H* ----------------------------- *L
H* pde-fsx2.nb
*L
H* ----------------------------- *L
Clear@"Global`*"D
H*
hHxL = 1
, sine series
L = Pi
*L
h@x_, n1_D := Sum@H4  HPi H2 n - 1LLL Sin@H2 n - 1L xD, 8n, n1<D
Plot@h@x, 5D, 8x, - 2 Pi, 2 Pi<D
1.0
0.5
-6
-4
2
-2
4
6
-0.5
-1.0
Plot@h@x, 30D, 8x, - 2 Pi, 2 Pi<D
1.0
0.5
-6
-4
2
-2
4
6
-0.5
-1.0
N@h@Pi  2, 30DD
0.989393
0.9893926136945959`
H* Pi4 = 1 -13 + 15 -17 .......
*L
0.989393
H*
fHxL = x^2
*L
f@x_, n1_D := Pi ^ 2  3 + Sum@4 H- 1L ^ n n ^ H- 2L Cos@n xD, 8n, n1<D
2
pde-fsx2.nb
Plot@f@x, 4D, 8x, - 2 * Pi, 2 * Pi<D
8
6
4
2
-6
-4
2
-2
4
6
4
6
Plot@f@x, 29D, 8x, - 2 * Pi, 2 * Pi<D
8
6
4
2
-6
-4
2
-2
f@0D
f@0D
N@%D
[email protected]
0.004750620377440562`
0.00475062
H*
gHxL = x for
x in H0 piL, and 0 for x in H-pi, 0L
*L
g@x_, n1_D := Pi  4 + Sum@HH- 1L ^ n - 1L Cos@n xD  Hn ^ 2 PiL - H- 1L ^ n Sin@n xD  n, 8n, n1<D
pde-fsx2.nb
Plot@g@x, 5D, 8x, - 2 Pi, 2 Pi<D
2.5
2.0
1.5
1.0
0.5
-6
-4
-2
2
4
6
2
4
6
Plot@g@x, 50D, 8x, - 2 Pi, 2 Pi<D
3.0
2.5
2.0
1.5
1.0
0.5
-6
-4
-2
g@Pi, 50D
22 954 154 121 234 309 743 026 936 868 465 336 261 378
Π
+
9 378 981 288 376 284 666 018 122 686 288 718 555 625 Π
4
N@%D
1.56443
N@Pi  2D
1.5708
H* plucked-string shape
L = pi
*L
ps@x_, n1_, k_, t_D :=
H4  Pi ^ 2L Sum@ HH- 1L ^ Hn + 1LL  H2 n - 1L ^ 2 Exp@- HH2 n - 1L ^ 2L k tD Sin@H2 n - 1L xD, 8n, n1<D
3
4
pde-fsx2.nb
Plot@ps@x, 24, 0.01, 0.0D, 8x, - Pi, Pi<D
0.4
0.2
-3
-2
1
-1
2
3
-0.2
-0.4
mesh@a_, b_, n_D := Table@a + Hi - 1L * Hb - aL  n, 8i, n + 1<D
mesh@1, 2, 5D
6
7
,
:1,
5
8
,
5
9
,
5
, 2>
5
Plot@ps@x, 20, 0.02, mesh@0, 50, 10DD, 8x, 0, Pi<D
0.5
0.4
0.3
0.2
0.1
0.5
H* Mixed BC
L = pi
1.0
1.5
2.0
2.5
*L
mbc@x_, n1_, k_, t_D := Sum@H2  Hn - 1  2L + H2  PiL * H- 1L ^ n Hn - 1  2L ^ H- 2LL
Exp@- HH n - 1  2L ^ 2L k tD Sin@Hn - 1  2L xD, 8n, n1<D
3.0
pde-fsx2.nb
Plot@mbc@x, 30, 0.02, 0D, 8x, - 2 Pi, 2 Pi<D
3
2
1
-6
-4
2
-2
4
6
-1
-2
-3
Plot@mbc@x, 40, 0.02, mesh@0, 50, 10DD, 8x, 0, Pi<, PlotRange ® 80, Pi<D
3.0
2.5
2.0
1.5
1.0
0.5
0.0
"
0.5
1.0
1.5
2.0
2.5
3.0
5