¿¾½ ©£¢(1) 2 ² «Fourier ¤ ¡¥¹ ³·« ¬. ¹ ¹: ¦

321
1.
& Fourier (1)
; (3) sinxax (a > 0)
(2)e−|x|
(1)sgnx;
2
iλ ;



 π, a > |λ|
π
(3)
2 , a = |λ|


 0, a < |λ|
2
(2) 1+λ
2;
& Fourier -'+.
(1) f (x) = e
(β > 0), -
2.
−β|x|
Z
+∞
cos ̟x
π −β|x|
d̟ =
e
2
2
β +̟
2β
0
(2)
f (x) =
(
sin t, |t| ≤ π
0,
|t| > π
Z
0
(1)F [f (x)] =
3.
1
2β
β 2 +λ2 ;
2
F −1 [e−a
(
0,
x < 0;
ex , x ≥ 0.
√
(
( π2 ) sin t, |t| ≤ π
0,
|t| > π
sin λπ
(2)F [f (t)] = − 2i1−λ
2 .
λ2 t
,f2 (x) =
f1 (x) ∗ f2 (x) =
5.
-
sin λπ sin λt
dλ =
1 − λ2
-
f (x) =
-:
4.
+∞
,
]=
(
x2
1
√ e− 4a2 t
2a πt
sin x, 0 ≤ x ≤
x ∈ (−∞, 0) ∪ ( π2 , ∞)
0,



 0,
1
2 (sin x − cos x
x
1 −x
(1 + e 2 ),
2e



1
2
2πe− 2 x =
Z
+∞
−∞
1
π
2
+e
−x
x≤0
), 0 ≤ x ≤
x>
e−|x−τ |ϕ(τ )dτ
π
2
π
2
ϕ(x) = p
6.
π
2 (2
2
1
− x2 )e− 2 x .
.%"
(
ut − a2 uxx = 0
, −∞ < x < ∞, t > 0
u|t=0 = cos x,
u(x, t) =
7.
1 − a12 t
2e
cos x
,Æ$ Fourier
/%"
(
ut − a2 (uxx + uyy ) = 0
u|t=0 = ϕ(x, y),
u(x, y, t) =
8.
RR +∞
1
4πa2 t
−∞
(x−ξ)2 +(y−η)2
4a2 t
dξdη
& Laplace (2)e−at sin kt;
(1) sin kt;
(1)
k
p2 +k2 (Re(p)
> 0);
(3) (p22kp
+k2 )2 (Re(p) > 0);
9.
ϕ(ξ, η)e−
, −∞ < x < ∞, t > 0
e2t
; (4) √ .
t
(3)t sin kt
k
(2)
2
2 (Re(p) > −a);
q (p+a) +k
π
(4) p−2 (Re(p) > 2)
, Laplace !&*
(1)
Z
+∞
te
−2t
(2)
dt;
0
(1)
10.
1
4;
Z
+∞
0
1
(2) p+1
−
1
p+2 (Re(p)
e−t − e−2t
dt
t
(3) π2
> −1);
L[f (t)] = F (p), -( #)0
L[tn f (t)] = (−1)n
11.
; (3)
,&%"




∂2 u
∂x∂y
dn F (p)
, n = 1, 2, · · ·
dpn
= 1,
x > 0, y > 0,
u|x=0 = y + 1,


 u|
y=0 = 1.
2
Z
0
+∞
sin2 t
dt
t2
u(x, y) = xy + y + 1
12.
,&%"
u(x, t) = 3e

2

, 0 < x < l, t > 0

 ut − a uxx = 0
u(0, t) = u(1, t) = 0,


 u(x, 0) = 3 sin 2πx.
−4π 2 a2 t
sin 2πx
3