Formula Sheet - Final Exam Page 2
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1
2
3L
2
nπx
2
nπx
Wave eq. u
= c
(u
+ u
), u(t = 0) = f , u
(t = 0) = g and 0-BC:
tt
xx
yy
t
Sf (x) =
b
sin(
),
b
=
f (x) sin(
) dx.
n
n
L
L
L
0
n=1
mπx
nπy
u(x, y, t) =
sin
sin
(A
cos µ
t + B
sin µ
t)
The coefficients are those for Fourier series multiplied by 2, and the integra-
m,n
m,n
m,n
m,n
a
b
tion is over [0, L]. Both Cf (x) and Sf (x) are defined for all real x and are
m,n=1
2L-periodic. Cf (x) is even while Sf (x) is odd.
4
mπx
2
2
m
n
µ
= cπ
+
, A
= sin[f, m, n] and µ
B
= sin[g, m, n].
m,n
m,n
m,n
m,n
Sine series on [0, L]:
1 =
sin
,
2
2
a
b
mπ
L
m is odd
d’Alembert’s formula
2
2L
mπx
8L
mπx
2
m+1
The solution to u
= c
u
, u(t = 0) = f , u
(t = 0) = g is
x =
( 1)
sin
,
x(L
x) =
sin
.
tt
xx
t
3
mπ
L
(mπ)
L
m=1
m is odd
x+ct
1
1
u(x, t) =
[f (x + ct) + f (x
ct)] +
g(s) ds
Heat and wave equations on a line [0, L]
2
2c
x ct
= U
(x + ct) + U
(x
ct)
For f (x) defined for x
[0, L], let sin[f, n] and cos[f, n] denote its sine series
left
right
and cosine series coefficients, respectively.
y
1
1
where U
(y), U
(y) =
f (y)
g(s) ds.
2
Heat eq. u
= c
u
, u(x, 0) = f (x) and 0-BC:
left
right
2
2c
0
t
xx
Laplace equation and Poisson equation
nπx
2
2
2
2
c
n
π
t/L
u(x, t) =
b
sin
e
,
b
= sin[f, n].
n
n
L
The solution u(x, y) for ∆u = 0 in Ω = (x, y) 0
x
a, 0
y
b with
n=1
BC u(a, y) = f
(y), u(0, y) = f
(y), u(x, 0) = 0 = u(x, b) is
R
L
2
Heat eq. u
= c
u
, u(x, 0) = f (x) and 0-flux BC:
t
xx
nπ(a x)
nπx
sinh
sinh
nπy
+ ˜ B
nπx
b
b
u(x, y) =
sin
B
2
2
2
2
c
n
π
t/L
n
n
u(x, t) = a
+
a
cos
e
,
a
= cos[f, n].
nπa
nπa
b
sinh
sinh
0
n
n
L
b
b
n=1
n=1
and ˜ B
where B
are coefficients of sine series of f
and f
, respectively.
2
Wave eq. u
= c
u
, u(x, 0) = f (x), u
(x, 0) = g(x) and 0-BC:
n
n
R
L
tt
xx
t
The eigenfunctions and eigenvalues for
∆u = λu in Ω with 0-BC are
nπx
cnπt
cnπt
mπx
nπy
u(x, t) =
sin
A
cos
+ B
sin
n
n
2
2
φ
= sin
sin
,
λ
= (mπ/a)
+ (nπ/b)
.
L
L
L
m,n
m,n
a
b
n=1
cnπ
where A
= sin[f, n] and
B
= sin[g, n].
If f (x, y) =
A
φ
in Ω, the solution for ∆u = f in Ω with 0-BC
n
n
L
m,n
m,n
m,n=1
A
is u =
m,n
φ
.
m,n
m,n=1
λ
Heat and wave equations on a rectangle [0, a]
[0, b]
m,n
Let sin[f, m, n] denote the double sine series coefficents of f (x, y),
Sturm-Liouville Eigenvalue Problems
ODE:
[p(x)y ] + [q(x) + λr(x)]y = 0,
a < x < b.
a
b
4
mπx
nπy
sin[f, m, n] =
f (x, y) sin
sin
dy dx.
BC:
c
y(a) + c
y (a) = 0,
d
y(b) + d
y (b) = 0.
1
2
1
2
ab
a
b
0
0
Hypothesis: p, p , q, r continuous on [a, b]. p(x) > 0 and r(x) > 0 for
2
2
2
2
2
Heat eq. u
= c
(u
+ u
), u(x, y, 0) = f (x, y) and 0-BC:
x
[a, b]. c
+ c
> 0. d
+ d
> 0.
t
xx
yy
1
2
1
2
Properties (1) The differential operator Ly = [p(x)y ] + q(x)y is symmetric
mπx
nπy
m 2
n 2
in the sense that (f, Lg) = (Lf, g) for all f, g satisfying the BC, where (f, g) =
2
2
c
π
(
+
)t
u(x, y, t) =
B
sin
sin
e
,
B
= sin[f, m, n].
a 2
b 2
m,n
m,n
b
a
b
f (x)g(x) dx. (2) All eigenvalues are real and can be ordered as λ
< λ
<
1
2
m,n=1
a
2
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