задачи_nl
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zADA^A 1 oPREDELITX, KAKOE IZ URAWNENIJ QWLQETSQ LINEJNYM, KWAZILINEJNYM, NE KWAZILINEJNYM.
1.1a) uxx + uyy + u2 = 0,
b)uxx + x2uxy + u = 0,
c)yuxx + uy + x2uyy = 0.
1.2a) u2x + uxx + uyy = 0,
b)u2xx + uyy + xux = 0,
c)uxy ux + uxx + u = 0.
1.3a) u2 + u2xx + uyuxy = 0,
b)ux2 + uxy + uy x = 0,
c)uux + uxx + uy = 0.
1.4a) uux + uxxx + uy = 0,
b)u;1ux + uxx + uy = 0,
c)(ux + uxx)2 = u2y.
1.5a) uy + u2ux + uxyx = 0,
b)uxx ; uyy = sin u,
c)12 (u2xx + u2yy) + uxxuyy = 8.
1.6a) u2xx ; uuxy + u2yy = 0,
b)3u3xx ; 6uxy + uyy ; 4 = 0,
c)(ux + uy)2 ; 2uxuxx ; 2uy uxx + u2xx = 0.
zADA^A 2 pROWERITX, BUDUT LI UKAZANNYE NIVE FUNKCII RE[E- NIQMI SOOTWETSTWU@]IH URAWNENIJ.
2.1u2xx + (uxx ; 2)uxy ; u2yy = 0,
a)u = x2 + y2,
b)u = 12 (x2 + y2).
2.2uy(ux ; uxxuy) + 3uxy + 12 uxx(ux ; uxxuy)2=0,
u = xy + 3 + x2.
2.3uxxuy(1 ; uy) + uyux ; u = 0, u = xy + x2.
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2.42uyy(u ; xux) = u2y, u = x + y2.
2.52uyy u ; 2xuxuyy ; u2y = 0, u = 2x ; y2.
2.6uyy ux ; uux + u = 0, u = x(1 + e;y).
2.74uyy + u2x ; 4u = 0, u = x2 + e;y.
2.8u2xx ; 4uxy + u2yy = 8,
a)u = x2 + y2,
b)u = 2p2xy.
2.9u2xx + 5u2xy + 6u2yy = 12,
a)u = 12 (x + y)2,
b)u = p3x2.
2.10uxx + uxyuyy + u2yy ; 4uyy = 0,
a)u = 2y2,
b)u = 5xy,
c)u = x.
w ZADA^AH (2.8) - (2.10) POKAZATX, ^TO PRINCIP SUPERPOZICII RE- [ENIJ NE WYPOLNQETSQ DLQ L@BYH RE[ENIJ, NO MOVET WYPOL- NQTXSQ DLQ NEKOTORYH ^ASTNYH RE[ENIJ.
zADA^A 3 dLQ UKAZANNYH NIVE URAWNENIJ ZAPISATX SISTEMU W WARIACIQH. pROWERITX, ^TO SOOTWETSTWU@]IE FUNKCII QWLQ@T- SQ RE[ENIQMI. oPREDELITX TIP URAWNENIQ DLQ UKAZANNOGO RE[E- NIQ.
3.1uxx + uyy ; 2u2xy + uxuy ; 2u2x + 2ux = 0,
u = x + y.
3.2u2xx + uyy + u2x ; uy = 1,
u = sin x.
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3.3 |
2 uxxuxy + xux ; yuy ; x2uxx ; u = 0, |
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u = xw(y) + yx2, w(y) | PROIZWOLXNAQ FUNKCIQ OT y. |
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3.4 |
uxxuxy |
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u = x sin y. |
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3.5 |
uyy + uxyuy + uxx2 + 4 = 0, |
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u = x ln y ; y2. |
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3.6 |
uuxy + uxx + uyy ; 2ux + u = 0, |
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u = ex + sin y. |
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3.7 |
u(uxx + uyy) ; 2u = 0, |
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3.8 |
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3.9 |
2yuyy + u2xx + uy = 0, |
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3.10 |
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dLQ DANNOGO URAWNENIQ NAJTI OPERATOR L I ZAPISATX URAWNENIE W WIDE
Lu(x y) + b = 0
GDE L = a11@2=@x2 + 2a12@2=@x@y + a22@2=@y2. oPREDELITX TIP OPERATORA L.
3.11xuxx + uxy ; 2uxy + u2x + u = 0.
3.12x2uxx + y2uyy + sin u = 0.
3.13xuuxx + uyy = 0.
3.14uxy + (x2 + y2)uxx = 0.
3.15uxx + 2uxy + x2uyy = 0.
3.16uxx + uy + u2 + u2x = 0.
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3.17uy + uux + uxx + u2 = 0.
3.182yuy + u2ux + uxx = 0.
3.192xyuyy + x2uxx + 4uxy + u2 + ux = 0.
3.20(x2 ; y2)uxx + uyy + u3 = 0.
zADA^A 4 kLASSIFIKACIQ KWAZILINEJNOGO URAWNENIQ WTOROGO PO- RQDKA (1.20) W SLU^AE, KOGDA aij ZAWISQT OT u x y. pOSTROENIE KANONI^ESKOJ FORMY.
4.1nAJTI FORMULY PEREHODA OT @=@x, @=@y K @=@ , @=@ W PREOB- RAZOWANII (1.32)
rE[ENIE: |
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D = 'x y ; x'y = x y ; x y :
4.2wYRAZITX ux uy ^EREZ PROIZWODNYE PO . rE[ENIE:
ux = D(u y ; u y )
uy = D(u x ; u x ):
4.3wYRAZITX HARAKTERISTI^ESKIE URAWNENIQ (1.31) ^EREZ PROIZWOD- NYE PO .
rE[ENIE:
y + (1)x = 0 y + (2)x = 0:
4.4wYRAZITX uxx ^EREZ PROIZWODNYE PO . rE[ENIE:
uxx = D(y @ ; y @ )D(y u ; y u ):
4.5wYRAZITX uxy ^EREZ PROIZWODNYE PO .
4.6wYRAZITX uyy ^EREZ PROIZWODNYE PO .
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2nELINEJNOE URAWNENIE TEPLOPROWODNOSTI
tERMODINAMI^ESKIE SWOJSTWA NERAWNOWESNYH SISTEM HARAKTERI- ZU@TSQ POLEM TEMPERATURY u(~r t) W KAVDYJ MOMENT WREMENI t. pONQTIE TEMPERATURY IMEET SMYSL W USLOWIQH LOKALXNOGO TERMO- DINAMI^ESKOGO RAWNOWESIQ, TO ESTX PRI
l u=jruj u=ut: (2.1) zDESX l | SREDNQQ DLINA SWOBODNOGO PROBEGA ^ASTIC SREDY, |
SREDNEE WREMQ SWOBODNOGO PROBEGA, u=jruj u=ut | HARAKTERNYE DLINA I WREMQ IZMENENIQ TEMPERATURY W SREDE, SOOTWETSTWENNO. nEODNORODNOSTX TEMPERATURNOGO POLQ POROVDAET TEPLOWYE PO- TOKI, KOTORYE QWLQ@TSQ WAVNEJ[EJ SOSTAWLQ@]EJ FIZI^ESKIH PRO- CESSOW W SPLO[NOJ SREDE. sWQZX TEPLOWOGO POTOKA S TEMPERATUROJ
USTANAWLIWAET ZAKON fURXE
~q = ; ru: |
(2.2) |
zDESX ~q | TEPLOWOJ POTOK, | KO\FFICIENT TEPLOPROWODNOSTI.
pOLU^ITX URAWNENIE TEPLOPROWODNOSTI W SLU^AE, KOG- DA KO\FFICIENT TEPLOPROWODNOSTI STEPENNYM OBRAZOM ZAWISIT OT TEMPERATURY, A ISTO^NIKI TEPLA OTSUTSTWU@T.
rE[ENIE. uRAWNENIE TEPLOPROWODNOSTI ESTX SLEDSTWIE ZAKONA SO- HRANENIQ \NERGII (I-GO NA^ALA TERMODINAMIKI) I ZAKONA fURXE (2.2). dLQ WYDELENNOGO OB_EMA V , OGRANI^ENNOGO POWERHNOSTX@ , I NA^ALO TERMODINAMIKI IMEET WID
dWdt = ;Q1 + Q2 |
(2.3) |
GDE W | WNUTRENNQQ \NERGIQ OB_EMA V , Q1 | TEPLOWOJ POTOK ^EREZ POWERHNOSTX , Q2 | KOLI^ESTWO TEPLOTY, WYDELQEMOE (PO- GLO]AEMOE) ISTO^NIKAMI, NAHODQ]IMISQ W OB_EME V . tAK KAK W USLOWII ZADA^I PREDPOLAGAETSQ, ^TO ISTO^NIKI OTSUTSTWU@T, TO Q2 = 0. oBOZNA^IM ^EREZ w(~r t) PLOTNOSTX \NERGII W TO^KE ~r, PRI- NADLEVA]EJ OB_EMU V W MOMENT WREMENI t, TOGDA
W (t) = ZV w(~r t)d~r:
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pLOTNOSTX \NERGII NESVIMAEMOJ SREDY ZAWISIT OT TEMPERATURY
u, w = w(u), PO\TOMU
@w
@t
oBOZNA^IM ^EREZ c~ = dw=du OB_EMNU@ TEPLOEMKOSTX SREDY, c~ =cV , | PLOTNOSTX SREDY, cV | UDELXNAQ TEPLOEMKOSTX SREDY. sOOTWETSTWENNO, PROIZWODNU@ OT WNUTRENNEJ \NERGII MOVNO ZA- PISATX SLEDU@]IM OBRAZOM:
dW (t) |
@u |
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= ZV cV @t dV: |
tEPLOWOJ POTOK Q1 ^EREZ POWERHNOSTX SOGLASNO ZAKONU fURXE (2.2) ZAPISYWAETSQ W WIDE
~ |
div( ru)dV: |
Q1 = ;Z rud = ;ZV |
s U^ETOM POLU^ENNYH WYRAVENIJ I-E NA^ALO TERMODINAMIKI (2.3) PRIMET WID URAWNENIQ TEPLOPROWODNOSTI
cV @u@t = div( ru): |
(2.4) |
pO USLOWI@ ZADA^I KO\FFICIENT TEPLOPROWODNOSTI ZAWISIT OT TEMPERATURY PO STEPENNOMU ZAKONU, = 0u , 0 | POSTOQNNAQ. tOGDA ISKOMOE URAWNENIE TEPLOPROWODNOSTI (2.4) ZAPISYWAETSQ SLE- DU@]IM OBRAZOM:
cV @u@t = div( 0u ru): |
(2.5) |
s^ITAQ POSTOQNNYMI cV , , URAWNENIE (2.5) ZAPISYWAETSQ W WIDE
@u@t = a2div(u ru): |
(2.6) |
zDESX a2 = 0=cV | KO\FFICIENT TEMPERATUROPROWODNOSTI.
zAPISATX URAWNENIE TEPLOPROWODNOSTI W SLU^AE, KOG- DA KO\FFICIENT TEPLOPROWODNOSTI STEPENNYM OBRAZOM ZAWISIT OT TEMPERATURY PRI NALI^II ISTO^NIKOW TEPLA S PLOTNOSTX@
F (~r t).
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rE[ENIE. w URAWNENII TEPLOWOGO BALANSA (2.3) W DANNOM SLU^AE Q2 6= 0. oBOZNA^IM ^EREZ F (~r t) | OB_EMNU@ PLOTNOSTX TEPLO- WYH ISTO^NIKOW, TOGDA KOLI^ESTWO TEPLOTY, WYDELQEMOE (POGLO- ]AEMOE) \TIMI ISTO^NIKAMI W NEKOTOROM OB_EME V , RAWNO Q2 = RV F (~r t)dV . uRAWNENIE TEPLOWOGO BALANSA (2.3) S U^ETOM OBOZNA- ^ENIJ PREDYDU]EJ ZADA^I 1 PRIMET WID
@u
ZV cV @t dV = ;ZV div(kru)dV + ZV F (~r t)dV:
pRINIMAQ WO WNIMANIE, ^TO KO\FFICIENT TEPLOPROWODNOSTI ZAWI- SIT OT TEMPERATURY PO STEPENNOMU ZAKONU k = k0u , ko = const, OTS@DA POLU^AEM ISKOMOE URAWNENIE W WIDE
cV @u@t = div(k0u ru) + F(~r t): pRI POSTOQNNYH cV , IMEEM
@u@t = a2div(u ru) + f(~r t)
GDE a2 = 0=cV | KO\FFICIENT TEMPERATUROPROWODNOSTI, f = = F= cV .
zADA^A 3 w ZADA^E O RASPROSTRANENII TEPLA W NELINEJNOJ SREDE
8 ut = a2 |
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> u(x 0) = Q (x) |
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POKAZATX, ^TO PRI ! 0 IZ RE[ENIQ ZADA^I (2.7) SLEDUET RE[E- |
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NIE ZADA^I DLQ LINEJNOGO RASPROSTRANENIQ TEPLA |
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rE[ENIE. rE[ENIE ZADA^I (2.7) POLU^ENO W WIDE |
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8 u(t) 21 |
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u(t) = 02= Q +2 |
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= 82I( ) |
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wY^ISLIM PREDEL I( ) PRI |
! 0. wOSPOLXZUEMSQ SWOJSTWOM ; - |
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FUNKCII 3 PRI jzj ! 1 |
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;(a + z) |
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;(z)eb ln z ! za;b: |
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oTS@DA IMEEM: |
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u^ITYWAQ POLU^ENNOE ASIMPTOTI^ESKOE WYRAVENIE DLQ I( ) PRI! 0, SOOTWETSTWU@]EE WYRAVENIE DLQ 0 MOVNO PREDSTAWITX W SLEDU@]EM WIDE:
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wY^ISLIM PREDEL u(t) PRI ! 0. |
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42( + 2)5
3gRAD[TEJN i.s., rYVIK i.m. tABLICY INTEGRALOW, SUMM, RQDOW I PROIZWEDENIJ. iZD.4. m.: nAUKA, 1963. 1099 S. s. 951.
18
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s U^ETOM POLU^ENNYH ASIMPTOTI^ESKIH PRI ! 0 WYRAVENIJ NETRUDNO NAJTI PREDEL u(x t) PRI ! 0 KAK
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^TO SOWPADAET S IZWESTNYM RE[ENIEM LINEJNOJ ZADA^I (2.8).
zADA^A 4 rE[ITX ZADA^U O RASPROSTRANENII TEPLA PRI RAZOGREWE PO STEPENNOMU ZAKONU.
8 ut = a2 |
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rE[ENIE I]EM W WIDE
u @x@u! t > 0 x > 0 |
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ET OPREDELITX W HODE RE[ENIQ ZADA^I, |
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KAK SLEDUET IZ GRANI^NOGO USLOWIQ (2.9). tAKIM OBRAZOM, WYRAVENIE (2.10) ZADAET AWTOMODELXNOE RE[ENIE URAWNENIQ (2.9).
POLU^IM
ut = |
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19
sOOTNO[ENIE (2.11) PEREHODIT W OBYKNOWENNOE DIFFERENCIALXNOE URAWNENIE, OPREDELQ@]EE FUNKCI@ , ESLI
x00 = const t;1: x0
sLEDOWATELXNO,
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oTS@DA NAHODIM: x0 = v0t, v0 = const. uRAWNENIE (2.11) PRINIMAET WID
1 u0 ; u0 0 = |
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wYBEREM v02 = u0 a , TOGDA DLQ POLU^AEM URAWNENIE |
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1 ; 00 = ( 0 )0: |
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pOKAVEM, ^TO RE[ENIEM \TOGO URAWNENIQ QWLQETSQ FUNKCIQ |
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pODSTAWIM FUNKCI@ (2.13) W (2.12), POLU^IM |
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1 (1 |
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OTKUDA I SLEDUET UTWERVDENIE. |
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tAKIM OBRAZOM, RE[ENIE ZADA^I |
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(2.9) DAETSQ WYRAVENIEM |
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u(x t) = u0t |
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rE[ENIE IMEET SMYSL PRI 0 x x0(t). pRI x x0(t) u = 0. sLEDOWATELXNO, SOOTNO[ENIE x0(t) = v0t PREDSTAWLQET SOBOJ ZAKON DWIVENIQ FRONTA TEPLOWOJ WOLNY.
zADA^A 5 wY^ISLITX PLOTNOSTX POTOKA TEPLOWOJ \NERGII W ZA- DA^E 4.
20