Math 257-316 PDE
R∞
√
m(m − 1)(m − 2) · · · 21 π. In general, Γ(m + 1) = 0 tm e−t dt for real
q
2
m ≥ 0. Special case: J1/2 (x) = πx
sin x.
Formula sheet - final exam
Power series and analytic functions:
(t can be a complex number)
Trigonometric identities
A function f (x) is analytic at a point x0 if it has a power series expansion
in powers of x − x0 with radius R > 0,
P∞
f (x) = n=0 an (x − x0 )n , |x − x0 | < R, R > 0.
sin(α ± β) = sin α cos β ± sin β cos α
sin2 t + cos2 t = 1
2
cos(α ± β) = cos α cos β ∓ sin β sin α.
sin t = 12 (1 − cos(2t))
1
cos2 t = 12 (1 + cos(2t))
sin α cos β = 2 [sin(α − β) + sin(α + β)]
1
cos α cos β = 2 [cos(α − β) + cos(α + β)]
sin(2t) = 2 sin t cos t
1
sin α sin β = 2 [cos(α − β) − cos(α + β)],Z cos(2t) = 2 cos2 t − 1 = 1 − 2 sin2 t
Z
x cos ωx sin ωx
x sin ωx cos ωx
,
x sin ωx dx = −
x cos ωx dx =
+
+
2
ω
ω
ω
ω2
Z
2
x cos ωx 2x sin ωx 2 cos ωx
+
+
x2 sin ωx dx = −
ω
ω2
ω3
The sum and product of analytic functions are still analytic. Their quotient is
also analytic if the denominator is nonzero. The new function has a radius of
convergence no smaller than the minimum of those of the original functions.
P∞
P∞ 1 k
1
m
et = k=0 k!
t .
m=0 t ,
1−t =
cos t =
(−1)m 2m
m=0 (2m)! t
P∞
cosh t =
P∞
=
1
2m
m=0 (2m)! t
=
eit +e−it
,
2
sin t =
et +e−t
,
2
sinh t =
(−1)m 2m+1
m=0 (2m+1)! t
P∞
P∞
=
1
2m+1
m=0 (2m+1)! t
=
Fourier, sine and cosine series
eit −e−it
.
2i
Suppose f (x) is a function defined in [−L, L]. Its Fourier series is
et −e−t
.
2
F f (x) = a0 +
ix
e
= cos x + i sin x,
cos t = cosh(it),
i sin t = sinh(it).
k−1
P∞
P∞
tk ,
ln(1 − t) = − k=1 k1 tk .
ln(1 + t) = k=1 (−1)k
Basic linear ODE’s with real coefficients
ODE
indicial eq.
r1 6= r2 real
r1 = r2 = r
r = λ ± iµ
constant coefficients
ay 00 + by 0 + cy = 0
ar2 + br + c = 0
y = Aer1 x + Ber2 x
y = Aerx + Bxerx
λx
e [A cos(µx) + B sin(µx)]
Bessel equations of order p
1
2L
where a0 =
an =
Euler eq
ax y + bxy 0 + cy = 0
ar(r − 1) + br + c = 0
y = Axr1 + Bxr2
y = Axr + Bxr ln |x|
λ
x [A cos(µ ln |x|) + B sin(µ ln |x|)]
2 00
1
L
Z
RL
−L
∞ n
X
nπx
nπx o
an cos(
) + bn sin(
)
L
L
n=1
f (x) dx,
L
f (x) cos(
−L
nπx
) dx,
L
bn =
1
L
(useful in polar coordinates)
x2 y 00 + xy 0 + (x2 − p2 )y = 0.
1
L
Bessel function of first kind of order p,
(−1)k
k=0 k! Γ(k+p+1)
P
x 2k+p
2
L
f (x) sin(
−L
nπx
) dx (n ≥ 1).
L
F f (x) is defined for all real x and is a 2L-periodic function.
Theorem (Pointwise convergence) If f (x) and f 0 (x) are piecewise continuous, then F f (x) converges for every x to 21 [f (x−) + f (x+)].
Theorem (Square norm convergence) If f (x) is square integrable,
RL
i.e., −L |f (x)|2 dx < ∞. Then F f (x) converge to f (x) in square norm, i.e.
RL
|f (x) − Fn f (x)|2 dx → 0 as n → ∞, here Fn f (x) denotes the partial sum
−L
of F f (x). Moreover, (Parseval’s indentity)
Eq:
Jp (x) =
Z
Z
L
|f (x)|2 dx = 2|a0 |2 +
−L
∞
X
|an |2 + |bn |2 .
n=1
For f (x) defined in [0, L], its cosine and sine series are (a0 =
.
Here Γ is the Gamma function, which generalizes the factorial function. If
m is an integer, Γ(m + 1) = m!; If m − 1/2 is an integer, Γ(m + 1) =
Cf (x) = a0 +
∞
X
n=1
1
an cos(
nπx
),
L
an =
2
L
Z
L
f (x) cos(
0
1
L
RL
0
f (x) dx)
nπx
) dx,
L
Sf (x) =
∞
X
bn sin(
n=1
nπx
),
L
bn =
2
L
Z
L
f (x) sin(
0
Wave eq. utt = c2 (uxx + uyy ), u(t = 0) = f , ut (t = 0) = g and 0-BC:
nπx
) dx.
L
u(x, y, t) =
The coefficients are those for Fourier series multiplied by 2, and the integration is over [0, L]. Both Cf (x) and Sf (x) are defined for all real x and are
2L-periodic. Cf (x) is even while Sf (x) is odd.
X
mπx
4
sin
,
Sine series on [0, L]: 1 =
mπ
L
2L
mπx
x=
(−1)m+1 sin
,
mπ
L
m=1
µm,n = cπ
x(L − x) =
m is odd
mπx
8L2
sin
.
(mπ)3
L
bn sin
n=1
∞
X
nπx −c2 n2 π2 t/L2
e
,
L
an cos
n=1
cnπ
L Bn
nπx −c2 n2 π2 t/L2
e
,
L
an = cos[f, n].
m,n=1
x+ct
g(s) ds
x−ct
1
2c
Ry
0
g(s) ds.
where Bn and B̃n are coefficients of sine series of fR and fL , respectively.
The eigenfunctions and eigenvalues for −∆u = λu in Ω with 0-BC are
φm,n = sin
nπy
mπx
sin
,
a
b
λm,n = (mπ/a)2 + (nπ/b)2 .
P∞
If f (x, y) = m,n=1 Am,n φm,n in Ω, the solution for ∆u = f in Ω with 0-BC
P∞
−Am,n
is u = m,n=1 λm,n
φm,n .
= sin[g, n].
mπx
nπy −c2 π2 ( m22 + n22 )t
a
b
sin
e
,
a
b
Z
The solution u(x, y) for ∆u = 0 in Ω = {(x, y)| 0 ≤ x ≤ a, 0 ≤ y ≤ b} with
BC u(a, y) = fR (y), u(0, y) = fL (y), u(x, 0) = 0 = u(x, b) is
!
∞
X
sinh nπx
sinh nπ(a−x)
nπy
b
b
u(x, y) =
sin
Bn
+ B̃n
b
sinh nπa
sinh nπa
b
b
n=1
Sturm-Liouville Eigenvalue Problems
Heat eq. ut = c2 (uxx + uyy ), u(x, y, 0) = f (x, y) and 0-BC:
Bm,n sin
1
1
[f (x + ct) + f (x − ct)] +
2
2c
Laplace equation and Poisson equation
bn = sin[f, n].
Let sin[f, m, n] denote the double sine series coefficents of f (x, y),
Z aZ b
mπx
4
nπy
f (x, y) sin
sin
dy dx.
sin[f, m, n] =
ab 0 0
a
b
∞
X
Am,n = sin[f, m, n] and µm,n Bm,n = sin[g, m, n].
where Uleft (y), Uright (y) = 21 f (y) ±
Heat and wave equations on a rectangle [0, a] × [0, b]
u(x, y, t) =
n2
b2 ,
= Uleft (x + ct) + Uright (x − ct)
Wave eq. utt = c2 uxx , u(x, 0) = f (x), ut (x, 0) = g(x) and 0-BC:
∞
X
cnπt
cnπt
nπx
An cos
+ Bn sin
u(x, t) =
sin
L
L
L
n=1
where An = sin[f, n] and
+
u(x, t) =
Heat eq. ut = c2 uxx , u(x, 0) = f (x) and 0-flux BC:
u(x, t) = a0 +
m2
a2
The solution to utt = c2 uxx , u(t = 0) = f , ut (t = 0) = g is
For f (x) defined for x ∈ [0, L], let sin[f, n] and cos[f, n] denote its sine series
and cosine series coefficients, respectively.
Heat eq. ut = c2 uxx , u(x, 0) = f (x) and 0-BC:
u(x, t) =
q
mπx
nπy
sin
(Am,n cos µm,n t + Bm,n sin µm,n t)
a
b
d’Alembert’s formula
X
Heat and wave equations on a line [0, L]
∞
X
sin
m,n=1
m is odd
∞
X
∞
X
Bm,n = sin[f, m, n].
2
ODE: [p(x)y 0 ]0 + [q(x) + λr(x)]y = 0, a < x < b.
BC:
c1 y(a) + c2 y 0 (a) = 0, d1 y(b) + d2 y 0 (b) = 0.
Hypothesis: p, p0 , q, r continuous on [a, b]. p(x) > 0 and r(x) > 0 for
x ∈ [a, b]. c21 + c22 > 0. d21 + d22 > 0.
Properties (1) The differential operator Ly = [p(x)y 0 ]0 + q(x)y is symmetric
in the sense that (f, Lg) = (Lf, g) for all f, g satisfying the BC, where (f, g) =
Rb
f (x)g(x) dx. (2) All eigenvalues are real and can be ordered as λ1 < λ2 <
a
· · · < λn < · · · with λn → ∞ as n → ∞, and each eigenvalue admits a unique
(up to a scalar factor) eigenfunction φn .
Rb
(3) Orthogonality: (φm , rφn ) = a φm (x)φn (x)r(x) dx = 0 if λm 6= λn .
(4) Expansion: If f (x) : [a, b] → R is square integrable, then
f (x) =
∞
X
Rb
cn φn (x), a < x < b , cn =
n=1
a
f (x)φn (x)r(x) dx
, n = 1, 2, . . .
Rb
φ2 (x)r(x) dx
a n
PDE in polar coordinates
∆u = uxx + uyy = urr + 1r ur + r12 uθθ .
Let Ba denote a ball of radius a centered at the origin.
Let αmn denote the n-th positive root of Jm (r) = 0. Let
φm,n (r, θ) = Jm (
αmn
r) cos mθ,
a
ψm,n (r, θ) = Jm (
αmn
r) sin mθ.
a
• Heat eq ut = c2 ∆u on Ba with 0-BC and u(r, θ, 0) = u0 (r, θ)
u(r, θ, t) =
∞ X
∞
X
m=0 n=1
R a R 2π
Jm (
2 2
2
αmn r) Am,n cos mθ + Bm,n sin mθ e−c αmn t/a
a
u (r,θ)φ
(r,θ)rdrdθ
R a R 2π
u (r,θ)ψ
(r,θ)rdrdθ
m,n
m,n
, Bm,n = 0 R a0 R 2π0[ψ (r,θ)]
,
where Am,n = 0 R a0 R 2π0[φ (r,θ)]
2 rdrdθ
2 rdrdθ
m,n
m,n
0
0
0
0
B0,n ≡ 0.
• Wave eq utt = c2 ∆u on Ba with 0-BC and u(r, θ, 0) = u0 , ut (r, θ, 0) = u1
∞ X
∞
X
t
αmn
[Am,n cos mθ + Bm,n sin mθ] cos cαmn
a
u(r, θ, t) =
Jm (
r) ·
t
∗
+[A∗m,n cos mθ + Bm,n
sin m] sin cαmn
a
a
m=0 n=1
cαmn
∗
Am,n , Bm,n and A∗m,n cαamn , Bm,n
are the coefficents of u0 and u1 .
a
(Am,n , Bm,n have the same formulas as for heat eq.)
• Laplace eq ∆u = 0 on Ba with BC u(a, θ) = f (θ):
P∞ r n u(r, θ) = a0 + n=1
an cos nθ + bn sin nθ
a
where an and bn are Fourier series coefficients of f (θ).
3