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AMATH 353 > Exam Prep > Flashcards

Exam Prep Flashcards

1
Q

Fourier series

A

Periodic with period 2pi

f(t) = sum(n=-inf, inf) c_n * e^(int)

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2
Q

Fourier coeffs

A

c_n = 1/2pi * int(-pi, pi) f(t)*e^(-int)dt

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3
Q

Fourier integral

A

Non periodic functions

f(t) = 1/2pi * int(-inf, inf) F(w)e^(iwt) dw

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4
Q

Fourier transform

A

F(w) = int(-inf, inf) f(t) e^(iwt) dt

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5
Q

Cosine series

A

f is even

a_n = 0, b_n = 2/pi * int(0, pi) f(x) sin(nx)dx
f(x) = 1/2 a_0 * sum(n=1, inf)a_n cos(nx)
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6
Q

Sine series

A

f is odd

a_n=0, b_n=2/pi int(0, pi) f(x) sin(nx) dx
f(x) = sum(n=1, inf) b_n sin(nx)

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7
Q

FT: bounded

A

int(-inf, inf) |f(x)|^2 dx < inf

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8
Q

FT: end behaviour

A

lim (x -> +/- inf) f(x) = 0

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9
Q

FT: linearity

A 
F(f+g) = F(f) + F(g)
F(cf) = cF(f)
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10
Q

FT: scaling

A

F(f(ax)) = 1/|a| F(w/a)

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11
Q

FT: shift

A

F(e^(iax) f(x)) = F(x-a)

F(f(x-c)) = e^(-icw)F(w)

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12
Q

FT: repeated FTs

A

F(F(x)) = 2pi * f(-w)

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13
Q

FT: derivative

A

F(f’(x)) = iwF(w)

F(x f(x)) = iF’(w)

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14
Q

Parseval’s formula

A

int(-inf, inf) f(x)^2 dx = 1/2pi * int(-inf, inf) |F(w)|^2 dw

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15
Q

Convolution

A

(f * g)(x) = int(-inf, inf) f(x-y)g(y)dy

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16
Q

Convolution: symmetry

A

(f*g)(x) = (g * f)(x)

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17
Q

Convolution: distributivity

A

(f * (g + h))(x) = (f * g)(x) + (f * h)(x)

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18
Q

Convolution theorem

A

F(f * g) = F(f) F(g)

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19
Q

Error function

A

erf(w) = 2/sqrt(Pi) int(0, w) e^(-z^2) dz for all wER

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20
Q

Laplace kernel

A

H(x, y) = 1/pi y/(x^2 + y^2)

1/2 plane

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21
Q

Diffusion kernel

A

K(x, t) = 1/sqrt(4 pi kt) exp(-x^2/(4kt))

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22
Q

Properties of the kernels

A
  • normalized (int(-inf, inf) dx = 1)
  • H has y, K has t, we will call them p and fnc P
  • p -> 0+, p -> 0 if x =/= 0, inf if x=0
  • P(x, 0) = delta(x)
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23
Q

How to use the fundamental sols

A

u = int(-inf, inf) P(x-s, p) f(s) ds

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24
Q

M(x, y)

A

M(x, y) = 1/pi * arctan(x/y)

  • lim(y -> 0+) M = 1/2, x>0; 0, x=0; -1/2, x<0
  • ue(x, y) = 1/2e (M(x+e, y) - M(x-e, y))
  • lim(y -> 0+) ue = 1/(2e), |x|e
  • lim(e -> 0+) ue = H
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25
Q

Q(x, t)

A

Q(x, t) = 1/2 * erf(x/sqrt(4pi)), t>0

  • lim(y -> 0+) Q = 1/2, x>0; 0, x=0; -1/2, x<0
  • ue(x, y) = 1/2e (Q(x+e, y) - Q(x-e, y))
  • lim(y -> 0+) ue = 1/(2e), |x|e
  • lim(e -> 0+) ue = Q
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26
Q

Conservation law (general form)

A

d/dt int(D) psi(x, t) dV = - int(del D) q•N dS

  • second term is double integral
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27
Q

Theorem abt conservation laws

A

Conservation law implies d(psi)dt + nabla •q = 0 in R

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28
Q

Diffusion eqn in 2D

A 
Psi = rho*cu
q = -k * nabla(u)
k = K/rho*c thermal diffusivity
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29
Q

Orthogonality

A

Int(D) vi*vj dV = 0 if i=/=j

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30
Q

What do we know from SL eigval problems?

A

1) countable set of positive eigvals
2) eigfunctions are orthogonal and each eigval has a finite number of indep eigfunctions
3) eigfunctions form basis

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31
Q

Wave equation frequency

A

v_n = c*sqrt(lambda_n)/2pi

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32
Q

Laplacian polar

A

1/r * dr(r dr) + 1/r^2 d2theta

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33
Q

Gradient polar

A

dr, 1/r dtheta

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34
Q

Bessel functions

A

Come from laplace’s eqn on 2D disk

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35
Q

Legendre polynomials

A

Come from laplace’s eqn on solid R3 sphere

36
Q

Raleigh quotient

A

R(v) = int(D) grad(v) • grad(v) dV / int(D) v^2 dV

  • lambda_1 <= R(v) if v=0 on delta D
  • lambda_1 = R(v) iff v is eigfnc for lambda
37
Q

Inclusion property

A

D, d E R2 and D in d -> lambda1D > lambda1d

38
Q

Faber-Krahn inequality

A

Lambda1(D) > pi(j0,1)^2/A ≈ pi/A * (2.4)^2

D has area A

39
Q

Nodal lines (def)

A

Eigfnc of laplace’s eqn = 0

40
Q

Nodal lines, square

A

Side a

x = ia/n, 1 <= i <= m-1
y = ja/n, 1 <= j <= n-1
41
Q

Nodal lines, disk

A

Radius b

r = b*j(m,i)/j(m,n)
1 <= i <= m-1, m>=2
Theta = 1/m * ((2i-1) pi/2 + delta), 1 <= i <= 2m

42
Q

Courant’s nodal line theorem

A

D in R2, nodal lines for nth eigfunction divide D into at most n regions

43
Q

Equilibrium solution (higher dimensions)

A

u = utrans + ueq

  • ueq: solve laplace’s eqn
  • utrans: solve normal eqn with all BCs = 0
44
Q

Method of undetermined coeffs two roots indep functions

A

Can also choose sinh(1x) + cosh(2x) or sinh(1(L-x)) + cosh(2(L-x))

45
Q

Isothermal curves

A

u(x, y) = constant

46
Q

Poisson’s formula

A

P(r, theta) = 1/2pi * (b^2 - r^2)/(b^2 - 2brcos(theta) + r^2)
AKA poisson kernel

47
Q

Mean value property

A

Steady state temperature distribution (solution to laplace’s eqn in D)
T(P) = mean value of T around any circle centred at P

P = point

48
Q

Uniqueness of higher dimension PDE sol

A

D bounded, boundary piecewise smooth

Then laplace’s eqn has a unique C2 solution on D (cts on boundary)

49
Q

Domain of validity

A

det(d(x,t)/d(r,s)) = 0 eqn dies

50
Q

Shock first time

A

t = 1/(2*max(f’(s)))

f is bounded

51
Q

Theorem abt quasilinear PDEs

A

ut + qx = 0 int(D) (u phi(t) 1 q phi(x)) dxdt = 0 for all C0 phi

The phi are derivatives

52
Q

Weak solution

A

u satisfied int(D) (u phi(t) + q phi(x)) dxdt = 0

The phi are derivatives

53
Q

Speed of discontinuity

A

xs’(t) = [q]/[u]

54
Q

Physical interpretation of weak solution discontinuities

A

Shock waves

55
Q

Weak solutions uniqueness

A

Might not be unique

56
Q

Wave eqn, what is c

A

c^2 = T/rho

T = tension at rest
Rho = linear density
57
Q

Wave KE

A

KE = 1/2 int(0, l) rho * (du/dt)^2 dx

58
Q

Wave PE

A

PE = 1/2 * int(0, l) tau * (du/dx)^2 dx

59
Q

Normal modes

A

Normal modes are the eigenfunctions

Frequencies / harmonics are the v_n = nc/2l

60
Q

d’Alembert’s formula, infinite

A

u = 1/2 * (f(x-ct) + f(x+ct)) + 1/2c * int(x-ct, x+ct) g(s) ds

61
Q

d’Alembert’s formula, finite

A

u(x, t) = 1/2 (ff(x-ct) + ff(x+ct))

ff = odd periodic f extension

62
Q

Series solution convergence

A

Coeffs bounded -> series converges absolutely for all t>0, xE[0, pi]

63
Q

Diffusion eqn: spatial T profiles

A

u(x, t0)

64
Q

Diffusion eqn: T profiles in time

A

u(x0, t)

65
Q

Diffusion eqn: curves if constant T

A

u(x, t) = C

66
Q

Steady state solution

A 
us = u0 Re(e^(iwt) V(x))
V(x) = A(x) e^(i psi(x))
67
Q

Diffusion eqn: general BCs

A

u(0, t) = b0
u(l, t) = b1
v(x, t) = u(x, t) - 1/l (l-x) b0 - 1/l x b1

Sub into pde and have fun

68
Q

Hadmard’s notion of well-posedness

A

1) at least one sol
2) at most one sol
3) sol continuously dependent on BCs and ICs

69
Q

Mean square temperature

A

U(t) = int(0, pi) u(x, t)^2 dx

- monotone non-increasing (U’ < 0)

70
Q

Uniqueness if diffusion eqn sol

A

Yep it’s unique

71
Q

W(x)

A

1, |x| < 1/2

0, |x| > 1/2

72
Q

T(x)

A

1-|x|, |x|<1

0, |x|>1

73
Q

FT: W(x)

A

sinc(1/2 w)

74
Q

FT: sinc(x)

A

Pi * W(1/2 w)

75
Q

FT: e^(-|x|)

A

2/(1 + w^2)

76
Q

FT: 1/(1+x^2)

A

Pi * e^(-|w|)

77
Q

FT: e^(-1/2 x^2)

A

sqrt(2pi) exp(-1/2 w^2)

78
Q

FT: T(x)

A

(sinc(1/2 w))^2

79
Q

FT: x^n

A

sqrt(2pi) (-i)^n delta(n)(w)

80
Q

FT: e^(ax)

A

Sqrt(2pi) delta(w-ia)

81
Q

FT: cos(w0 x)

A

Pi (delta(w-w0) + delta(w+w0))

82
Q

FT: sin(w0 x)

A

Pi (delta(w+w0) - delta(w-w0))

83
Q

FT: 1/x

A

i sqrt(pi/2) sgn(w)

84
Q

Inverse FT: e^(aw)

A

Sqrt(2pi) delta(ia+x)

85
Q

Inverse FT: cos(w)

A

Sqrt(pi/2) delta(t-1) + sqrt(pi/2) delta(t+1)

86
Q

Inverse FT: sin(w)

A

i sqrt(pi/2) delta(t+1) - i sqrt(pi/2) delta(t-1)

AMATH 353 (1 decks)

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