Tutorial 3-10 - FDFDSFSAFSFSDFSDFSAFASDFSFSADFSDFDS FSDF DSD DF SDF SDF ASF DSAF PDF

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FDFDSFSAFSFSDFSDFSAFASDFSFSADFSDFDS FSDF DSD DF SDF SDF ASF DSA...

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∂ 2u , ∂x2 ∂t ∂ 2u ∂u (b) = 2, ∂t ∂x ∂ 2u ∂u (c) = 2, ∂x ∂t = 81

u(0, t) = u(1, t) = 0, ∂u ∂u (1, t) = 0 (0, t) = ∂x ∂x u(0, t) = 10,

∞ X k=1

2

1

Z

+

1

−1 Z 1

| sin(πx)| cos(kπx)dx = 2

u(x, 0) = sin(πx).

u(10, t) = 30,

u(x, 0) = x(10 − x).

  bk sin(kπx) exp −81k2 π 2 t .

∞ X k=1

Z

u(x, 0) = sin(2πx) − sin(9πx).

  ak cos(kπx) exp −k2 π 2 t .

1

sin(πx) cos(kπx)dx 0

1  cos((k + 1)πx) cos((k − 1)πx) 1 =2 (sin((k + 1)πx) − sin((k − 1)πx)) dx = − + (k − 1)π (k + 1)π 0 2 0 1 − cos((k + 1)π ) cos((k − 1)π) − 1 2(1 + cos(kπ)) 4 = + = = (k + 1)π (k − 1)π (1 − k2 )π Z 1 Z 1 1 4 | sin(πx)|dx = 2 a0 = sin(πx)dx = π 1 −1 0 ∞ X  2 2 2 u(x, t) = + cos(kπx ) exp −k π t . (1 − k2 )π π

∂t

=

∂ 2w , ∂x2

w(0, t) = 0,

w(10, t) = 0,

w(x, 0) = u(x, 0) − us (x) = −x2 + 8x − 10.

∞ X

  2 2  k π t kπx bk sin exp − . 10 100 k=1   Z 2 10 kπx 2 dx = . . . bk = (−x + 8x − 10) sin 10 10 0      20 π 2 3(−1)k − 1 k2 − 20 (−1)k − 1 ... = 3 3  π2 k       2 2  ∞ X 20 π 3(−1)k − 1 k2 − 20 (−1)k − 1 kπx k π t u(x, t) = 10 + 2x + sin exp − 3 3 π k 100 10 k=1 

∞ X n=1

4L2

X

=





 (2n − 1)πx sin . 2L

1 T′ = −λ. α2 T

√ √ X = A cos( λx) + B sin( λx). √ X(0) = A = 0 ⇒ X = B sin( λx). √ √ √ (2n − 1)π X ′ (L) = B λ cos( λ) = 0 ⇒ λ= n = 1, 2, 3, . . .   2L (2n − 1)π x n = 1, 2, 3, . . . X = B sin 2L

∂t

= D∆C

=⇒

0 = D∆C

C(r = b) = 0

∂ r ∂r

  ∂C r = 0. ∂r

ln

∂ r 2 ∂r

 a ln b

r  b

  2 ∂P r =0 ∂r ∂P r2 = C1 ∂r P (r) = C2 −

r

.

.

C1 r

6πµr

6πµ(5 ×

=⇒

10−7 cm)

kB T = 2.5 × 103 cm3 s−1 6πµ

1 1 = 2.5 × 103 cm3 s−1 = 1 × 10−4 cm 2 s−1 . 6πµ r 2.5 × 107 cm

h2

h2new

∂t

k

1 = 2



=D

=

<

1 2

k 1 1 < . 2 −4 2 h 10

∂ 2u . ∂x2

wp+1,q − 2wp,q + wp−1,q wp+1,q+1 − 2wp,q+1 + wp−1,q+1 + 2 h h2



∞ X

ck eikπx/L .

k=−∞

2

+

∞ X k=1

ak cos



kπx L



+ bk sin



kπx L



  k k2 3 w + kw˙ + w¨ + O(k ) − w = w˙ + w¨ + O(k2 ). 2 k 2     3 2 h ′′ h ′′′ h2 ′′ h3 ′′′ 4 4 ′ ′ w + hw + w + w + O(h ) + k w˙ + hw˙ + w˙ + w˙ + O(h ) 6 2h2 6 2 2   2 3   h h −2 w + kw˙ + O(k2 ) + w − hw′ + w′′ − w′′′ + O(h4 ) 6 2      2 3 h ′′ h ′′′ 1 h2 ′′ h3 ′′′ ′ 4 2 4 ′ +k w˙ − hw˙ + w˙ − w˙ + O(h ) + O(k ) + 2 w + hw + w + w + O(h ) − 6 6 2 2h 2   2 3 h h 2w + w − hw′ + w′′ − w′′′ + O(h4 ) 6 2   1  1  2 ′′ h w + kh2 w˙ ′′ + O(h4 ) + O(kh4 ) + O(k2 ) + 2 h2 w′′ + O(h4 ) = 2 2h 2h 1 = w′′ + kw˙ ′′ + O(k2 ) + O(h2 ) 2 1 w¨ + O(k2 ) = w′′ + kw˙ ′′ + O(k2 ) + O(h2 ) 2 2

k

=

eiβ(p+1)h ξ q+1 − 2eiβph ξ q+1 + eiβ(p−1)h ξ q+1 eiβ(p+1)h ξ q − 2eiβph ξ q + eiβ(p−1)h ξ q + 2h2 2h2

eiβh ξ − 2ξ + e−iβh ξ eiβh − 2 + e−iβh + k 2h2 2h2 2ξ(−1 + cos βh) + 2(−1 + cos βh) ξ−1 = 2h2 k k ξ − 1 = 2 (ξ(−1 + cos βh) + (−1 + cos βh)) h     −(h2 /k) + (1 − cos βh)  (h2 /k) − 2 sin2 (βh/2)  ≤ 1.  = |ξ| =  (h2 /k) + (1 − cos βh)   (h2 /k) + 2 sin2 (βh/2) =

∂u ∂ 2u = ∂t ∂x2

k

=

wp+1,q+1 − 2wp,q+1 + wp−1,q+1 h2

  k2 k 3 w + kw˙ + w¨ + O(k ) − w = w˙ + w¨ + O(k2 ). k 2 2   3 2 h h w + hw′ + w′′ + w′′′ + O(h4 ) + O(k) 2 2 h 6      h2 ′′ h3 ′′′ 2 ′ 4 −2 w + kw˙ + O(k ) + w − hw + w − w + O(h ) + O(k) 2 6  1  2 ′′ = 2 h w + O(h4 ) + O(k) h = w′′ + O(k) + O(h2 )

k

=

eiβ(p+1)h ξ q+1 − 2eiβph ξ q+1 + eiβ(p−1)h ξ q+1 h2

eiβh ξ − 2ξ + e−iβh ξ k h2 2ξ(−1 + cos βh) ξ−1 = 2h2 k ξk ξ − 1 = 2 (−1 + cos βh) h   2   (h /k)  ≤ 1. |ξ| =  2 (h /k) + (1 − cos βh)  =...