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Flashcards in Fourier Series and PDES Deck (57)
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1
Q

f : R→R is a periodic function if…

A

∃p >0 s.t. f(x+p) = f(x) ∀x∈R.

2
Q

The smallest p for which f is p-periodic is called…

A

The prime period.

3
Q

The periodic extension F : R→R of f: (α,α+p]→R is defined by…

A

F(x) = f(x−mp), where for each x∈R, m is the unique integer such that x−mp∈(α,α+p].

4
Q

5 Properties of periodic functions: If f and g are p–periodic, then:

A

(1) f,g are np–periodic ∀n∈N{0};
(2) αf+βg are p–periodic ∀α,β∈R;
(3) fg is p–periodic;
(4) f(λx) is p/λ–periodic ∀λ >0;
(5) ᵖ∫₀ f(x) dx = ᵃ⁺ᵖ∫ₐ f(x) dx ∀α∈R.

5
Q

f : R→R is odd if…

A

f(x) = −f(−x) ∀x∈R.

6
Q

f : R→R is even if…

A

f(x) = f(−x) ∀x∈R.

7
Q

4 Properties of odd/even functions: If f, f1 are odd and g, g1 are even, then:

A

(1) f(0) = 0;
(2) ᵃ∫₋ₐ f(x) dx = 0 ∀α∈R;
(3) ᵃ∫₋ₐ g(x) dx = 2ᵃ∫₀ g(x) dx ∀x∈R;
(4) fg is odd, ff1 is even, and gg1 is even.

8
Q

Let f : R→R be a periodic function of period 2π. We say a Fourier expansion for f is of the form

A

f(x) ∼ a₀/2 + ∞∑ₙ₌₁(aₙcos(nx) +bₙsin(nx)).

9
Q

What is a₀ of a 2π-periodic Fourier series for f(x)?

A

a₀ = 1/π ᵖᶦ∫₋ₚᵢ f(x) dx

10
Q

Let m,n∈N{0}. Then we have the orthogonality relations:
ᵖᶦ∫₋ₚᵢ cos(mx)cos(nx) dx =
ᵖᶦ∫₋ₚᵢ cos(mx)sin(nx) dx =
ᵖᶦ∫₋ₚᵢ sin(mx)sin(nx) dx =

A

ᵖᶦ∫₋ₚᵢ cos(mx)cos(nx) dx = πδₘₙ
ᵖᶦ∫₋ₚᵢ cos(mx)sin(nx) dx = 0
ᵖᶦ∫₋ₚᵢ sin(mx)sin(nx) dx = πδₘₙ

where δₘₙ is Kronecker’s delta.

11
Q

Prove what is aₘ of a 2π-periodic Fourier series for f(x), given the series. m =/= 0

A

ᵖᶦ∫₋ₚᵢ f(x)cos(mx) dx = 1/2 a₀ᵖᶦ∫₋ₚᵢ cos(mx) dx + ∞∑ₙ₌₁ (aₙᵖᶦ∫₋ₚᵢ cos(mx)cos(nx) dx + bₙᵖᶦ∫₋ₚᵢ cos(mx)sin(nx) dx)
giving ᵖᶦ∫₋ₚᵢ f(x)cos(mx) dx = 1/2 a₀·0 +∞∑ₙ₌₁(aₙπδₘₙ+bₙ·0) = πaₘ,
so that aₘ = 1/π ᵖᶦ∫₋ₚᵢ f(x)cos(mx) dx for m∈N{0}.

12
Q

What is bₘ of a 2π-periodic Fourier series for f(x)? m =/= 0

A

bₘ = 1/π ᵖᶦ∫₋ₚᵢ f(x)sin(mx) dx for m∈N{0}.

13
Q

What is the Fourier Sine series expansion for f(x)?

A

f(x) ∼ ∞∑ₙ₌₁ bₙsin(nx),

14
Q

What does bₙ equal in a 2π-periodic Fourier Sine series for f(x)?

A

bₙ = 2/π ᵖᶦ∫₀ f(x)sin(nx) dx

15
Q

What is the Fourier Cosine series expansion for f(x)?

A

f(x) ∼ 1/2 a₀ + ∞∑ₙ₌₁ aₙcos(nx),

16
Q

What does aₙ equal in a 2π-periodic Fourier Cosine series for f(x)?

A

aₙ = 2/π ᵖᶦ∫₀ f(x)cos(nx) dx

17
Q

The right-hand limit of f at c is…

A

f(c₊) = lim₍ₕ→₀, ₕ>₀₎ f(c+h) if it exists.

18
Q

The left-hand limit of f at c is…

A

f(c₋) = lim₍ₕ→₀, ₕ

19
Q

f is continuous at c iff…

A

f(c₋) = f(c) = f(c₊).

20
Q

f is piecewise continuous on (a,b)⊆R if…

A

There exists a finite number of points x₁,…,xₘ∈R with a = x₁< x₂< … < xₘ = b such that:

(i) f is defined and continuous on (xₖ, xₖ₊₁) for all k = 1,…,m−1;
(ii) f(xₖ₊) exists for k = 1,…,m−1;
(iii) f(xₖ₋) exists for k = 2,…,m.

21
Q

What is the Fourier Convergence Theorem?

A

Let f : R→R be 2π-periodic, with f and f′ piecewise continuous on (−π, π). Then, the Fourier coefficients
aₙ = 1/π ᵖᶦ∫₋ₚᵢ f(x)cos(nx) dx (n∈N),
bₙ = 1/π ᵖᶦ∫₋ₚᵢ f(x)sin(nx) dx for n∈N{0} exist,
and 1/2 (f(x₊) + f(x₋)) = a₀/2 + ∞∑ₙ₌₁(aₙcos(nx) + bₙsin(nx)) for x∈R.

[Proof not examinable, so fun times :-) ]

22
Q

Say the Fourier Convergence Theorem is true for 2π periodic f, then what is the formula for g, the even part of f, and h, the odd part of f. and their repective fourier series?

A
g(x) = 1/2 (f(x) + f(−x))
h(x) = 1/2 (f(x) − f(−x))

1/2 (g(x₊) + g(x₋)) = a₀/2 + ∞∑ₙ₌₁aₙcos(nx), for x∈R,
1/2 (h(x₊) + h(x₋)) = ∞∑ₙ₌₁bₙsin(nx) for x∈R,

23
Q

What conditions allow the Fourier Series to be integrated? And what does the integration equal?

A

If f is only 2π-periodic and piecewise continuous on (−π,π), then
ˣ∫₀ f(s) ds = 1/2 a₀x + ∞∑ₙ₌₁(aₙˣ∫₀ cos(ns) ds + bₙˣ∫₀ sin(ns) ds) for x∈R.

24
Q

What conditions allow the Fourier Series to be differentiated? And what does the differentiation equal?

A

If f is 2π-periodic and continuous on R with both f′ and f′′ piecewise continuous on (−π,π), then
1/2 (f′(x₊) +f′(x₋)) = ∞∑ₙ₌₁ (aₙ d/dx (cos(nx)) + bₙ d/dx (sin(nx))) for x∈R

25
Q

The smoother f, i.e.the more continuous derivatives it has…

A

the faster the convergence of the Fourier series for f.

26
Q

If the first jump discontinuity is in the pth derivative of f, with the convention that p = 0 if there is a jump discontinuity in f, then typically the slowest decaying aₙ and bₙ decay like…

A

1/(nᵖ⁺¹) as n→∞.

27
Q

Gibb’s Phenomenon is…

A

The persistent overshoot near a jump discontinuity. It happens whenever a jump discontinuity exists.

28
Q

(Gibb’s Phenomenon) As the number of terms in the partial sum tends to ∞, the width of the overshoot region…

while the total height of the overshoot region…

A

the width of the overshoot region tends to 0 (by the Fourier Convergence Theorem),

the total height of the overshoot region approaches γ|f(x₊) − f(x₋)|, where γ = 1/π ᵖᶦ∫₋ₚᵢ sin(x)/x dx ≈ 1.18,

[I think we just need to know the overshoot is 9% or something, 1.18 is 9% top and bottom overshoot]

29
Q

What is the Fourier series for a function of period 2L?

A

f(x) ~ a₀/2 + ∞∑ₙ₌₁(aₙcos(nπx/L) + bₙsin(nπx/L)),

30
Q

What are the Fourier coefficients of a Fourier series of period 2L?

A
aₙ = 1/L ᴸ∫₋ₗ f(x)cos(nπx/L) dx
bₙ = 1/L ᴸ∫₋ₗ f(x)sin(nπx/L) dx.
31
Q
What are the orthogonality relations equal to for 
ᴸ∫₋ₗ cos(mπx/L)cos(nπx/L) dx = 
ᴸ∫₋ₗ cos(mπx/L)sin(nπx/L) dx =
ᴸ∫₋ₗ sin(mπx/L)sin(nπx/L) dx = 
where n,m∈N\{0}.
A

ᴸ∫₋ₗ cos(mπx/L)cos(nπx/L) dx = Lδₘₙ
ᴸ∫₋ₗ cos(mπx/L)sin(nπx/L) dx = 0,
ᴸ∫₋ₗ sin(mπx/L)sin(nπx/L) dx = Lδₘₙ

32
Q

The even 2L-periodic extension fₑ : R→R of f : [0, L] → R is defined by…

A

fₑ(x) =
{ f(x) for 0≤x≤L,
{ f(−x) for −L

33
Q

What is the Fourier Cosine series for f : [0, L]→R,

and what are the coefficients equal to?

A

The Fourier Cosine series for f is equal to the Fourier series for fₑ,
fₑ(x) ~ a₀/2 + ∞∑ₙ₌₁(aₙcos(nπx/L))

aₙ = 2/L ᴸ∫₀ f(x)cos(nπx/L) dx, (n∈N)

34
Q

The odd 2L-periodic extension fₒ : R→R of f : [0, L] → R is defined by…

A

fₒ(x) =
{ f(x) for 0≤x≤L,
{ −f(−x) for −L

35
Q

What is the Fourier Sine series for f : [0, L]→R,

and what is the coefficients equal to?

A

bₙ = 2/L ᴸ∫₀ f(x)sin(nπx/L) dx, (n∈N{0})

36
Q

Note that the odd 2L-periodic extension fₒ : R→R of f : [0, L] → R is odd for x/L∈R\Z and odd (on R) iff…

A

f(0) = f(L) = 0.

37
Q

Consider the function f: [0,L]→R defined by f(x) = x for 0≤x≤L, why does the truncated cosine series give a better approximation to f on [0,L] than the truncated sine series?

[fₑ(x) ∼ L/2 − ∞∑ₘ₌₀ (4L)/((2m+1)²π²)) * cos((2m+1)πx/L).]
[fₒ(x) ∼∞∑ₙ₌₁ (2L(−1)ⁿ⁺¹)/(nπ) * sin(nπx/L).]

A

(1) it converges everywhere on [0,L];
(2) it converges more rapidly;
(3) it does not exhibit Gibb’s phenomenon.

38
Q

What is the Fundamental Theorem of Calculus?

A

If f(x) is continuous in a neighbourhood of a, then 1/h ᵃ⁺ʰ∫ₐ f(x) dx → f(a) as h→0.

39
Q

Leibniz’s Integral Rule: Let F(x,t) and ∂F/∂t be continuous in both x and t in some region R of the (x,t) plane containing the region S = {(x,t) : a(t)≤x≤b(t), t₀≤t≤t₁}, where the functions a(t) and b(t) and their derivatives are continuous for t∈[t₀,t₁]. Then d/dt ᵇ⁽ᵗ⁾∫ₐ₍ₜ₎ F(x,t) dx =

A

d/dt ᵇ⁽ᵗ⁾∫ₐ₍ₜ₎ F(x,t) dx = ᵇ⁽ᵗ⁾∫ₐ₍ₜ₎ ∂F/∂t(x,t) dx + db(t)/dt F(b(t),t)− da(t)/dt F(a(t),t).

40
Q

What is the Heat equation? And derive it in a rigid isotropic conducting rod of constant cross-sectional area A lying along the x-axis.

A

∂T/∂t = κ ∂²T/∂x²,

Derived using the whole entire pages of 2 and 3 lectures 5-6

41
Q

Notation: We denote by [p]…

A

the dimension of the quantity p in fundamental dimensions (M,L,T,Θ etc) or e.g. SI units (kg,m,s,K etc).

42
Q

Both sides of an equation modelling a physical process must have the same…

A

dimensions

43
Q

What do these symbols stand for and what are their SI units?

x, t, T, q, A, ρ, c, k, κ (kappa)

A
x Axial distance m 
t Time s
T Absolute temperature K
q Heat flux in positive x-direction Jm⁻²s⁻¹
A Cross-sectional area m²
ρ Rod density kgm⁻³
c Rod specific heat Jkg⁻¹K⁻¹
k Rod thermal conductivity JK⁻¹m⁻¹s⁻¹
κ Rod thermal diffusivity m²s⁻¹
44
Q

What is nondimensionalisation?

A

Method of scaling variables with typical values to derive dimension-less equations. These usually contain dimensionless parameters that characterise the relative importance of the physical mechanisms in the model.

45
Q

How would you typically nondimensionalise a problem?

Here the heat equation is used as an example

A

We nondimensionalise the variables by scaling x = Lxₕₐₜ, t = τtₕₐₜ, T(x,t) = T₂Tₕₐₜ(xₕₐₜ,tₕₐₜ),

By the chain rule, ∂T/∂t = T₂(∂Tₕₐₜ/∂tₕₐₜ)(dtₕₐₜ/dt) = (T₂/τ)(∂Tₕₐₜ/∂tₕₐₜ),
∂T/∂x = T₂(∂Tₕₐₜ/∂xₕₐₜ)(dxₕₐₜ/dx) = (T₂/L)(∂Tₕₐₜ/∂xₕₐₜ), etc.

The conditions can then be modified into their dimensionless counterparts.
Then heat problems can be compared on different scales, pg 5 lectures 5-6 for more info.

46
Q

What is Fourier’s Method for solving PDEs?

A

(I) Use the method of separation of variables to find the countably infinite set of nontrivial separable solutions satisfying the partial differential equation and boundary conditions, each containing an arbitrary constant

(II) Use the principle of superposition — that the sum of any number of solutions of a linear problem is also a solution (assuming convergence) — to form the general series solution that is the infinite sum of the separable solutions of the partial differential equation and boundary conditions.

(III) Use the theory of Fourier series to determine the constants in the general series solution for which it satisfies the initial condition

47
Q

Solve the heat equation with homogenous Dirichlet boundary conditions, where ∂T/∂t = κ(∂²T/∂x²) for 0< x < L, t >0,
with the boundary conditions T(0,t) = 0, T(L,t) = 0 for t >0,
and the initial condition T(x,0) = f(x) for 0< x < L,
where the initial temperature profile f(x) is given.

A

Lectures 5-6 bottom of page 6 to halfway down page 8.
(hence why its not on here)

Its the pretty standard method though, sep of variables, using boundary conditons here,making sure T is non-trivial, finding solutions that depend on n, superpositioning these solutions, then using the initial conditions to solve the constants Fourierly.

48
Q

What are some important implications of the general solution of the heat equation, where initial temperatures at the end of the rod are fixed to 0?
(Bad flashcard I know, but there might be some things worth remembering)

A

–the heat equation smoothes out instantaneously even irregular initial temperature profiles;
–as soon as t >0, most of the high frequency terms Tn(x,t) for n»1 will be extremely small, so that the solution may be well approximated by only a handful of terms;
–the temperature tends to zero exponentially quickly as κt/L² → ∞, i.e. on the timescale of heat conduction, with the thermal energy initially stored in the rod being conducted out of the ends of the rod on this timescale.

49
Q

How do you solve for uniqueness for the heat equation solutions?

A

We suppose that T₁(x,t) and T₂(x,t) are solutions to the heat equation: Let W(x,t) = T₁(x,t) − T₂(x,t) be their difference.
Work out the boundary/initial conditions for W.
Work out W = 0:
For example:

I(t) := 1/2 ᴸ∫₀ W(x,t)² dx.
Therefore I(t) ≥0 for t≥0 and I(0) = 0
[W(x,0) = T₁(x,0) − T₂(x,0) = f(x) − f(x) = 0]

dI/dt = ᴸ∫₀ W(∂W/∂t) dx
= ᴸ∫₀ Wκ(∂²W/∂x²) dx
= [κW(∂W/∂x)]ᴸ₀ 
− κ ᴸ∫₀ (∂W/∂x)(∂W/∂x) dx
= −κᴸ∫₀ (∂W/∂x)²dx ≤0

which means that I(t) cannot increase, so that I(t) ≤ I(0) = 0 for t≥0. So I(t) = 0 for all t≥0, and hence W(x,t) = 0 for 0≤x≤L, t ≥ 0

50
Q

Solve the heat equation with inhomogenous Dirichlet boundary conditions, where
∂T/∂t = κ(∂²T/∂x²) for 00,
with the inhomogeneous Dirichlet boundary conditions
T(0,t) = Tₗ, T(L,t) = Tᵣ for t >0,
and the initial condition T(x,0) = 0 for 0< x < L, where Tₗ and Tᵣ are prescribed constant temperatures, not both zero.

A

Lectures 7-8 bottom of page 2 to bottom of page 3

(On physical grounds, we conjecture that T(x,t) → S(x) as t→ ∞, where S(x) is the afore-mentioned steady-state solution, which satisfies 0 = κ(d²S/dx²) for 0< x < L, with S(0) = Tₗ and S(L) = Tᵣ. Thus, S(x) has the linear temperature profile given by S(x) = Tₗ(1−x/L) + Tᵣ(x/L); we note that in steady state thermal energy is conducted steadily along the rod with constantheat flux q = −k(∂T/∂x) = k(Tₗ − Tᵣ)/L,so that heat flows steadily in the positive x-direction for Tₗ> Tᵣ.

We now observe that if we let T(x,t) = S(x) + U(x,t), U has homogenous Dirichlet boundary conditions, and so can be solved normally.)

51
Q

Solve the heat equation with homogenous Neumann boundary conditions, where
∂T/∂t = κ(∂²T/∂x²) for 00,
with the homogeneous Neumann boundary conditions Tₓ(0,t) = 0, Tₓ(L,t) = 0 for t >0,
and the initial condition T(x,0) = f(x) for 0< x < L

A

Lectures 7-8 middle of page 4 to top of page 5

Solved pretty normally here, just note the ends must be thermally insulated due to the boundary conditions.

52
Q

Solve the heat equation with inhomogenous Neumann boundary conditions, where
ρc∂T/∂t = k(∂²T/∂x²) + Q(x,t) for 0< x < L, t >0,
with the inhomogeneous Neumann boundary conditions Tₓ(0,t) = φ(t), Tₓ(L,t) = ψ(t) for t >0,
and the initial condition T(x,0) = f(x) for 0< x < L
where the functions Q(x,t), φ(t), ψ(t) and f(x) are given.

A

Lectures 7-8 middle of page 5 to middle of page 7

(if we let T(x,t) = S(x,t) +U(x,t), where S(x,t) = −φ(t)(x−L)²/2L + ψ(t)x²/2L, then U(x,t) has homogenous boundary conditions and so can be solved normally.
If the new Q function for U is not identically 0, U should be solved as a cosine series however:
U(x,t) = U₀(t)/2 + ∞∑ₙ₌₁ Uₙ(t)cos(nπx/L), then by computing ρc(dUₙ/dt), and substituting in the integral to have Uₓₓ inside, integration by parts can be used to get ρc(dUₙ/dt) + kn²π²/L² Uₙ = newQₙ(t).
As the newQₙ(t) coeffiecient can be determined by its the fourier expansion of newQ(x,t), then Uₙ can be determined.
53
Q

Derive the wave equation in one dimension: Consider the small transverse vibrations of a homogeneous extensible elastic string stretched initially along the x-axis at time t = 0.
A point at xi at time t = 0 is displaced to r(x,t) = xi + y(x,t)j at time t >0, where the transverse displacement y(x,t) is to be determined.

(There’s a lot more conditions, look at lectures 7-8 bottom of page 7 to middle of page 8)

A

Lectures 7-8 middle of page 8 to bottom of page 9.

(Combine linear momentum, the fact that the tension is parallel to direction of string, the string has constant linear density, look at string between x = a + h and x = a, ignore gravity, air resistance, resisitence to bending by the string, use Newton’s second law, use the fundamental theorem of calculus, and assume |yₓ| &laquo_space;1)
c = √(T/ρ) is the wave speed

54
Q

On what timescale does a displacement travel a distance L when it follows the wave equation
yₜₜ = c²yₓₓ? (Non-dimensionalise)

A

The terms balance giving
(∂²yₕₐₜ/∂tₕₐₜ²) = (∂²yₕₐₜ/∂xₕₐₜ²) provided t₀ = L/c, which is therefore the timescale for a displacement to travel a distance L.

55
Q

Solve the wave equation yₜₜ = c²yₓₓ for 0 < x < L, with the Dirichlet boundary conditions y(0, t) = 0 and y(L, t) = 0, and the initial conditions y(x, 0) = f(x), (∂y/∂t)(x, 0) = g(x)

A

Lectures 9-10 bottom of page 1 to middle of page 2

(Use seperable solutions y = F(x)G(t),  then F''(x)/F(x) = G''(t)/c²G(t) = -λ ∈R.  
As F(0)G(t) = 0, and y non-trivial, F(0) = 0, and also F(L) = 0. Then F(x) can be solved, thus enabling the solving of G, and then all the solutions can be superposed together. Then the theory of Fourier series can be used with the initial conditions to find the constants aₙ and bₙ
56
Q

What are the normal modes for the wave equation yₜₜ = c²yₓₓ for 0 < x < L, with the boundary conditions y(0, t) = 0 and y(L, t) = 0?
And what is the prime period and frequency of the normal modes?
And what is the first normal mode, what is it called, and what’s special about it?

A

A normal mode is a solution for yₙ (a solution for a specific value of n)
Its prime period is p =
2π/(nπc/L) = 2L/nc
Its frequency (or pitch) is 1/p = nc/2L

The first normal mode y₁ is called the fundamental mode, with associated fundamental frequency c/2L. All of the other modes have a frequency that is an integer multiple of the fundamental frequency.

57
Q

What is the kinetic energy of the string whose motion is governed by the wave equation yₜₜ = c²yₓₓ for 0 < x < L, with the Dirichlet boundary conditions y(0, t) = 0 and y(L, t) = 0, and the initial conditions y(x, 0) = f(x), (∂y/∂t)(x, 0) = g(x), and also with line denisty ρ and tension T?

Also what is its elastic potential energy, to a first approximation?

And so what is the total energy of the string?

Prove that if y(x, t) satifies this wave equation, then E(t) (the energy) is constant.

A

pg 6 lectures 9-10