to a difference equation (in
terms of un, un+1, un−1, ∆h …).
Compute A14 where A = (
0
1
1
0
).
Mention two advantages of using the finite element method. (Three lines max. Penalty 50%.)
Obtain an ON set from {1, x} over [0, 2] by the Gram-Schmidt method.2
When solving a diffusion equation by the finite difference
method, what kind of precaution needs to be taken for the
time step size when the space step size is refined to 1/3 of the previous step size ?
What is the equation of the streamline for f(z) = eiz ?
What is the definition of "symmetry" for a linear operator, L ? (Three lines max. Penalty 50%.)
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SOLUTION
1.
(x (x−1) − cx, x) = 0, or
⌠ ⌡
1
0
(x (x−1) − cx) xdx = 0,
from which
c = −
1
4
.
2.
u(x) = A sin
√
λ
x + B cos
√
λ
x, u′(x) = A
√
λ
cos
√
λ
x − B
√
λ
sin
√
λ
x,
From u(0) = 0, B = 0. From u′(π) = 0, cos √{λ} π = 0 from which it follows
√
λ
π =
π
2
+ n π,
so the smallest eigenvalue is
λ1 =
1
4
.
3
||L|| =
max
||e||=1
||Le||, Le =
⎛ ⎜
⎜ ⎝
1
1
2
0
⎞ ⎟
⎟ ⎠
⎛ ⎜
⎜ ⎝
cos θ
sin θ
⎞ ⎟
⎟ ⎠
=
⎛ ⎜
⎜ ⎝
cos θ+sin θ
2cos θ
⎞ ⎟
⎟ ⎠
.
||Le ||2 = (cos θ+sin θ)2 +(2 cos θ)2 = 3+2 cos 2 θ+sin 2θ ≤ 3 +√5.
Hence,
|| L || =
√
3+√5
∼ 2.28825.
4.
un−1 − 2 un + un+1
(∆h)2
= c.
5. Note that AA = I.
A14=(A2)7=I =
⎛ ⎜
⎜ ⎝
1
0
0
1
⎞ ⎟
⎟ ⎠
.
You can also use the eigenvector method.
6.
(1) The matrix to invert is a sparce matrix (tri-diagonal). (2) The coefficient of each base function directly
represents the value at each node.
7
a1 = 1, a2 = x,
As
||a1||2 = (1, 1) =
⌠ ⌡
2
0
12dx = 2,
e1 =
a1
||a1||
=
1
√2
.
Next,
e2′ = a2 − (a2, e1) e1 = x −
⎛ ⎝
⌠ ⌡
2
0
x
√2
dx
⎞ ⎠
1
√2
= x − 1.
||e2′|| = (x − 1, x − 1) =
⌠ ⌡
2
0
(x − 1)2dx =
2
3
,
so
e1 =
1
√2
, e2 =
⎛ √
3
2
(x − 1).
8.
∆t needs to be 1/9 of the previous step size.
9.
f(z)=eiz = ei (x+iy) = e−yeix=e−y(cos x+ i sin x),