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} } {\*\listtable{\list\listtemplateid1\listsimple{\listlevel\levelnfc23\leveljc0\levelfollow0\levelstartat1 {\leveltext\'01\u-3913 ?;}{\levelnumbers;}\f1\fbias0 }{\listname ;}\listid1}{\list\listtemplateid2\listsimple{\listlevel\levelnfc23\leveljc0\levelfollow0\levelstartat1 {\leveltext\'01-;}{\levelnumbers;}\f0\fbias0 }{\listname ;}\listid2}{\list\listtemplateid3\listsimple{\listlevel\levelnfc23\leveljc0\levelfollow0\levelstartat1 {\leveltext\'01 ;}{\levelnumbers;}\f0\fbias0 }{\listname ;}\listid3}} {\*\listoverridetable{\listoverride\listid1\listoverridecount0\ls1}{\listoverride\listid2\listoverridecount0\ls2} {\listoverride\listid3\listoverridecount0\ls3}} \titlepg\pgnstart1 \pgnrestart {\headerf \pard\plain\s1\qr \cs17\f0\fs24\cf2\i0 \b0 \ul0 Page {\field{\*\fldinst{ PAGE }}{\fldrslt { 1}}} } {\header\pard\plain\s1\qr \cs17\f0\fs24\cf2\i0 \b0 \ul0 Page {\field{\*\fldinst{ PAGE }}{\fldrslt { 1}}} } {\info{\title 3_9_РешУрТеплВ_Maple.mws - [Server 1]}{\creatim\yr2014\mo11\dy25\hr23 \min44}} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Файл: "...КомпТехн\\Maple\\УМФ\\3_9_РешУрТеплВ_Maple.mws" Титов К.В. 12.11.12 } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Гл.6, §4, п.7 } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Реквизиты, отмеченные звездочкой *), необходимо задать самостоятельно. } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 restart; } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs196\f0\fs24\cf2\i0 \b \ul0 Решение однородного уравнения диффузии с краевым и условиями в среде Maple } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {(задача №3, вар.9, Покровский). } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Рассмотрим решение однородного уравнения с краевыми условиями, описывающего } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {процесс диффузии в бесконечном слое толщиной } {\cs247\f0\fs24\cf2\i \b0 \ul0 L } {, которое будем искать в виде } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs108\f0\fs24\cf2\i \b0 \ul0 u } {( } {\cs109\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs110\f0\fs24\cf2\i \b0 \ul0 t } {)= } {\cs112\f0\fs24\cf2\i \b0 \ul0 z } {( } {\cs111\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs113\f0\fs24\cf2\i \b0 \ul0 t } {)+ } {\cs117\f0\fs24\cf2\i \b0 \ul0 w } {( } {\cs118\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs119\f0\fs24\cf2\i \b0 \ul0 t } {)+ } {\cs114\f0\fs24\cf2\i \b0 \ul0 v } {( } {\cs115\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs116\f0\fs24\cf2\i \b0 \ul0 t } {). } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 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} \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Здесь -infinity< } {\cs253\f0\fs24\cf2\i \b0 \ul0 y } {, } {\cs254\f0\fs24\cf2\i \b0 \ul0 z } {<+infinity, 0< } {\cs255\f0\fs24\cf2\i \b0 \ul0 x } {< } {\cs256\f0\fs24\cf2\i \b0 \ul0 L } {, } {\cs259\f0\fs24\cf2\i \b0 \ul0 u } {( } {\cs260\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs261\f0\fs24\cf2\i \b0 \ul0 t } {) - концентрация диффундирующего вещества. На- } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {чальные и граничные условия, а также плотность мощности источников, не зависят от } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs257\f0\fs24\cf2\i \b0 \ul0 y } { и } {\cs258\f0\fs24\cf2\i \b0 \ul0 z } {. Таким образом, задача сводится к одномерной. } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs35\f0\fs24\cf2\i0 \b \ul0 Д а н о: } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {*) Установим параметры конкретной решаемой задачи. Пусть таковыми будут: } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 A1:=1:C0:=1:L:=1:KD:=1: a:=sqrt(KD): } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Зададим краевые условия исходного дифференциального уравнения в общем виде: } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {- укажем функцию, задающую начальное условие } {\cs198\f0\fs24\cf2\i \b0 \ul0 u } {( } {\cs199\f0\fs24\cf2\i \b0 \ul0 x } {,0)= } {\cs201\f0\fs24\cf2\i \b0 \ul0 f } {( } {\cs200\f0\fs24\cf2\i \b0 \ul0 x } {): } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 f:=x->x*C0/2; } \par} {\pard\plain \s266 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn23 \picw2888 \pich1036\wmetafile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} \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {- функции } {\cs223\f0\fs24\cf2\i \b0 \ul0 phi } {( } {\cs221\f0\fs24\cf2\i \b0 \ul0 t } {) и } {\cs224\f0\fs24\cf2\i \b0 \ul0 psi } {( } {\cs222\f0\fs24\cf2\i \b0 \ul0 t } {), задающие правые части граничных условий решения } {\cs218\f0\fs24\cf2\i \b0 \ul0 u } {( } {\cs220\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs219\f0\fs24\cf2\i \b0 \ul0 t } {): } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 phi:=t->A1*t; psi:=t->0; u(x,t): } \par} {\pard\plain \s266 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn7 \picw2480 \pich495\wmetafile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} \par} {\pard\plain \s266 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn7 \picw2091 \pich495\wmetafile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} \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {*) Последнее отразим в виде четырех равенств, которые записаны ниже. Нужное из них } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {следует открыть, сняв апострофы. } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 `u(0,t):=phi(t); u(L,t):=psi(t); `: } {1) } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 `D[1](u)(0,t):=phi(t); D[1](u)(L,t):=psi(t);`: } {2) } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 `D[1](u)(0,t):=phi(t); u(L,t):=psi(t);`: } {3) } {\cs12\f2\fs24\cf1\i0 \b \ul0 } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 u(0,t):=phi(t); D[1](u)(L,t):=psi(t); } { 4) } {\cs12\f2\fs24\cf1\i0 \b \ul0 } \par} {\pard\plain \s266 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn7 \picw2073 \pich495\wmetafile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} \par} {\pard\plain \s266 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn15 \picw2995 \pich619\wmetafile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} \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Отметим, что в структуре решения функция } {\cs226\f0\fs24\cf2\i \b0 \ul0 z } {( } {\cs225\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs227\f0\fs24\cf2\i \b0 \ul0 t } {) подбирается таким образом, чтобы гра- } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {ничные условия решения } {\cs228\f0\fs24\cf2\i \b0 \ul0 u } {( } {\cs229\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs230\f0\fs24\cf2\i \b0 \ul0 t } {) совпадали со значениями } {\cs232\f0\fs24\cf2\i \b0 \ul0 z } {( } {\cs231\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs233\f0\fs24\cf2\i \b0 \ul0 t } {) в аналогичных точках, то } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {есть их разность равнялась нулю. } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Сформируем логический блок выбора типа граничных условий, которые будут определ ять } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {(задача Штурма-Лиувилля) собственные значения } {\cs207\f0\fs24\cf2\i \b0 \ul0 b } {( } {\cs208\f0\fs24\cf2\i \b0 \ul0 n } {) и собственные функции } {\cs209\f0\fs24\cf2\i \b0 \ul0 Xn } {( } {\cs210\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs211\f0\fs24\cf2\i \b0 \ul0 n } {), } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {а также подбираемую к граничным условиям функцию } {\cs212\f0\fs24\cf2\i \b0 \ul0 z } {( } {\cs213\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs214\f0\fs24\cf2\i \b0 \ul0 t } {): } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 if u(0,t)=phi(t)and u(L,t)=psi(t) then } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 b:=n->Pi*n/L: Xn:=(x,n)->sin(b(n)*x): } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn23 \picw6519 \pich1036\wmetafile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} \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 elif D[1](u)(0,t)=phi(t) and D[1](u)(L,t)=psi(t) t hen } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 b:=n->Pi*n/L: Xn:=(x,n)->cos(b(n)*x): } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn21 \picw6979 \pich1106\wmetafile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00002105010079008501EB11040000002D010100070000002105010074008501ED13040000002D01 0000070000002105010028008501CD170700000021050100290085014119040000002D0102000700 00002105010066008501E716040000002D0101000700000021050100740085019018040000002D01 000007000000210501003200A203E413040000002D01010007000000210501004C00A20323150300 00000000 } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 elif D[1](u)(0,t)=phi(t) and u(L,t)=psi(t) then } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 b:=n->Pi*(2*n-1)/2/L: Xn:=(x,n)->cos(b(n)*x): } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn7 \picw6749 \pich495\wmetafile8 0100090000037F01000003001C00000000000400000003010800050000000B020000000005000000 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74003F012D0C040000002D010000070000002105010028003F01270F070000002105010029003F01 5319040000002D0102000800000021050300202D20203F012B13040000002D010000070000002105 010028003F012911070000002105010029003F019D12040000002D01020007000000210501007900 3F01EA0F040000002D010100070000002105010074003F01EC1107000000210501004C003F01E614 040000002D010000070000002105010028003F011C17070000002105010029003F01901804000000 2D010200070000002105010066003F013616040000002D010100070000002105010074003F01DF17 030000000000 } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 elif u(0,t)=phi(t) and D[1](u)(L,t)=psi(t) then } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 b:=n->Pi*(2*n-1)/2/L: Xn:=(x,n)->sin(b(n)*x): } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn7 \picw4836 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5709040000002D010100070000002105010074003F01000B070000002105010078003F01FA0D0400 00002D010000070000002105010028003F016610070000002105010029003F01DA11040000002D01 0200070000002105010079003F01270F040000002D010100070000002105010074003F0129110300 00000000 } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 fi: } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Проведем визуализацию выбранных собственных значений } {\cs202\f0\fs24\cf2\i \b0 \ul0 b } {( } {\cs203\f0\fs24\cf2\i \b0 \ul0 n } {), собственных функций } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs204\f0\fs24\cf2\i \b0 \ul0 Xn } {( } {\cs205\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs206\f0\fs24\cf2\i \b0 \ul0 n } {) и функции } {\cs215\f0\fs24\cf2\i \b0 \ul0 z } {( } {\cs216\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs217\f0\fs24\cf2\i \b0 \ul0 t } {). } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 b(n);Xn(x,n); z(x,t); } \par} {\pard\plain \s266 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn23 \picw2250 \pich1036\wmetafile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} \par} {\pard\plain \s266 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn23 \picw3453 \pich1036\wmetafile8 0100090000034B01000004001C00000000000400000003010800050000000B020000000005000000 0C020C047D0D050000002E0118000000050000000201010000000500000009020000FF001C000000 FB025CFE0000000000009001000000020700001253796D626F6C0000980A0A89C4880A00E8B1F377 F1B1F3772040F577040B6651040000002D0100000700000021050100E600A9014802070000002105 0100E800C60348020700000021050100E7007D0248020700000021050100E700F202480207000000 21050100F600A901620C0700000021050100F800C603620C0700000021050100F7007D02620C0700 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0100090000035000000001001C00000000000400000003010800050000000B020000000005000000 0C02EF016201050000002E0118000000050000000201010000000500000009020000FF001C000000 FB025CFE0000000000009001010000000400001254696D6573204E657720526F6D616E00E8B1F377 F1B1F3772040F577260A6651040000002D010000070000002105010074003F016A00030000000000 } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Проверим на однородность полученных граничных условий для вводимой в последующе м } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {функции W(x,t), выбрав ниже нужную строчку и сняв апострофы. } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 Ww(x,t): Ww(0,t)=u(0,t)-z(0,t); Ww(L,t)=u(L,t)-z(L ,t); } {1) } \par} {\pard\plain \s266 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn7 \picw2534 \pich495\wmetafile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} \par} {\pard\plain \s266 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn7 \picw4040 \pich495\wmetafile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} \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 D[1](Ww)(0,t)=D[1](u)(0,t)-D[1](z)(0,t); D[1](Ww)( L,t)=D[1](u)(L,t)-D[1](z)(L,t); } {2) } \par} {\pard\plain \s266 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn15 \picw5581 \pich619\wmetafile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} \par} {\pard\plain \s266 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn15 \picw3278 \pich619\wmetafile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} \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 D[1](Ww)(0,t)=D[1](u)(0,t)-D[1](z)(0,t); Ww(L,t)=u (L,t)-z(L,t); } {3) } \par} {\pard\plain \s266 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn15 \picw5581 \pich619\wmetafile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} \par} {\pard\plain \s266 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn7 \picw4040 \pich495\wmetafile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} \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 Ww(0,t)=u(0,t)-z(0,t); D[1](Ww)(L,t)=D[1](u)(L,t)- D[1](z)(L,t); } {4) } {\cs12\f2\fs24\cf1\i0 \b \ul0 } \par} {\pard\plain \s266 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn7 \picw2534 \pich495\wmetafile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} \par} {\pard\plain \s266 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn15 \picw3278 \pich619\wmetafile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} \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {*) Зададим конкретную функцию возмущения, например: } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 g:=(x,t)->0; } \par} {\pard\plain \s266 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn7 \picw2782 \pich495\wmetafile8 010009000003D600000003001C00000000000400000003010800050000000B020000000005000000 0C02EF01DE0A050000002E0118000000050000000201010000000500000009020000FF001C000000 FB025CFE0000000000009001000000000400001254696D6573204E657720526F6D616E00E8B1F377 F1B1F3772040F577040B665B040000002D0100000800000021050400203A3D203F013F011C000000 FB025CFE0000000000009001010000000400001254696D6573204E657720526F6D616E00E8B1F377 F1B1F3772040F577040B665B040000002D010100070000002105010067003F016A001C000000FB02 5CFE0000000000009001000000020700001253796D626F6C0000980A0A91D0880A00E8B1F377F1B1 F3772040F577040B665B040000002D010200080000002105030020AE20203F010F07040000002D01 0000070000002105010028003F017603070000002105010029003F01810607000000210501002C00 3F01FC04040000002D010100070000002105010078003F013904070000002105010074003F01D005 040000002D010000070000002105010030003F018D09030000000000 } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Решение исходного дифференциального уравнения, удовлетворяющего выше сформули- } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {рованным краевым условиям, как было сказано, будем искать в виде: } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs150\f0\fs24\cf2\i \b0 \ul0 u } {( } {\cs151\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs152\f0\fs24\cf2\i \b0 \ul0 t } {)= } {\cs153\f0\fs24\cf2\i \b0 \ul0 z } {( } {\cs154\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs155\f0\fs24\cf2\i \b0 \ul0 t } {)+ } {\cs161\f0\fs24\cf2\i \b0 \ul0 w } {( } {\cs156\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs157\f0\fs24\cf2\i \b0 \ul0 t } {)+ } {\cs158\f0\fs24\cf2\i \b0 \ul0 v } {( } {\cs159\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs160\f0\fs24\cf2\i \b0 \ul0 t } {) и находить из уравнения: } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn29 \picw8572 \pich1176\wmetafile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} \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Для каждой компоненты решения } {\cs172\f0\fs24\cf2\i \b0 \ul0 u } {( } {\cs173\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs174\f0\fs24\cf2\i \b0 \ul0 t } {), кроме } {\cs234\f0\fs24\cf2\i \b0 \ul0 z } {( } {\cs235\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs236\f0\fs24\cf2\i \b0 \ul0 t } {), запишем своё дифференциальное } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {уравнение с частными производными, из которого она должна определяться, в удобн ом } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {для дальнейших вычислений виде. Одна из компонент } {\cs138\f0\fs24\cf2\i \b0 \ul0 z } {( } {\cs139\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs140\f0\fs24\cf2\i \b0 \ul0 t } {) решения } {\cs141\f0\fs24\cf2\i \b0 \ul0 u } {( } {\cs142\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs143\f0\fs24\cf2\i \b0 \ul0 t } {) уже известна } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {и её запись приводится ниже. Две другие будем искать из приводимых ниже диффере нци- } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {альных уравнений. Компоненту } {\cs149\f0\fs24\cf2\i \b0 \ul0 w } {( } {\cs144\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs145\f0\fs24\cf2\i \b0 \ul0 t } {) из уравнения: } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn29 \picw7455 \pich1176\wmetafile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{\cs147\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs148\f0\fs24\cf2\i \b0 \ul0 t } {) из уравнения, для которого } {\cs162\f0\fs24\cf2\i \b0 \ul0 g } {1( } {\cs163\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs164\f0\fs24\cf2\i \b0 \ul0 t } {) будет определена ниже. } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn29 \picw7810 \pich1176\wmetafile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Запишем 1-ю компоненту } {\cs68\f0\fs24\cf2\i \b \ul0 z } {\cs71\f0\fs24\cf2\i0 \b \ul0 ( } {\cs69\f0\fs24\cf2\i \b \ul0 x } {\cs72\f0\fs24\cf2\i0 \b \ul0 , } {\cs70\f0\fs24\cf2\i \b \ul0 t } {\cs73\f0\fs24\cf2\i0 \b \ul0 ) решения } {\cs39\f0\fs24\cf2\i \b \ul0 u } {\cs47\f0\fs24\cf2\i0 \b \ul0 ( } {\cs40\f0\fs24\cf2\i \b \ul0 x } {\cs48\f0\fs24\cf2\i0 \b \ul0 , } {\cs41\f0\fs24\cf2\i \b \ul0 t } {\cs49\f0\fs24\cf2\i0 \b \ul0 ) } {: } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 z(x,t); } \par} {\pard\plain \s266 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn7 \picw354 \pich495\wmetafile8 0100090000035000000001001C00000000000400000003010800050000000B020000000005000000 0C02EF016201050000002E0118000000050000000201010000000500000009020000FF001C000000 FB025CFE0000000000009001010000000400001254696D6573204E657720526F6D616E00E8B1F377 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ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00030000000000} \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Проведем коррекцию возмущения, если таковая появилась в результате подстановки } {\cs104\f0\fs24\cf2\i \b0 \ul0 z } {( } {\cs105\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs106\f0\fs24\cf2\i \b0 \ul0 t } {) } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {в исходное дифференциальное уравнение } {\cs175\f0\fs24\cf2\i \b0 \ul0 de } {. } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 g1:=(x,t)->g(x,t)-D[2](z)(x,t)+a^2*D[1,1](z)(x,t): g1(x,t); } \par} {\pard\plain \s266 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn7 \picw584 \pich495\wmetafile8 0100090000035000000001001C00000000000400000003010800050000000B020000000005000000 0C02EF014802050000002E0118000000050000000201010000000500000009020000FF001C000000 FB025CFE0000000000009001000000000400001254696D6573204E657720526F6D616E00E8B1F377 F1B1F3772040F5778A0A66FE040000002D01000007000000210502002D313F016A00030000000000 } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Проведем коррекцию краевых условий для функции } {\cs165\f0\fs24\cf2\i \b0 \ul0 w } {( } {\cs82\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs83\f0\fs24\cf2\i \b0 \ul0 t } {), которая также связана с подста- } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {новкой } {\cs74\f0\fs24\cf2\i \b0 \ul0 z } {( } {\cs75\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs76\f0\fs24\cf2\i \b0 \ul0 t } {)+ } {\cs166\f0\fs24\cf2\i \b0 \ul0 w } {( } {\cs77\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs78\f0\fs24\cf2\i \b0 \ul0 t } {)+ } {\cs79\f0\fs24\cf2\i \b0 \ul0 v } {( } {\cs80\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs81\f0\fs24\cf2\i \b0 \ul0 t } {) в исходное дифференциальное уравнение, полагая нулевыми } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {краевые условия функции } {\cs84\f0\fs24\cf2\i \b0 \ul0 v } {( } {\cs85\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs86\f0\fs24\cf2\i \b0 \ul0 t } {). В этом предположении вычислим начальные условия для } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {функции } {\cs167\f0\fs24\cf2\i \b0 \ul0 w } {( } {\cs107\f0\fs24\cf2\i \b0 \ul0 x } {,0)= } {\cs238\f0\fs24\cf2\i \b0 \ul0 f } {1( } {\cs237\f0\fs24\cf2\i \b0 \ul0 x } {), где: } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 f1:=x->f(x)-z(x,0): f1(x); } \par} {\pard\plain \s266 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn23 \picw549 \pich1036\wmetafile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} \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs42\f0\fs24\cf2\i \b0 \ul0 Примечание } {. При этом, как было сказано ранее, } {\cs264\f0\fs24\cf2\i \b0 \ul0 w } {( } {\cs262\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs263\f0\fs24\cf2\i \b0 \ul0 t } {) имеет однородные (нулевые) гранич- } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {ные условия, что очень важно для дальнейшего построения алгоритма. } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 assume(n,integer): N:=infinity: } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs50\f0\fs24\cf2\i0 \b \ul0 2. Запишем 2-ю компоненту } {\cs89\f0\fs24\cf2\i \b \ul0 w } {\cs171\f0\fs24\cf2\i0 \b \ul0 ( } {\cs87\f0\fs24\cf2\i \b \ul0 x } {\cs90\f0\fs24\cf2\i0 \b \ul0 , } {\cs88\f0\fs24\cf2\i \b \ul0 t } {\cs91\f0\fs24\cf2\i0 \b \ul0 ) решения } {\cs43\f0\fs24\cf2\i \b \ul0 u } {\cs51\f0\fs24\cf2\i0 \b \ul0 ( } {\cs44\f0\fs24\cf2\i \b \ul0 x } {\cs52\f0\fs24\cf2\i0 \b \ul0 , } {\cs45\f0\fs24\cf2\i \b \ul0 t } {\cs53\f0\fs24\cf2\i0 \b \ul0 ) } {. } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Начнем с вычисления коэффициентов ряда Фурье: } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {- запишем коэффициент разложения в ряд Фурье по собственным функциям } {\cs243\f0\fs24\cf2\i \b0 \ul0 Xn } {( } {\cs244\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs245\f0\fs24\cf2\i \b0 \ul0 n } {) функ- } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {ции } {\cs242\f0\fs24\cf2\i \b0 \ul0 f } {1( } {\cs241\f0\fs24\cf2\i \b0 \ul0 x } {), задающей начальное условие } {\cs240\f0\fs24\cf2\i \b0 \ul0 w } {( } {\cs239\f0\fs24\cf2\i \b0 \ul0 x } {,0): } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn26 \picw7023 \pich1867\wmetafile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Запишем 3-ю компоненту } {\cs95\f0\fs24\cf2\i \b \ul0 v } {\cs98\f0\fs24\cf2\i0 \b \ul0 ( } {\cs96\f0\fs24\cf2\i \b \ul0 x } {\cs99\f0\fs24\cf2\i0 \b \ul0 , } {\cs97\f0\fs24\cf2\i \b \ul0 t } {\cs100\f0\fs24\cf2\i0 \b \ul0 ) решения } {\cs54\f0\fs24\cf2\i \b \ul0 u } {\cs58\f0\fs24\cf2\i0 \b \ul0 ( } {\cs55\f0\fs24\cf2\i \b \ul0 x } {\cs59\f0\fs24\cf2\i0 \b \ul0 , } {\cs56\f0\fs24\cf2\i \b \ul0 t } {\cs60\f0\fs24\cf2\i0 \b \ul0 ) } {. } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Компоненту } {\cs176\f0\fs24\cf2\i \b0 \ul0 v } {( } {\cs177\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs178\f0\fs24\cf2\i \b0 \ul0 t } {) следует определять из уравнения: } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn29 \picw8979 \pich1176\wmetafile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} \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Будем вычислять эту компоненту, как было сказано выше, полагая нулевыми краевые } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {условия для } {\cs120\f0\fs24\cf2\i \b0 \ul0 v } {( } {\cs121\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs122\f0\fs24\cf2\i \b0 \ul0 t } {), что также предопределено выбором собственных функций. } {\cs123\f0\fs24\cf2\i \b0 \ul0 } { } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {3.1. Вычислим коэффициенты ряда Фурье: } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {- коэффициент ряда Фурье функции, задающей возмущение, может быть представлен в } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {виде: } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn26 \picw8351 \pich1867\wmetafile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} \par} {\pard\plain \s266 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn23 \picw2816 \pich1036\wmetafile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} \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {3.2. И, наконец, запишем саму 3-ю компоненту } {\cs101\f0\fs24\cf2\i \b0 \ul0 v } {( } {\cs102\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs103\f0\fs24\cf2\i \b0 \ul0 t } {): } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn90 \picw12828 \pich1643\wmetafile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01007400C909B404040000002D01050007000000210501002800C9097E0D07000000210501002900 C909FF130A0000002105080073696D706C696679C9092A0807000000210501002800C90931100700 0000210501002900C9093C130700000021050200556EC909410E07000000210501002C00C909B711 040000002D01080007000000210501007800C909F41007000000210501007400C9098B1203000000 0000 } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Заметим, что значение этого предела limit( } {\cs186\f0\fs24\cf2\i \b0 \ul0 Un } {( } {\cs187\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs188\f0\fs24\cf2\i \b0 \ul0 t } {), } {\cs185\f0\fs24\cf2\i \b0 \ul0 NN } {=infinity)= } {\cs182\f0\fs24\cf2\i \b0 \ul0 U } {( } {\cs183\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs184\f0\fs24\cf2\i \b0 \ul0 t } {) равно функции } {\cs179\f0\fs24\cf2\i \b0 \ul0 U } {( } {\cs180\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs181\f0\fs24\cf2\i \b0 \ul0 t } {). } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs64\f0\fs24\cf2\i0 \b \ul0 5. Проверка } {. Подставим полученное решение } {\cs124\f0\fs24\cf2\i \b0 \ul0 U } {( } {\cs125\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs126\f0\fs24\cf2\i \b0 \ul0 t } {) в исходное дифференциальное урав- } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {нение } {\cs18\f0\fs24\cf2\i \b0 \ul0 de } {: } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 re:=subs(u(x,t)=Un(x,t),de): } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs19\f0\fs24\cf2\i0 \b \ul0 5.1 } {. В правильности полученного решения можно убедиться, запустив процедуру упроще - } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {ния, в результате которой должны получить тождество или из-за ограниченного чис ла } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {членов ряда только приблизительное тождество. } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 simplify(re): } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Полученное тождество (его левую часть) проиллюстрируем графически. Для этого во зьмем } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {функцию } {\cs128\f0\fs24\cf2\i \b0 \ul0 Un } {( } {\cs129\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs130\f0\fs24\cf2\i \b0 \ul0 t } {) с ограниченным числом слагаемых. } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {В результате получим поверхность } {\cs189\f0\fs24\cf2\i \b0 \ul0 lhs } {( } {\cs190\f0\fs24\cf2\i \b0 \ul0 re } {), которую представим графически } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 plot3d(lhs(re), x=0..L, t=0..1,axes=boxed,orientat ion=[-33,56],title=`Рис.1. 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\pich495\wmetafile8 0100090000035500000001001C00000000000400000003010800050000000B020000000005000000 0C02EF01730A050000002E0118000000050000000201010000000500000009020000FF001C000000 FB025CFE0000000000009001000000000400001254696D6573204E657720526F6D616E00E8B1F377 F1B1F3772040F5778A0A660F040000002D0100000C00000021050C00302E32353030303030303030 3F016A00030000000000 } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Согласно краевым условиям функция } {\cs36\f0\fs24\cf2\i \b0 \ul0 S } {( } {\cs25\f0\fs24\cf2\i \b0 \ul0 x } {)= } {\cs131\f0\fs24\cf2\i \b0 \ul0 U } {( } {\cs28\f0\fs24\cf2\i \b0 \ul0 x } {,0) должна равняться } {\cs27\f0\fs24\cf2\i \b0 \ul0 f } {( } {\cs26\f0\fs24\cf2\i \b0 \ul0 x } {) рис.3,4. Проверим } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {это равенство графически. Построим графики двух функций и сравним их. При этом один } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {из них, например, } {\cs132\f0\fs24\cf2\i \b0 \ul0 f } {( } {\cs133\f0\fs24\cf2\i \b0 \ul0 x } {) для отличия от } {\cs67\f0\fs24\cf2\i \b0 \ul0 S } {(x) на графике умножим на коэффициент } {\cs66\f0\fs24\cf2\i \b0 \ul0 k } {. Для } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {построения графика возьмем } {\cs134\f0\fs24\cf2\i \b0 \ul0 Un } {( } {\cs135\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs136\f0\fs24\cf2\i \b0 \ul0 t } {), используя } {\cs137\f0\fs24\cf2\i \b0 \ul0 NN } {-ю частичную сумму полученного } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {решения. } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 Sn:=x->simplify(Un(x,0)): } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Строим график } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 k:=1.1: plot(\{Sn(x),k*f(x)\},x=0..L+1,title=`Рис. 3.`); } \par} {\pard\plain \s268 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl240 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ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff000300 00000000} \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Построенные графики убеждают нас в их равестве } {\cs37\f0\fs24\cf2\i \b0 \ul0 Sn } {( } {\cs20\f0\fs24\cf2\i \b0 \ul0 x } {)= } {\cs21\f0\fs24\cf2\i \b0 \ul0 k } {* } {\cs23\f0\fs24\cf2\i \b0 \ul0 f } {( } {\cs22\f0\fs24\cf2\i \b0 \ul0 x } {), если } {\cs24\f0\fs24\cf2\i \b0 \ul0 k } {->1 на отрезке } {\cs252\f0\fs24\cf2\i \b0 \ul0 L } {. При } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {этом } {\cs38\f0\fs24\cf2\i \b0 \ul0 Sn } {( } {\cs32\f0\fs24\cf2\i \b0 \ul0 x } {) является разложением } {\cs34\f0\fs24\cf2\i \b0 \ul0 f } {( } {\cs33\f0\fs24\cf2\i \b0 \ul0 x } {) в ряд Фурье по собственным функциям и этот ряд } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {равен } {\cs195\f0\fs24\cf2\i \b0 \ul0 S } {( } {\cs194\f0\fs24\cf2\i \b0 \ul0 x } {). } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Запишем задание на построение графика найденного решения } {\cs29\f0\fs24\cf2\i \b0 \ul0 Un } {( } {\cs30\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs31\f0\fs24\cf2\i \b0 \ul0 t } {) } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 plot3d(Un(x,t), x=0..L, t=0..1,axes=boxed,orientat ion=[-33,56],title=`Рис.4.Решение u(x,t)`); } \par} {\pard\plain \s268 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn0 \picw16894 \pich8376\wmetafile8 0100090000033884030000001e84030000000400 000003010800050000000b020000000005000000 0c02ef00e2011e840300430f2000cc000000ef00 e20100000000ef00e2010000000028000000e201 0000ef0000000100200000000000f80707000000 0000000000000000000000000000ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 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ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00 ffffff00ffffff00ffffff00ffffff00ffffff00030000000000} \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {Построенный график рис.4. позволяет лучше представить динамику температуры объе кта в } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {каждой точке пространства. } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {В качестве проверки вычислим значение решения } {\cs249\f0\fs24\cf2\i \b0 \ul0 Un } {( } {\cs250\f0\fs24\cf2\i \b0 \ul0 x } {, } {\cs251\f0\fs24\cf2\i \b0 \ul0 t } {) в заданной точке: } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\cs16\f2\fs24\cf2\i0 \b \ul0 Un(1,0.5); } \par} {\pard\plain \s266 \f0 \fs24 \cf2 \i0 \b0 \ul0 \qc \sl360 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 {\pict \dn7 \picw2675 \pich495\wmetafile8 0100090000035500000001001C00000000000400000003010800050000000B020000000005000000 0C02EF01730A050000002E0118000000050000000201010000000500000009020000FF001C000000 FB025CFE0000000000009001000000000400001254696D6573204E657720526F6D616E00E8B1F377 F1B1F3772040F577010A6613040000002D0100000C00000021050C00302E32363833353931353133 3F016A00030000000000 } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 > {\cs12\f2\fs24\cf1\i0 \b \ul0 } \par} {\pard\plain \s265 \f0 \fs24 \cf2 \i0 \b0 \ul0 \ql \sl240 \slmult1\sb0\sa0\pagebb0 \li0\fi0\ri0 > {\cs12\f2\fs24\cf1\i0 \b \ul0 } \par} }