数学物理方程与特征函数-05

0 x l,t 0 t0 0 xl
n
n
l
2
,
n
1,2,3,
Xn
Bn sin
n
l
x
T a2 1 T 0
Tn
a
2
n 2
l2
2
1Tn
0
T A e
1
a
2 n 2 l2
2
t
n
n
un
X nTn
A e
1
a
2n2 l2
n
2
t
Bn
sin
n l
x
C e
1
a
2n2 l2
2
n
t
sin
n l
0
n
n
vn (t) Acosa l t B sin a l t
n2
vn (t) 0
v2(t)
a2
4 2
l2
v2 (t)
sin
2a
l
t
2u u(t02,
t)
a
2 2u
x 2 u (l , t )
f( 0,
x,
t
)
,
u ( x,0)
u ( x,0) t
0,
0 x l,t 0
u
vn (t) sin
n 1
n
l
x
t0 0 xl
v n (t )
a2
n 2
l2
2
vn (t)
f n (t)
0
u(x,0)

vn (0) sin
n1
n
l
x0
u ( x,0) t
vn (0) sin
u(x, y,0) (x, y),
0 x p,0 y q,t 0
X X 0
0 y q,t 0 0 x p,t 0
Y Y 0
0 x p,0 y q
T ( )a2T 0
X (0) X ( p) 0 Y (0) Y (q) 0
2 0 X 2 X 0 X Aex Bex
n1
Emn
sin
n
p
x sin m
q
y
q p sin n x sin m y sin r x sin s xdxdy
00
p
q
p
q
q p sin2 n x sin2 m ydxdy p sin2 n xdx q sin2 m ydy pq
00
p
q
0
p
0
q
4
p sin n x sin r xdx q sin m
p,
2u y 2 y,t)
,
0,
u(x,0,t) u(x, q,t) 0,
u(x, y,0) (x, y),
0 x p,0 y q,t 0
0 y q,t 0 0 x p,t 0 0 x p,0 y q
T ( )a2T 0
n
n2 2
p2
,n
1,2,3,
x
u
un
n1
C e
1
a 2 n 2 l2
2
t
n
n1
sin
n l
x
u(x,0)
Cn sin
n1
n
l
x
(x)
Cn
2 l
l (x) sin n
0
l
xdx
2u u(t02,
t)
a
2 2u
x 2 u (l , t )
f( 0,
x,
t
),
0 x l,t 0 t0
u ( x,0)
u(x,0) u(x,t) X (x)T (t)
t
0,
0 x l,t 0
t0 0 xl
X X 0
X (0) 0,
2 0
X 2 X 0 X Aex Bex
0 xl X (l) 0
X (0) A B 0 X (l) Ael Bel
AB0
X 0
0
X 0
X Ax B X 0 B0
2 0
X 2 X 0 X Asin x B cosx
AB0
X (x) 0
当 0 时
X 0 AB0
X (x) Ax B X (x) 0
当 2 0 时 X 2 X 0 X (x) Acosx Bsin x
X (0) A 0 , X (l) Bsin l 0
n n / l, n 1,2,3, n n2 n / l2
X 0
0
X 0
X Ax B
X 0
2 0 X 2 X 0
X Acosx Bsin x
X (0) A 0
X ( p) B sin p 0
n
n
p
n
2 n
n2
p2
2
,n
1,2,3,
Xn
Bn
sin
n
p
x
m
m2
q2
2
,m
1,2,3,
Ym
Cm sin
m
q
y
u(ut0,ya, t2)xu2 u2(
t
x 2
v a 2 2v
t
x 2
u(0,t) etv(0,t) 0
v(0,t) 0
u(l,t) etv(l,t) 0 u(x,0) v(x,0)
v(l,t) 0
v(x,0) (x)
u(ut0,t)a2
2u
x 2 u(l,
t)
u, 0,
u(x,0) (x),
0 x l,t 0 t0 0 xl
u(0,t) X (0)T (t) 0 x
T X a2TX
T a 2T
X X
u(l,t) X (l)T (t) 0 x X (0) 0, X (l) 0
X X 0 T a2T 0
X X 0 0 x l
X (0) 0,
X (l) 0
u(uutx(,0x0,)at)2 0x,2u2u(lx,sti)n
u
vn (t) sin
n1
n
l
x
vn(t)
n1
a2
n2 2
l2
vn (t) sin
n
l
x sin
2
l
x sin
2a
l
t
n
u(x,0) vn (0) sin
n0
l
x0
vn (0) 0
u(x,0)
t
vn (0) sin
n0
n
l
x
0
vn (0) 0
n2
vn(t)
a2
n2
l2
2
vn (t)
X 0
0
X 0
X Ax B
X 0
2 0
X 2 X 0 X Asin x B cosx
X (0) B 0
X (l) Asin l 0
n
n
a
n
2 n
n
l
2
, n
1,2,3,
Xn
Bn sin
n
l
x
u(ut0,t
a2 )
2u x 2 u(l , t )
u, 0,
u(x,0) (x),
X (0) A 0 X (l) B sin l 0
n
n
l
n
2 n
n
l
2
,
n
1,2,3,
n
X n Bn cos l x
u(uutx(,0x0,)at)2 0x,2u2u(lx,sti)n
t
0,
0 x l,t 0
t0 0 xl
Xn
Bn
cos n
l
x, n
0,1,2,3,
u
vn (t) cos
u XT XT a2 X T XT
T a 2T
1 a2
X
X
X X 0
T a2 1 T 0
u(0,t) X (0)T (t) 0 u(l,t) X (l)T (t) 0
X (0) 0, X (l) 0
X X 0,
0 xl
X
(0)
0,
X
(l)
0
u(ut0,t
2
t
vn (t) 0
u 1 1 cost
2u u(t02,
t)
a
2 2u
x 2 u(l , t )
sin 0,
2
l
x sin
2a
l
t
u ( x,0)
0,
u ( x,0) t
0,
0 x l,t 0 t0 0 xl
u(x,t) X (x)T (t)
XT a2 X T
X 1 T
分离变量法
u(ut0,ya, t2)xu2 u2(
p,
2u y 2 y,t)
,
0,
u(x,0,t) u(x, q,t) 0,
u(x, y,0) (x, y),
0 x p,0 y q,t 0
0 y q,t 0 0 x p,t 0 0 x p,0 y q
u(x, y,t) X (x)Y ( y)T (t)
u(x, y,0) (x, y),
0 x p,0 y q,t 0
0 y q,t 0 0 x p,t 0 0 x p,0 y q
u
umn
m1 n1
n
Emn sin
m1 n1
p
x sin m
q
ye(
n
2 p2
2
m2 q2
2
)
a
2t
u(x,
y,0)
(x,
y)
m1
XYT a2 X YT XY T
X Y 1 T 0 X Y a2 T
X
X
Y
Y
1 T ( )
a2 T
X X 0 Y Y 0
T ( )a2T 0
u(ut0,ya, t2)xu2 u2(
p,
2u y 2 y,t)
,
0,
u(x,0,t) u(x, q,t) 0,
0, 0 x l
t XT a 2 X T f
T a2 X f
T
X XT
u
(C n
n 1
cos
na
l
t
Dn
sin
na
l
t) sin
n
l
x
u
vn (t) sin
n 1
n
l
x
vn(t) sin
n1
n
l
x
a 2 n1
n2 2
l2
vn (t) sin
n
l
x
f
(x, t)
mn
Emn
sin
n
p
x sin m
q
ye(
n
2 p2
2
m2 q2
2
)
a
2t
u
umn
m1 n1
m1
n1
Emn
sin
n
p
x sin m
q
ye(
n
2 p2
2
m2 q2
2
)
a
2t
u(ut0,ya, t2)xu2 u2(
p,
2u y 2 y,t)
,
0,
u(x,0,t) u(x, q,t) 0,
n
X n Bn sin p x
m
m2
q2
2
,m
1,2,3,
Ym
Cm
s
in
m
q
y
Tmn
(
n
2
p2
2
m2
q2
2
)a
2Tmn
0
( n2 2 m2 2 )a2t
Tmn Dmne p2 q2
umn
X nYmTmn
n
Bn sin p
m
x Cm sin q
y D e(
n
2 p2
2
m2 q2
2
)
a
2t
a2 )
2u x 2 u(l , t )
u, 0,
u(x,0) (x),
0 x l,t 0 t0 0 xl
X X 0,
0 xl
X
(0)
0,
X
(l)
0
2 0 X 2 X 0 X (0) A B 0 AB0
X Aex Bex X (a) Aea Bea 0
k sin kt p2 k2
1
p2
a2
n2
l2
2
l sin na t na l
l2
vn (t)
l
na
sin
na
l
t
fn (t)
l
na
t 0
fn
(
)
sin
na
l
(t
)d
u(uutx(,0x0,)at)2 0x,2u2u(lx,sti)n
t
0,
0 x l,t 0
t0 0 xl
u(x,t) X (x)T (t)
a
2
n1
n2 2
l2
vn (t) sin
n
l
x
n1
fn (t) sin
n
l
x
f
( x, t )
n1
f n (t) sin
n
l
x
fn
(t)
2 l
l 0
f (x, t) sin n
l
xdx
v n (t )
a2
n 2
l2
2
vn (t)
f n (t)
0
2u u(t02,
t)
a
2 2u
X
a2
T
X X 0
T a2T 0
u(0,t) X (0)T (t) 0 u(l,t) X (l)T (t) 0 X (0) 0, X (l) 0
X X 0, 0 x l
X (0) 0,
X (l) 0
2u u(t02,
t)
a
2 2u
x 2 u(l , t )
sin 0,
2
l
x sin
2a
l
t
u ( x,0)
0,
u ( x,0) t
0,
0 x l,t 0
t 0
0 x
l
X X
(0)
X
0,
0,
当 2 0 时 X 2 X 0 X (x) Ae x Be x
0 xl X (l) 0
X (0) A B 0 , X (l) Ael Bel 0
n1
n
l
x
0
vn (0) 0
vn (0) 0
vn (t) Vn ( p) f n (t) Fn ( p)
vn(t) p2Vn ( p) pvn (0) vn (0) p2Vn ( p)
p 2Vn ( p)
a2
n 2
l2
2
Vn ( p)
Fn ( p)
0
1
Vn ( p) p 2 a 2 n 2 2 Fn ( p)
n0
n