- :
. 1 .
{ T/t = aDT; T½L=f; T(M,0) = j(M) }
S . y D=Dy=h) , const) - Ti j. / / /
. : Ti j /t = (L1+L2)Tij (*)
L1Tij = a(Ti +1,j - 2Tij + Ti-1,j ) / h2
L2Tij = (a/h2)(Ti,j+1 - 2Tij + Ti-1,j)
(*) t tk tk+1 Dt: (Tij k+1 Tijk )/Dt = (L1+L2)T~ij
T ~ij = (1/Dt) òtktk+1 Tij dt / . T ~ - -. / 2 :
1) T ~ij1=Tijk : Tk+1ij=Dt(L1+L2) Tkij+ Tkij k=0 - - - k+1. - - .. -. 1- : . (Dta) / h 2 < 1 . Dt/h2. . .
2) T ~ij = Tk+1ij - .
T k+1ij - Dt(L1+L2)T k+1ij = T kij (" t, h)
k=0 , T 1ij . .
- . .
. . . . . - . . .. ..
{ DU=0 S, UçL = f } (*)
/ () :
{ V/t = DV (**)
{VçL = f: V(M,0)=g(M)
g(M) . - - f. t ¥ (**) (*) .. V(M,t)U(M), t ¥
(*) (**) - j(M,t) = V - U ..: { j/t = Dj; (3*)
{ jçs= 0: j(M,0) = g(M)-U(M)
j (M,0)- -, 0.
(3*) :
j (M,t) = S¥k=1Cke --ltjk(M), jk lk - { Djk = -lkjk
{ jk|L = 0
, Ck = òS j(M,0) jk(M)dS
., lk -
l1 £ l2 £ l3 £ - jk . -: òS jk jndS = {1, k = n
{0, k ¹ n.
Use - - - j (M,t) òS j2dS = åkC2k exp(-2lkt) £ exp(-2l1t)åkCk2 <¥
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t¥ òS j2dS0 , j(M,t)0 .. V(M,t)U(M) t¥.
exp- .
.
8.5 () -
-. R, 0.
t=0 - - T1 < T0. - - : T(M,t)=? : . Þ / T(r,t).
/ . - , t0- - T(R,t) = T1
t0-: DT = - (1/a)T/t = 0
: 1/r2 (/r)(r2 T/r) - 1/a(T/t) = 0
- - j(r,t)=T(r,t)-T1
{(1/r2)/r(r2 j/r) - 1/a(j/t)=0
| j(R,t) = 0
| j(r,0) = T0 - T1
{j (r,t) < ¥
. - -, . . - . - , . -.
1/r (/r)(r2 j/r) - 1/a(/t)(rj) = 0
2/r2(rj) - 1/a(/t)(rj) = 0. / -. y(r,t)=rj. . . . -.
{ (2y/r2) - 1/a(y/t) = 0
{ y(R,t) = 0: y(r,0) = r(T0-T1)
{ y(r,t) / r < ¥.
. . : y(r,t)=R(r)Q(t)
R``-mR = 0: Q`-maQ = 0 - m=const . : Q(t)=C1emat. - Þ m = -l2. :
Q(t)=C1exp(-l2at); R(r) = C2 sin lr + C3 cos lr. , - R(r)/r < ¥ : C3 = 0. . .: R | S = 0 sin lR0=0 lk = kp/R0 , k = 1,2. . yk :
yk (r,t) = Dk exp(-l2k at)sin lk r. . : y(r,t)=åk Dk exp(-lk2at)sin lkr. . . .: r (T0 - T1) = åk Dk sin rlk
. Dk: Dk = (T0 - T1) 2R0 / [kp(-1)k-1]
/ T(r,t) .:
{ T(r,t) = T1 + 2R0(T0-T1)/(pr) *
*åk[ (-1)k-1/ k]exp(-lk2at) sin lk r
{ lk = kp / R0.
o . - . - .
- . - - . . . . . , . . / . - -:
{Dj - 1/a(j/t)=f , - .
{j (M,0) = j0 (M) . -
{j (M,t)|MÎt = j1(M,t). . - . t - . , use - . -.
{Dj 0 - P/a (j0) = F0 - j0(M)/a;
{j0(M,P)|MÎS = j01(M,P).
, . . . -. . . , . . - ò-:
j (M,t) = 1/2pi òs-i¥s+i¥j0(N,P)eptdP Re p =s
ò- .
---------------
---------------
.
. , . . . . . - ò- . . . :
{ T1o(x,p)=(1/p) exp{-x Öp/a}
{. . . . . f(t) = T(0,t)
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{ To(x,p)=Fo(p) exp{-x Ö(p/a)}
. exp : To(x,t)=pFo(p)To(x,p)
- ò- .
T(x,t) = f(0)T1(x,t) + ò0t T1`(x,t)F(t-t)dt =
= T1(0,t) f(t) + ò0t f `(t) T1(x, t-t)dt =
= ò0tf `(t)Erf [ x / 2Öa(t-t)| ]dt. .