热传导方程.pdf

1、 c .Z 1f u = u(t;x1;x2; ;xn) .Z / T ut = k4u k . B bn = 1 H Ls N Z n = 3 H ZbN V H9 C Zbc | K ZBt QaZE TbB n = 3 .Z#MHqb = p .ZCauchy5(95)FourierMEb p .ZH5s M Eb .Z #5Bb .ZLi-Yau Hanarck Tb T+ s 1Tb B ) HWt t k H .ZH5#Cauchy5v b c) KB bWM Z bWM f V ) V AF. JohnIuPartial Difierential Equationsv, Spri

2、nger-Verlag, 1982. x1: .Z# Hq | I .Z# MHqb 1.1Z Ba .Z I bW D .5bf u(t;x;y;z)V UD(x;y;z)# H Ytb G . Fourier L p kl H dt =ELZ_n VB kl dS dQ w dSELZ_Z_ un1 dQ = k(x;y;z)undSdt; (1.1) k(x;y;z)(x;y;z) . |b(1.1) T| y 9 VB _ByNdQuns|b 1 D = |B w S u:(1.1) TV H Yt1t2 N w Q = Z t2 t1 ZZ S k(x;y;z)undS dt; (1

3、.2) unV UuS ELZ_nZ_ b P =? 3M HWW(t1;t2)Vu(t1;x;y;z)M u(t2;x;y;z) l ZZZ ”(x;y;z)(x;y;z)u(t2;x;y;z)u(t1;x;y;z)dxdydz; ”1 byN Z t2 t1 ZZ S kundSdt = ZZZ ”u(t2;x;y;z)u(t1;x;y;z)dxdydz: (1.3) L !f u1M x;y;z = 1t B Green T V(1.3) T Z t2 t1 ZZZ x kux + y kuy + z kuz dxdydzdt = ZZZ ” Z t2 t1 u tdt dxdydz;

4、s Z t2 t1 ZZZ ”ut x kux y kuy z kuz dxdydzdt = 0: (1.4) t1t2 u i ”ut = x kux + y kuy + z kuz : (1.5) (1.5) Td ( _: :.:Z:b T ( N Hk”# ( :k=” = c2 u t = c 2 2u x2 + 2u y2 + 2u z2 : (1.6) T I = ( Y Q)5 . Z(1.5)w1 I n Yb ! HW =8 3 2 F(t;x;y;z)5N H Z Z t2 t1 ZZ S kundSdt+ Z t2 t1 ZZZ F(t;x;y;z)dxdydzdt =

5、 ZZZ ”u(t2;x;y;z)u(t1;x;y;z)dxdydz: M(1.5) .Z ”ut = x kux + y kuy + z kuz +F(t;x;y;z): (1.7) MN HZ(1.6) u t = c 2 2u x2 + 2u y2 + 2u z2 +f(t;x;y;z); (1.8) f(t;x;y;z) = F(t;x;y;z)” : (1.9) (1.6): :Q: :.:Z:7(1.8):d: :Q: :.:Z:b =a Z s0 V9 Zb 8 A8 1i 8 b/ y I V sZ b Z .Z BVb1| V ? p .V ? pT 1 Z 4 b w .Z

6、V T Fourier p o pZ (1.1)Z(1.3) Tb I n V H M p o p dm = (x;y;z)UndSdt; (1.10) Z t2 t1 ZZ S UndSdt = ZZZ U(t2;x;y;z)U(t1;x;y;z)dxdydz; (1.11) UV U idmV U kl H dt =ELZ_nVB kl dS T (x;y;z): : : |(1.1)a(1.3)cl Mb 3 |(1.10)a(1.11)(1.1)a(1.3)1 ?C T d b I .Z QauaksYM V maUa 7C(1.3) Ty 0” 5M 1b Z V U t = x U

7、x + y Uy + z Uz : (1.12) T : = c25 Z(1.12) .Z(1.6) M Tb 1.2Hq V A T IH ( ) S H Y V H Yb .Z K1 H9KB5 X SHqHHq/ p5 b 1 SHq4E u(0;x;y;z) = (x;y;z); (1.13) (x;y;z)Xf V Ut = 0 H Ysb / IHHq4Eb c s f ) B HHqKe fV XHq Vr T u(t;x;y;z)j(x;y;z)2S = g(t;x;y;z); (1.14) SV UHg(t;x;y;z) l0;TS Xf T B bHHq .Z:B: :H:

8、H:q:DirichletH:H:qb = HHq I 6B f V V 7 V 9 V HW = V Q Xb Fourier p dQ = kundSdt VHHq L= V UuV E_ X u n flfl flfl (x;y;z)2S = g(t;x;y;z); (1.15) 4 unV UuHS ELZ_nZ_ 7g(t;x;y;z) l0;T S Xf bHHq .Z:=: :H:H:q:NeumannH :H:qb HHq Ib 6B1 f ? I()1u1 IV u iMbu1X HHHq4EA 6B . L p d pV I 1 1 dQ = (uu1)dSdt; (1.1

9、6) 1 : : |b I V IV S V I = A Fourier p 7V1Z A5 d p % yN kundSdt = (uu1)dSdt; u+kun = u1: k yNHHq V u n + u flfl flfl (x;y;z)2S = g(t;x;y;z); (1.17) unV UuHS ELZ_nZ_ 7g(t;x;y;z) l0;T S Xf X bHHq .Z: : :H:H:qb oZM1 HHq V sYB T Bb T I8v7 HW lS =M f HHq 3Y V H V I j b W75M:Cauchy:5N H SHq u(0;x;y;z) = (

10、x;y;z) (1 x;y;z 1): (1.18) N +YoZ f .Z5 SHq ?Bb 5 a f / .Z bWUSM Vh b 8 ( % HL ! 9 3 L !s B M5f uUSx HWt1 :B: :.: :Z:u t = c 2 2u x2: (1.19) I . V:=: :.:Z: u t = c 2 2u x2 + 2u y2 : (1.20) .Z V 4 Cauchy5H5b Z ) b b 5 1.B ( %LL ! B MV ? 3 iV? p dQ = (uu1)dSdt: L ! 1 ” . k kN Hu Zb 2. kw V sZb 3. r =%

11、 vbb % 1bQ(t)V U 8 % Q0 S H Y % 5dQdt = flQ fl bL ! r1 ” . k p u Zb 6 x2: Cauchy5 FourierMs 1 sMB s#/ Es1b FourierM p . ZCauchy5b n5 TwZEVr T YV pCauchy5b 2.1 FourierM il(1;1) V O Vf f(x)l FourierM Ff() , g() = Z 1 1 f(x)eixdx: (2.1) f(x)(1;1) V O V H FourierM ib 6BZ il(1;1) V O Vf g()l Fourier IM f

12、(x) = 12 Z 1 1 g()eixd: (2.2) Y g()f(x):FourierM:Ff() H9:f() 6BZ f(x)g():Fourier I:M:F1g(x)b Y L f(x)(1;1) V O V H FourierMFf()i i O V FourierM IMf(x)9 F1 Ff = f: yN V Fourier IM L FourierMo Ipb FourierM p .ZCauchy5# p Bt Z 5 / Bt1FourierMb L !/ T C FourierM9 ib FourierM/ +Z 2.1 FourierM LM9 i fi1fi

13、2#f f1f2 Ffi1f1 +fi2f2 = fi1Ff1+fi2Ff2: (2.3) l2.1 Tf1(x)f2(x)x 2R H f(x) = Z 1 1 f1(xy)f2(y)dy (2.4) 1 i5f(x)f1(x)f2(x): :b f1(x)f2(x) f(x)Y:f1 f2bA f1 f2 = f2 f1; (2.5) Vb 1 FourierM 2.2 f1(x)f2(x) FourierMf1(x)f2(x)FourierM Ff1 f2 = Ff1Ff2: (2.6) i Ff1 f2 = Z 1 1 eixdx Z 1 1 f1(xy)f2(y)dy #f1f2(1

14、;1) VasQ V Y L Ff1 f2 = Z 1 1 f2(y)dy Z 1 1 eixf1(xy)dx = Z 1 1 f2(y)dy Z 1 1 f1()ei(y+)d = Z 1 1 eiyf2(y)dy Z 1 1 f1()eid = Ff1Ff2: T 1(2.6) Tb8b 2.3 g1()g2() Fourier IMg1()g2()Fourier IM 2 F1g1 g2 = 2F1g1F1g2: (2.7) 2.2 F1g1 g2 = 12 Z 1 1 eixd Z 1 1 g1()g2()d = 12 Z 1 1 g2()d Z 1 1 g1()eixd = 12 Z

15、 1 1 g2()d Z 1 1 g1()ei(+)xd = 12 Z 1 1 eixg2()d Z 1 1 eixg1()d = 2F1g1F1g2: 2 T 1(2.7) Tb8b 2.3 2.4 f1(x)f2(x)FourierMf1(x)f2(x)FourierM (2)1 Ff1 f2 = 12Ff1Ff2: (2.8) : g1 = Ff1; g2 = Ff2: (2.7) T F1g1 g2 = 2F1g1F1g2 = 2F1Ff1F1Ff2 = 2f1 f2: SFourierM V g1 g2 = 2Ff1 f2: T 1(2.8) Tb8b 2.5 Tf(x)f0(x)F

16、ourierMii Ojxj!1 Hf(x) ! 05 Ff0 = iFf: (2.9) 9 Ff0() = Z 1 1 eixf0(x)dx = eixf(x)“flflx=1x=1 + Z 1 1 ieixf(x)dx = i Z 1 1 eixf(x)dx = iFf(): 2.6 Tf(x)xf(x)FourierM (i5 Fixf(x) = ddFf: (2.10) YV9 V Fixf(x) = Z 1 1 (ix)f(x)eixdx = dd Z 1 1 f(x)eixdx = ddFf: 3 8b Vl1M FourierM Ff() = g(1; ;n) = Z 1 1 Z

17、 1 1 ei Pn k=1 kxkf(x1; ;xn)dx1dxn: (2.11) MFourier IMl f(x1; ;xn) = (2)n Z 1 1 Z 1 1 ei Pn k=1 kxkg(1; ;n)d1dn: (2:11a) 1M f FourierM9 2.12.6 bW%) =b :2.1 Tf(x) l(1;1) Lf 5 FourierM9 (1;1) Lf b Y L cosx f 7sinx f yN Ff = Z 1 1 eixf(x)dx = Z 1 1 (cosx+isinx)f(x)dx = Z 1 1 f(x)cosxdx: TV Ff() B Lf b

18、:2.2FourierMl V ) f 1F BtKHqy7 sB Kb 7 9 V: 0 Hs(2.20) Vrf u(t;x) Z(2.14)b n5 i/ f 1 2cpte (x)24c2t (2.29) M tx7( | A )t 0 H Z(2.14)byN t 0 HCZ(2.14) V(2.20) Ts|/ pZE 9 b(2.20) TsK k YVs| p V ?A s|/ p s B l B 4xB |(2.20) T xBQ p s 1 2cp Z 1 1 (x)() 2c2t32 e (x)24c2t d; t t0 0( t0 i )S =9 B l byNt 0

19、H u(t;x) x = 1 2cp Z 1 1 (x)() 2c2t32 e (x)24c2t d; xBQ p ?YVs|b V(2.20) 9 ?s|/ p 7byNt 0 Hs(2.20) Vrf u(t;x) Z(2.14)b :/1 (2.20) T lf u(t;x) SHq(2.15)1 x0t ! 0;x ! x0 Hu(t;x) ! (x0)bN i 0 Vs al 0 Pjxx0j6 ;t 6 H ju(t;x)(x0)j6 : Y L (2.20) T 7 = x2cpt u(t;x) = 1p Z 1 1 (x+2cpt)e2d: (2.30) 6BZ (x0) V

20、(x0) = 1p Z 1 1 (x0)e2d: (2.31) yN u(t;x)(x0) = 1p Z 1 1 (x+2cpt)(x0)e2d: (2.32) 0 |N 0v P 1p Z 1 N e2d 6 6M; 1p Z N 1 e2d 6 6M: (2.33) 7 %N(x) Vs 0 Pjxx0j6 ;0 0); (2) eajxj (a 0); (3) x(a2 +x2)k; 1(a2 +x2)k (a 0;k1 ): 2. f(x)(1;1) V HFf f b 3.FourierM p .ZCauchy5 8 : ut = c2(uxx +uyy +uzz); ujt=0 = (x;y;z): 4. (2.27) lf d QZ(2.12)# SHq(2.13)b 5. p .Z(2.14)Cauchy5X (1) ujt=0 = sinx; 8 (2)E pL .Z(2.14)L ! 8 : u(0;x) = (x) (0 x : u t = c 2 2u x2 + 2u y2 ; ujt=0 = 1(x)2(y) b 8.= .ZCauchy5 8 : u t = c 2 2u x2 + 2u y2 ; ujt=0 = (x;y) Vr T u(t;x;y) = 14c2t Z 1 1 Z 1 1 (;)e(x) 2+(y)2 4c2t dd: 9 x3: