaIXaaabaGaaGOmaaaadaWdXbqabSqaaiaaicdaaeaaca WG0baaniabgUIiYdGccaWGKbGaeqiXdq3aa8qCaeqaleaacaaIWaaa baGaaGymaaqdcqGHRiI8aOGaaGiFaiaadghacaaIOaGaeq4TdGMaaG ykaiaaiYhacaaI8bGaamyyamaaBaaaleaacaWGUbaabeaakiaaiIca cqaH3oaAcaaISaGaeqiXdqNaaGykaiaaiYhacaaMi8UaamizaiabeE 7aOjabgsMiJcaa@76B4@

                    2m+1 2 0 t dτ 0 1 |q(η)| M 1 M 2 2 n1 τ n1 (n1)! dη= M 1 M 2 2 n t n n! . MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqGHKjYOdaWcaaqaaiaaikdacaWGTb Gaey4kaSIaaGymaaqaaiaaikdaaaWaa8qCaeqaleaacaaIWaaabaGa amiDaaqdcqGHRiI8aOGaamizaiabes8a0naapehabeWcbaGaaGimaa qaaiaaigdaa0Gaey4kIipakiaaiYhacaWGXbGaaGikaiabeE7aOjaa iMcacaaI8bGaamytamaaBaaaleaacaaIXaaabeaakmaabmaabaWaaS aaaeaacaWGnbWaaSbaaSqaaiaaikdaaeqaaaGcbaGaaGOmaaaaaiaa wIcacaGLPaaadaahaaWcbeqaaiaad6gacqGHsislcaaIXaaaaOWaaS aaaeaacqaHepaDdaahaaWcbeqaaiaad6gacqGHsislcaaIXaaaaaGc baGaaGikaiaad6gacqGHsislcaaIXaGaaGykaiaaigcaaaGaaGjcVl aadsgacqaH3oaAcaaI9aGaamytamaaBaaaleaacaaIXaaabeaakmaa bmaabaWaaSaaaeaacaWGnbWaaSbaaSqaaiaaikdaaeqaaaGcbaGaaG OmaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaad6gaaaGcdaWcaaqa aiaadshadaahaaWcbeqaaiaad6gaaaaakeaacaWGUbGaaGyiaaaaca aIUaaaaa@6ADA@

Тем самым оценка (26) установлена. Оценим M 1 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGnbWaaSbaaSqaaiaaigdaaeqaaa aa@337A@ . Имеем

             | a 1 (x,t)| 1 2 0 t dτ xt+τ x+tτ | f ˜ 0 (η,τ)|dη 1 2 0 T dτ m m+1 | f ˜ 0 (η,τ)|dη= MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaaI8bGaamyyamaaBaaaleaacaaIXa aabeaakiaaiIcacaWG4bGaaGilaiaadshacaaIPaGaaGiFaiabgsMi JoaalaaabaGaaGymaaqaaiaaikdaaaWaa8qCaeqaleaacaaIWaaaba GaamiDaaqdcqGHRiI8aOGaamizaiabes8a0naapehabeWcbaGaamiE aiabgkHiTiaadshacqGHRaWkcqaHepaDaeaacaWG4bGaey4kaSIaam iDaiabgkHiTiabes8a0bqdcqGHRiI8aOGaaGiFaiqadAgagaacamaa BaaaleaacaaIWaaabeaakiaaiIcacqaH3oaAcaaISaGaeqiXdqNaaG ykaiaaiYhacaaMi8UaamizaiabeE7aOjabgsMiJoaalaaabaGaaGym aaqaaiaaikdaaaWaa8qCaeqaleaacaaIWaaabaGaamivaaqdcqGHRi I8aOGaamizaiabes8a0naapehabeWcbaGaeyOeI0IaamyBaaqaaiaa d2gacqGHRaWkcaaIXaaaniabgUIiYdGccaaI8bGabmOzayaaiaWaaS baaSqaaiaaicdaaeqaaOGaaGikaiabeE7aOjaaiYcacqaHepaDcaaI PaGaaGiFaiaayIW7caWGKbGaeq4TdGMaaGypaaaa@7E38@

                    = 2m+1 2 0 T dτ 0 1 | f 0 (η,τ)|dη 2m+1 2 0 1 |q(η)|dη T T+1 | φ ˜ (τ)|dτ MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaaI9aWaaSaaaeaacaaIYaGaamyBai abgUcaRiaaigdaaeaacaaIYaaaamaapehabeWcbaGaaGimaaqaaiaa dsfaa0Gaey4kIipakiaadsgacqaHepaDdaWdXbqabSqaaiaaicdaae aacaaIXaaaniabgUIiYdGccaaI8bGaamOzamaaBaaaleaacaaIWaaa beaakiaaiIcacqaH3oaAcaaISaGaeqiXdqNaaGykaiaaiYhacaaMi8 UaamizaiabeE7aOjabgsMiJoaalaaabaGaaGOmaiaad2gacqGHRaWk caaIXaaabaGaaGOmaaaadaWdXbqabSqaaiaaicdaaeaacaaIXaaani abgUIiYdGccaaI8bGaamyCaiaaiIcacqaH3oaAcaaIPaGaaGiFaiaa yIW7caWGKbGaeq4TdG2aa8qCaeqaleaacqGHsislcaWGubaabaGaam ivaiabgUcaRiaaigdaa0Gaey4kIipakiaaiYhacuaHgpGAgaacaiaa iIcacqaHepaDcaaIPaGaaGiFaiaayIW7caWGKbGaeqiXdqNaeyizIm kaaa@766A@

                         2m+1 2 0 1 |q(η)|dη m m+1 | φ ˜ (τ)|dτ= (2m+1) 2 2 q 1 φ 1 . MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqGHKjYOdaWcaaqaaiaaikdacaWGTb Gaey4kaSIaaGymaaqaaiaaikdaaaWaa8qCaeqaleaacaaIWaaabaGa aGymaaqdcqGHRiI8aOGaaGiFaiaadghacaaIOaGaeq4TdGMaaGykai aaiYhacaaMi8UaamizaiabeE7aOnaapehabeWcbaGaeyOeI0IaamyB aaqaaiaad2gacqGHRaWkcaaIXaaaniabgUIiYdGccaaI8bGafqOXdO MbaGaacaaIOaGaeqiXdqNaaGykaiaaiYhacaaMi8Uaamizaiabes8a 0jaai2dadaWcaaqaaiaaiIcacaaIYaGaamyBaiabgUcaRiaaigdaca aIPaWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGOmaaaarqqr1ngBPrgi fHhDYfgaiqaacqWFLicucaWGXbGae8xjIa1aaSbaaSqaaiaaigdaae qaaOGae8xjIaLaeqOXdOMae8xjIa1aaSbaaSqaaiaaigdaaeqaaOGa aGOlaaaa@6D60@

Отсюда вытекает требуемая оценка для M 1 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGnbWaaSbaaSqaaiaaigdaaeqaaa aa@337A@ .

Таким образом, ряд A 1 (x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGbbWaaSbaaSqaaiaaigdaaeqaaO GaaGikaiaadIhacaaISaGaamiDaiaaiMcaaaa@3789@  сходится абсолютно и равномерно в Q T MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGrbWaaSbaaSqaaiaadsfaaeqaaa aa@339C@ .

Теорема 4 Уравнение (24) имеет единственное решение u(x,t)=A(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaadIhacaaISaGaam iDaiaaiMcacaaI9aGaamyqaiaaiIcacaWG4bGaaGilaiaadshacaaI Paaaaa@3C6A@ , где A(x,t)= a 0 (x,t)+ A 1 (x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGbbGaaGikaiaadIhacaaISaGaam iDaiaaiMcacaaI9aGaamyyamaaBaaaleaacaaIWaaabeaakiaaiIca caWG4bGaaGilaiaadshacaaIPaGaey4kaSIaamyqamaaBaaaleaaca aIXaaabeaakiaaiIcacaWG4bGaaGilaiaadshacaaIPaaaaa@43F0@ , получаемое по методу последовательных подстановок.

Proof. Положим v(x,t)=u(x,t) a 0 (x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG2bGaaGikaiaadIhacaaISaGaam iDaiaaiMcacaaI9aGaamyDaiaaiIcacaWG4bGaaGilaiaadshacaaI PaGaeyOeI0IaamyyamaaBaaaleaacaaIWaaabeaakiaaiIcacaWG4b GaaGilaiaadshacaaIPaaaaa@4373@ . Тогда из (24) получаем для v(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG2bGaaGikaiaadIhacaaISaGaam iDaiaaiMcaaaa@36CD@  интегральное уравнение

                                                    v(x,t)= a 1 (x,t)+Bv. MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG2bGaaGikaiaadIhacaaISaGaam iDaiaaiMcacaaI9aGaamyyamaaBaaaleaacaaIXaaabeaakiaaiIca caWG4bGaaGilaiaadshacaaIPaGaey4kaSIaamOqaiaadAhacaaIUa aaaa@40D8@                                                        (27)

Так как a 1 (x,t)C[ Q T ] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGHbWaaSbaaSqaaiaaigdaaeqaaO GaaGikaiaadIhacaaISaGaamiDaiaaiMcacqGHiiIZcaWGdbGaaG4w aiaadgfadaWgaaWcbaGaamivaaqabaGccaaIDbaaaa@3DA6@ , то уравнение (27) рассматриваем в C[ Q T ] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGdbGaaG4waiaadgfadaWgaaWcba GaamivaaqabaGccaaIDbaaaa@363A@ . По методу последовательных подстановок из (27) получаем ряд A 1 (x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGbbWaaSbaaSqaaiaaigdaaeqaaO GaaGikaiaadIhacaaISaGaamiDaiaaiMcaaaa@3789@ . Поскольку B MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGcbaaaa@3288@   MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefmuySLMyYL gaiuaajugGbabaaaaaaaaapeGaa8hfGaaa@3A95@ линейный и ограниченный оператор в C[ Q T ] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGdbGaaG4waiaadgfadaWgaaWcba GaamivaaqabaGccaaIDbaaaa@363A@  и B A 1 (x,t)= n=2 a n (x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGcbGaamyqamaaBaaaleaacaaIXa aabeaakiaaiIcacaWG4bGaaGilaiaadshacaaIPaGaaGypamaaqaha beWcbaGaamOBaiaai2dacaaIYaaabaGaeyOhIukaniabggHiLdGcca WGHbWaaSbaaSqaaiaad6gaaeqaaOGaaGikaiaadIhacaaISaGaamiD aiaaiMcaaaa@456B@ , то A 1 (x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGbbWaaSbaaSqaaiaaigdaaeqaaO GaaGikaiaadIhacaaISaGaamiDaiaaiMcaaaa@3789@ есть решение (27). Докажем, что уравнение (27) имеет единственное решение. Допустим, что кроме A 1 (x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGbbWaaSbaaSqaaiaaigdaaeqaaO GaaGikaiaadIhacaaISaGaamiDaiaaiMcaaaa@3789@ есть еще другое решение w(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG3bGaaGikaiaadIhacaaISaGaam iDaiaaiMcaaaa@36CE@  этого уравнения. Тогда z(x,t)= A 1 (x,t)w(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG6bGaaGikaiaadIhacaaISaGaam iDaiaaiMcacaaI9aGaamyqamaaBaaaleaacaaIXaaabeaakiaaiIca caWG4bGaaGilaiaadshacaaIPaGaeyOeI0Iaam4DaiaaiIcacaWG4b GaaGilaiaadshacaaIPaaaaa@435A@  - решение уравнения z(x,t)=Bz(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG6bGaaGikaiaadIhacaaISaGaam iDaiaaiMcacaaI9aGaamOqaiaadQhacaaIOaGaamiEaiaaiYcacaWG 0bGaaGykaaaa@3D6F@ , а, значит, и z(x,t)= B n z(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG6bGaaGikaiaadIhacaaISaGaam iDaiaaiMcacaaI9aGaamOqamaaCaaaleqabaGaamOBaaaakiaadQha caaIOaGaamiEaiaaiYcacaWG0bGaaGykaaaa@3E99@  при любом натуральном n MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGUbaaaa@32B4@ . Заметим, что оценка (26) в лемме 5 остается верной, если в качестве a 1 (x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGHbWaaSbaaSqaaiaaigdaaeqaaO GaaGikaiaadIhacaaISaGaamiDaiaaiMcaaaa@37A9@  взять любую функцию из C[ Q T ] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGdbGaaG4waiaadgfadaWgaaWcba GaamivaaqabaGccaaIDbaaaa@363A@ . Возьмем в качестве такой функции функцию z(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG6bGaaGikaiaadIhacaaISaGaam iDaiaaiMcaaaa@36D1@ . Тогда из оценки (26) получаем следующую оценку:

             z(x,t) C[ Q T ] = B n1 z(x,t) C[ Q T ] z(x,t) C[ Q T ] M 2 2 n1 q 1 T n1 (n1)! . MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaarqqr1ngBPrgifHhDYfgaiqaacqWFLi cucaWG6bGaaGikaiaadIhacaaISaGaamiDaiaaiMcacqWFLicudaWg aaWcbaGaam4qaiaaiUfacaWGrbWaaSbaaeaacaWGubaabeaacaaIDb aabeaakiaai2dacqWFLicucaWGcbWaaWbaaSqabeaacaWGUbGaeyOe I0IaaGymaaaakiaadQhacaaIOaGaamiEaiaaiYcacaWG0bGaaGykai ab=vIiqnaaBaaaleaacaWGdbGaaG4waiaadgfadaWgaaqaaiaadsfa aeqaaiaai2faaeqaaOGaeyizImQae8xjIaLaamOEaiaaiIcacaWG4b GaaGilaiaadshacaaIPaGae8xjIa1aaSbaaSqaaiaadoeacaaIBbGa amyuamaaBaaabaGaamivaaqabaGaaGyxaaqabaGcdaqadaqaamaala aabaGaamytamaaBaaaleaacaaIYaaabeaaaOqaaiaaikdaaaaacaGL OaGaayzkaaWaaWbaaSqabeaacaWGUbGaeyOeI0IaaGymaaaakiab=v IiqjaadghacqWFLicudaWgaaWcbaGaaGymaaqabaGcdaWcaaqaaiaa dsfadaahaaWcbeqaaiaad6gacqGHsislcaaIXaaaaaGcbaGaaGikai aad6gacqGHsislcaaIXaGaaGykaiaaigcaaaGaaGOlaaaa@743F@

Отсюда в силу произвольности n MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGUbaaaa@32B4@  получаем z(x,t)=0 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG6bGaaGikaiaadIhacaaISaGaam iDaiaaiMcacaaI9aGaaGimaaaa@3852@ , и единственным решением уравнения (27) является ряд A 1 (x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGbbWaaSbaaSqaaiaaigdaaeqaaO GaaGikaiaadIhacaaISaGaamiDaiaaiMcaaaa@3789@ , а уравнение (24) MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefmuySLMyYL gaiuaajugGbabaaaaaaaaapeGaa8hfGaaa@3A95@ ряд A(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGbbGaaGikaiaadIhacaaISaGaam iDaiaaiMcaaaa@3698@ .

Для сравнения приведем следующие результаты из [6] и [4].

Теорема 5 (см. [4, теорема 6]).  Если функции φ(x) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqaHgpGAcaaIOaGaamiEaiaaiMcaaa a@35E0@ , φ (x) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacuaHgpGAgaqbaiaaiIcacaWG4bGaaG ykaaaa@35EC@  абсолютно непрерывны и φ(0)=φ(1)=0 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqaHgpGAcaaIOaGaaGimaiaaiMcaca aI9aGaeqOXdOMaaGikaiaaigdacaaIPaGaaGypaiaaicdaaaa@3BC2@ , то сумма ряда A(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGbbGaaGikaiaadIhacaaISaGaam iDaiaaiMcaaaa@3698@ представляет собой классическое решение задачи (21) MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefmuySLMyYL gaiuGajugGbabaaaaaaaaapeGaa83eGaaa@3A96@ (23) при условии, что 2 u(x,t)/ t 2 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqGHciITdaahaaWcbeqaaiaaikdaaa GccaWG1bGaaGikaiaadIhacaaISaGaamiDaiaaiMcacaaIVaGaeyOa IyRaamiDamaaCaaaleqabaGaaGOmaaaaaaa@3D26@  класса Q MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGrbaaaa@3297@ (уравнение удовлетворяется почти всюду).

Теорема 6 (см. [6, теорема 5]).  Если φ(x)L[0,1] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqaHgpGAcaaIOaGaamiEaiaaiMcacq GHiiIZcaWGmbGaaG4waiaaicdacaaISaGaaGymaiaai2faaaa@3C2C@ , φ h (x) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqaHgpGAdaWgaaWcbaGaamiAaaqaba GccaaIOaGaamiEaiaaiMcaaaa@3703@  удовлетворяет условиям теоремы 5, φ h φ 1 0 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaarqqr1ngBPrgifHhDYfgaiqaacqWFLi cucqaHgpGAdaWgaaWcbaGaamiAaaqabaGccqGHsislcqaHgpGAcqWF LicudaWgaaWcbaGaaGymaaqabaGccqGHsgIRcaaIWaaaaa@41B6@  при h0 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGObGaeyOKH4QaaGimaaaa@3555@ , то соответствующее φ h (x) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqaHgpGAdaWgaaWcbaGaamiAaaqaba GccaaIOaGaamiEaiaaiMcaaaa@3703@  классическое решение u h (x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bWaaSbaaSqaaiaadIgaaeqaaO GaaGikaiaadIhacaaISaGaamiDaiaaiMcaaaa@37EF@  задачи (21) MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefmuySLMyYL gaiuGajugGbabaaaaaaaaapeGaa83eGaaa@3A96@ (23) сходится при h0 MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGObGaeyOKH4QaaGimaaaa@3555@  по норме L[ Q T ] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGmbGaaG4waiaadgfadaWgaaWcba GaamivaaqabaGccaaIDbaaaa@3643@  к A(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGbbGaaGikaiaadIhacaaISaGaam iDaiaaiMcaaaa@3698@ .

Утверждение теоремы следует из линейности A(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGbbGaaGikaiaadIhacaaISaGaam iDaiaaiMcaaaa@3698@  по φ(x) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqaHgpGAcaaIOaGaamiEaiaaiMcaaa a@35E0@  и леммы 5.

Таким образом, классическое решение задачи (21) MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefmuySLMyYL gaiuaajugGbabaaaaaaaaapeGaa83eGaaa@3A94@ (23) и решение ее, приводимое в статье, выражаются одной и той же формулой: u(x,t)=A(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWG1bGaaGikaiaadIhacaaISaGaam iDaiaaiMcacaaI9aGaamyqaiaaiIcacaWG4bGaaGilaiaadshacaaI Paaaaa@3C6A@ , и A(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGbbGaaGikaiaadIhacaaISaGaam iDaiaaiMcaaaa@3698@  в случае φ(x)L[0,1] MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacqaHgpGAcaaIOaGaamiEaiaaiMcacq GHiiIZcaWGmbGaaG4waiaaicdacaaISaGaaGymaiaai2faaaa@3C2C@  играет роль обобщенного решения, понимаемого как предел классических.

Отметим еще, что ряд A(x,t) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGabiWadaaakeaacaWGbbGaaGikaiaadIhacaaISaGaam iDaiaaiMcaaaa@3698@  в [6] получается иным приемом с более активным использованием обобщенной смешанной задачи.

×

Об авторах

А. П. Хромов

Саратовский национальный исследовательский государственный университет имени Н. Г. Чернышевского

Автор, ответственный за переписку.
Email: khromovap@info.sgu.ru
Россия, Саратов

Список литературы

  1. Корнев В. В., Хромов А. П. Сходимость формального решения по методу Фурье в смешанной задаче для простейшего неоднородного волнового уравнения Мат. Мех. 2017 19 41–44
  2. Натансон И. П. Теория функций вещественной пееменной М.-Л. ГИТТЛ 1957
  3. Харди Г. Расходящиеся ряды М. ИЛ 1951
  4. Хромов А. П. Необходимые и достаточные условия существования классического решения смешанной задачи для однородного волнового уравнения в случае суммируемого потенциала Диффер. уравн. 2019 55 5 717–731
  5. Хромов А. П. Расходящиеся ряды и обобщенная смешанная задача для волнового уравнения Мат. 21 Междунар. конф. <<Современные проблемы теории функций и их приложения>> Саратов, 31 января "— 4 февраля 2022 г. Саратов 2022 319–324
  6. Хромов А. П., Корнев В. В. Расходящиеся ряды в методе Фурье для волнового уравнения Тр. ин-та мат. мех. УрО РАН. 2021 27 4 215–238
  7. Эйлер Л. Дифференциальное исчисление М.-Л. ГИТТЛ 1949

Дополнительные файлы

Доп. файлы
Действие
1. JATS XML

© Хромов А.П., 2023

Согласие на обработку персональных данных с помощью сервиса «Яндекс.Метрика»

1. Я (далее – «Пользователь» или «Субъект персональных данных»), осуществляя использование сайта https://journals.rcsi.science/ (далее – «Сайт»), подтверждая свою полную дееспособность даю согласие на обработку персональных данных с использованием средств автоматизации Оператору - федеральному государственному бюджетному учреждению «Российский центр научной информации» (РЦНИ), далее – «Оператор», расположенному по адресу: 119991, г. Москва, Ленинский просп., д.32А, со следующими условиями.

2. Категории обрабатываемых данных: файлы «cookies» (куки-файлы). Файлы «cookie» – это небольшой текстовый файл, который веб-сервер может хранить в браузере Пользователя. Данные файлы веб-сервер загружает на устройство Пользователя при посещении им Сайта. При каждом следующем посещении Пользователем Сайта «cookie» файлы отправляются на Сайт Оператора. Данные файлы позволяют Сайту распознавать устройство Пользователя. Содержимое такого файла может как относиться, так и не относиться к персональным данным, в зависимости от того, содержит ли такой файл персональные данные или содержит обезличенные технические данные.

3. Цель обработки персональных данных: анализ пользовательской активности с помощью сервиса «Яндекс.Метрика».

4. Категории субъектов персональных данных: все Пользователи Сайта, которые дали согласие на обработку файлов «cookie».

5. Способы обработки: сбор, запись, систематизация, накопление, хранение, уточнение (обновление, изменение), извлечение, использование, передача (доступ, предоставление), блокирование, удаление, уничтожение персональных данных.

6. Срок обработки и хранения: до получения от Субъекта персональных данных требования о прекращении обработки/отзыва согласия.

7. Способ отзыва: заявление об отзыве в письменном виде путём его направления на адрес электронной почты Оператора: info@rcsi.science или путем письменного обращения по юридическому адресу: 119991, г. Москва, Ленинский просп., д.32А

8. Субъект персональных данных вправе запретить своему оборудованию прием этих данных или ограничить прием этих данных. При отказе от получения таких данных или при ограничении приема данных некоторые функции Сайта могут работать некорректно. Субъект персональных данных обязуется сам настроить свое оборудование таким способом, чтобы оно обеспечивало адекватный его желаниям режим работы и уровень защиты данных файлов «cookie», Оператор не предоставляет технологических и правовых консультаций на темы подобного характера.

9. Порядок уничтожения персональных данных при достижении цели их обработки или при наступлении иных законных оснований определяется Оператором в соответствии с законодательством Российской Федерации.

10. Я согласен/согласна квалифицировать в качестве своей простой электронной подписи под настоящим Согласием и под Политикой обработки персональных данных выполнение мною следующего действия на сайте: https://journals.rcsi.science/ нажатие мною на интерфейсе с текстом: «Сайт использует сервис «Яндекс.Метрика» (который использует файлы «cookie») на элемент с текстом «Принять и продолжить».