summary of important ideas

Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Math/Mthe 338 Course Summary
November 28, 2014
Fourier Integrals and Transforms
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Topics
a) Fourier Series;
b) Sturm-Liouville Problems;
c) Linear PDE and Principle of Superposition;
d) Separation of Variables;
e) Fourier Integrals and Transforms;
Separation of Variables
Fourier Integrals and Transforms
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Fourier Series
Separation of Variables
Fourier Integrals and Transforms
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Fourier Series
∞
f ∼
∞
nπ X
nπ a0 X
+
an cos
x +
bn sin
x
2
L
L
n=1
n=1
a) Be able to find Fourier series on [−L, L] and sine/cosine series on [0, L];
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Fourier Series
∞
f ∼
∞
nπ X
nπ a0 X
+
an cos
x +
bn sin
x
2
L
L
n=1
n=1
a) Be able to find Fourier series on [−L, L] and sine/cosine series on [0, L];
b) Understand how series extend the function f to the real line through
periodicity and symmetry;
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Fourier Series
∞
f ∼
∞
nπ X
nπ a0 X
+
an cos
x +
bn sin
x
2
L
L
n=1
n=1
a) Be able to find Fourier series on [−L, L] and sine/cosine series on [0, L];
b) Understand how series extend the function f to the real line through
periodicity and symmetry;
c) Have a rough understanding of Gibb’s phenomenon and pointwise vs.
uniform convergence;
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Fourier Series
∞
f ∼
∞
nπ X
nπ a0 X
+
an cos
x +
bn sin
x
2
L
L
n=1
n=1
a) Be able to find Fourier series on [−L, L] and sine/cosine series on [0, L];
b) Understand how series extend the function f to the real line through
periodicity and symmetry;
c) Have a rough understanding of Gibb’s phenomenon and pointwise vs.
uniform convergence;
Functions whose extensions to the real line are continuous and have a piecewise
continuous derivative have uniformly convergent Fourier series.
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Sturm-Liouville Problems
Fourier Integrals and Transforms
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Sturm-Liouville Problems
Aϕ0
0
0
+ Bϕ = ±λC ϕ.
a0 ϕ(a) + a1 ϕ (a) = 0
b0 ϕ(b) + b1 ϕ0 (b) = 0
a) The problem is regular if A and C are nonzero on [a, b]. The ± is chosen so
that C (x) > 0.
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Sturm-Liouville Problems
Aϕ0
0
+ Bϕ = ±λC ϕ.
0
b0 ϕ(b) + b1 ϕ0 (b) = 0
a0 ϕ(a) + a1 ϕ (a) = 0
a) The problem is regular if A and C are nonzero on [a, b]. The ± is chosen so
that C (x) > 0.
b) Eigenfunctions for distinct eigenvalues are orthogonal for
Z b
hf , g i =
C (x)f (x)g (x) dx
a
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Sturm-Liouville Problems
Aϕ0
0
+ Bϕ = ±λC ϕ.
0
b0 ϕ(b) + b1 ϕ0 (b) = 0
a0 ϕ(a) + a1 ϕ (a) = 0
a) The problem is regular if A and C are nonzero on [a, b]. The ± is chosen so
that C (x) > 0.
b) Eigenfunctions for distinct eigenvalues are orthogonal for
Z b
hf , g i =
C (x)f (x)g (x) dx
a
c) The collection of eigenfunctions {ϕn }∞
n=1 is complete:
f ∼
∞
X
hf , ϕn i
ϕn ,
hϕ
n , ϕn i
n=1
provided f is piecewise continuous;
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Sturm-Liouville Problems
Aϕ0
0
+ Bϕ = ±λC ϕ.
0
b0 ϕ(b) + b1 ϕ0 (b) = 0
a0 ϕ(a) + a1 ϕ (a) = 0
a) The problem is regular if A and C are nonzero on [a, b]. The ± is chosen so
that C (x) > 0.
b) Eigenfunctions for distinct eigenvalues are orthogonal for
Z b
hf , g i =
C (x)f (x)g (x) dx
a
c) The collection of eigenfunctions {ϕn }∞
n=1 is complete:
f ∼
∞
X
hf , ϕn i
ϕn ,
hϕ
n , ϕn i
n=1
provided f is piecewise continuous;
d) Notice the zero boundary conditions.
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Sturm-Liouville Problems and Fourier Series
∞
f ∼
nπ a0 X
+
an cos
x
2
L
n=1
a) What Sturm-Liouville problem leads to cosine series?
Fourier Integrals and Transforms
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Sturm-Liouville Problems and Fourier Series
∞
f ∼
nπ a0 X
+
an cos
x
2
L
n=1
a) What Sturm-Liouville problem leads to cosine series?
b) What are the eigenvalues and eigenfunctions here?
Fourier Integrals and Transforms
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Sturm-Liouville Problems and Fourier Series
∞
f ∼
nπ a0 X
+
an cos
x
2
L
n=1
a) What Sturm-Liouville problem leads to cosine series?
b) What are the eigenvalues and eigenfunctions here?
c) Why is a0 divided by 2?
Fourier Integrals and Transforms
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Linear PDE and Principle of Superposition
Fourier Integrals and Transforms
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Linear PDE
utt = c 2 ux x
vs
ut + 3uux = 0.
a) Be able to tell a linear PDE from a nonlinear PDE;
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Linear PDE
utt = c 2 ux x
vs
ut + 3uux = 0.
a) Be able to tell a linear PDE from a nonlinear PDE;
b) Be able to use principle of superposition strategically
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Linear PDE
utt = c 2 ux x
vs
ut + 3uux = 0.
a) Be able to tell a linear PDE from a nonlinear PDE;
b) Be able to use principle of superposition strategically
ztt = c 2 zxx + f
z(0, x) = g (x)
zt (0, x) = h(x)
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Linear PDE
utt = c 2 ux x
vs
ut + 3uux = 0.
a) Be able to tell a linear PDE from a nonlinear PDE;
b) Be able to use principle of superposition strategically
ztt = c 2 zxx + f
z(0, x) = g (x)
zt (0, x) = h(x)
Can be treated as
utt = c 2 uxx + f
u(0, x) = 0
ut (0, x) = 0
vtt = c 2 vxx
v (0, x) = g (x)
vt (0, x) = 0
z = u + v + w.
wtt = c 2 wxx
w (0, x) = 0
wt (0, x) = h(x)
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Separation of Variables
Fourier Integrals and Transforms
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Separation of Variables
Main Idea
Reduce a partial differential equation to ordinary differential equations.
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Separation of Variables
Main Idea
Reduce a partial differential equation to ordinary differential equations.
Reduce the “boundary value part” to a Sturm-Liouville problem (for which
there is a very solid theory)
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Separation of Variables
Boundary conditions
Because of the boundary conditions in Sturm-Liouville problems, boundary
conditions in the PDE need to be treated carefully.
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Separation of Variables
Boundary conditions
Because of the boundary conditions in Sturm-Liouville problems, boundary
conditions in the PDE need to be treated carefully.
Example
The boundary conditions here
ut = κuxx
u(t, 0) = A
u(t, L) = B
are inhomogeneous and need to be made homogeneous.
u(0, x) = f (x)
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Separation of Variables
Boundary conditions
Because of the boundary conditions in Sturm-Liouville problems, boundary
conditions in the PDE need to be treated carefully.
Example
The boundary conditions here
ut = κuxx
u(t, 0) = A
u(t, L) = B
u(0, x) = f (x)
are inhomogeneous and need to be made homogeneous.
Example
The boundary conditions for ∆u = 0 given by
u(x, 0) = f (x)
u(x, M) = g (x)
u(0, y ) = h(y )
can be treated individually using superposition.
u(L, y ) = i(y )
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Two Perspectives in Separation of Variables
Fully homogeneous problems
For L(u) = 0, for example
utt − c 2 uxx = 0
ut − κ∆u = 0
one can separate variables as T (t)X (x) or T (t)X (x)Y (y ) find ODE by
plugging into the PDE.
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Two Perspectives in Separation of Variables
Inhomogeneous problems
For L(u) = f , for example
utt − κuxx = f
one represents u as
u∼
∞
X
bn (t)Xn (x)
n=1
where Xn are eigenfunctions for the associated Sturm-Liouville problem, which
comes from the PDE without f .
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Two Perspectives in Separation of Variables
Inhomogeneous problems
For L(u) = f , for example
utt − κuxx = f
one represents u as
u∼
∞
X
bn (t)Xn (x)
n=1
where Xn are eigenfunctions for the associated Sturm-Liouville problem, which
comes from the PDE without f .
ODE for bn are found by Fourier expanding f in terms of Xn and plugging the
representation for u into the PDE.
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Coordinate Systems
Polar Coordinates
In polar coordinates
1
1
ur + 2 uθθ .
r
r
Be able to solve ∆u = 0 with different boundary conditions in this coordinate
system.
∆u = urr +
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Coordinate Systems
Polar Coordinates
In polar coordinates
1
1
ur + 2 uθθ .
r
r
Be able to solve ∆u = 0 with different boundary conditions in this coordinate
system.
∆u = urr +
Spherical Coordinates
In spherical coordinates, for functions that do not depend on φ, θ
∆u =
1 ∂2
(ru) .
r ∂r 2
Be able to solve ∆u = 0 and ut = ∆u with different boundary conditions in
this coordinate system.
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Fourier Integrals and Transforms
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Fourier Integrals and Transforms
∞
Z
f (x) =
Z
A(α) cos(αx) dα +
0
A(α) =
1
π
Z
∞
B(α) sin(αx) dα
0
∞
f (x) cos(αx) dx
−∞
B(α) =
1
π
Z
∞
f (x) sin(αx) dx
−∞
a) Be able to solve PDE with unbounded domains using Fourier integral and
separation of variables;
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Fourier Integrals and Transforms
∞
Z
f (x) =
Z
A(α) cos(αx) dα +
0
A(α) =
1
π
Z
∞
B(α) sin(αx) dα
0
∞
f (x) cos(αx) dx
B(α) =
−∞
1
π
Z
∞
f (x) sin(αx) dx
−∞
a) Be able to solve PDE with unbounded domains using Fourier integral and
separation of variables;
b) Understand the analogy with Fourier series;
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Fourier Integrals and Transforms
∞
Z
f (x) =
Z
A(α) cos(αx) dα +
0
A(α) =
1
π
Z
∞
B(α) sin(αx) dα
0
∞
f (x) cos(αx) dx
B(α) =
−∞
1
π
Z
∞
f (x) sin(αx) dx
−∞
a) Be able to solve PDE with unbounded domains using Fourier integral and
separation of variables;
b) Understand the analogy with Fourier series;
c) If f is defined only on [0, ∞) and I want to represent f using a sine integral,
how do I do it?
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Final Exam
A major focus of the course has been on solving applied problems and the final
will reflect this.
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Final Exam
A major focus of the course has been on solving applied problems and the final
will reflect this.
I will not ask you questions that involve Duhamel’s principle, the d’Alembert
solution to the wave equation, or resonance.
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Final Exam
A major focus of the course has been on solving applied problems and the final
will reflect this.
I will not ask you questions that involve Duhamel’s principle, the d’Alembert
solution to the wave equation, or resonance.
I do not expect you to memorize trigonometric identities.
Fourier Series
Sturm-Liouville Problems
Linear PDE and Principle of Superposition
Separation of Variables
Fourier Integrals and Transforms
Final Exam
A major focus of the course has been on solving applied problems and the final
will reflect this.
I will not ask you questions that involve Duhamel’s principle, the d’Alembert
solution to the wave equation, or resonance.
I do not expect you to memorize trigonometric identities.
Fourier formulas (for coefficients and for integrals) should be memorized.

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