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39 Cards in this Set
- Front
- Back
- 3rd side (hint)
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Fourier coefficients a_n & b_n |
a_n = 2/L ∫ cos (2πx/L) u(x) dx b_n = 2/L ∫ sin (2πx/L) u(x) dx
integral from -L/2 to L/2 |
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û(x) = ? |
û(x) = a_0/2 + ∑(n=1 to ∞) [a_n cos(2πx/L) + b_n sin(2πx/L)] |
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Fourier transform of u(x) |
F(w) = ∫ e^-iwx u(x) dx
integral from -∞ to ∞ |
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Inverse Fourier transform of u(x) |
u~(x) = 1/2π ∫ e^-iwx F(w) dx
integral from -∞ to ∞ |
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Fourier inversion theorem |
u~(x) = ∫ ∫ e^iw(x-y) u(y) dydx
double integral, both from -∞ to ∞ |
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First Shift Theorem |
f(x) --> F(w) g(x) = e^iαx f(x) --> G(w)
Then G(w) = F(w-α) |
g(x) has α>0 |
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Second Shift Theorem |
f(x) --> F(w) g(x) = f(x-α) --> G(w)
Then G(w) = e^-iαx F(w) |
both α <0 |
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Derivative Theorem (1st order) |
u(x) --> F(u)(w) and |u(x)|->0 as |x|->∞
Then u'(x) --> F(u')(w) = iw F(u)(w) |
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Derivative Corollary (2nd order) |
F(u'')(w) = -w² F(u)(w) |
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For u(x) = e^-α|x|, what is F(w)? |
F(w) = 2α / (α² + w²) |
dirac delta function |
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If u(x) = 1, what is F(w)? |
F(w) = 2π δ(w) |
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convolution formula |
h = f * g = ∫ f(y)g(x-y)dy
integral from -∞ to ∞ need to check all cases! |
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Convolution Theorem |
f(x) --> F(w) g(x) --> G(w) Then H(w) = F(w)G(w) and h(x) = f * g |
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FT of 1st order equations |
au' + bu = h(x)
aiwF + bF = H
F(w) = H(w) / (b + iaw) |
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FT of 2nd order equations |
au'' + bu' + cu' = h(x)
-w²aF + iwbF + cF = H
F(w) = H(w) / (-aw² + iwb + c) |
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If G(w) = 1 / (b + iaw), what is g(x)? |
g(x) = 1/a e^-bx/a, x>0 = 0, x<0 = 1/2a, x=0 |
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square integral of u(x) |
‖u‖₂² = ∫|u(x)|²dx
‖F‖₂² = ∫|F(w)|²dw
integral from -∞ to ∞ |
like the 2-norm... |
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Plancheval's Theorem |
‖u‖₂² < ∞ ↔ ‖F‖₂² <∞ and ‖u‖₂² = 1/2π ‖F‖₂² |
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Steps for solving linear PDEs |
1. Take PDE for function h(x,y) with -∞ 2. Take FT wrt x: u(x,y) --> F(w,y) 3. All derivatives wrt x become multiples of wⁿF. F(w,y) then satisfies an ODE in y. 4. Solve the ODE applying initial/boundary conditions. 5. Transform back to give u(x,y) (often as a convolution)
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generalised Sturm-Louisville equation |
(p(x)u')' -q(x)u + λr(u) = 0 & BCs |
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Sturm-Louisville operator |
Lu ≡ (pu')' - qu |
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Dirichlet boundary conditions |
u(a) = u(b) = 0 |
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Neumann boundary conditions |
u'(a) = u'(b) = 0 |
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eigenvalue - eigenfunction pair |
(λ,θ(x)) s.t. Lθ + λrθ = 0 and θ satisfies αu(a) + βu'(a) = 0 and γu(b) + µu'(b) = 0 |
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weighted inner product |
_r = ∫ vur dx = _r
integral from a to b r(x) >0, u(x), v(x) functions defined on I = (a,b) |
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showing orthogonality wrt inner product |
for u and v if _r = 0 |
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Orthogonality Theorem |
L a SL operator with weight r and (λn,θn) satisfies Lθn + λnθn = 0. Then λn∈R ∀n and <θn,θm>_r = 0 for n ≠ m |
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Green's Identity |
u,v functions defined over I and L a SL-operator Then ∫(v.Lu - u.Lv)dx = [p(vu'-uv')] integral from a to b |
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Lagrange's Corollary |
vLu - uLv = (p(vu' - uv') |
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square norm of a function θ |
‖θ‖²_r = ∫|θ|²_r dx
integral from a to b |
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orthonormal set |
a set Θ={θ_n(x)} is orthonormal if Θ is orthogonal and ‖θ_n‖_r = 1 ∀n |
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error, E_N |
the error of the approximation of a function is E_N = ‖f - f_N‖²_r |
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When is E_N minimised? What is the minimum value? |
∀α_n ∈R when α_n = _r
E_N = _r - ∑²_r sum from n=0 to N |
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generalised Fourier coefficient |
α_n = _r |
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Bessel's inequality |
∑²_r ≤ _r sum from n=0 to N |
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complete orthonormal set |
an orthonormal set Θ is complete if E_N = →0 as N→∞ for all functions f where ‖f‖_r<∞ |
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Parseval's Theorem |
If Θ is complete then ∑<θ_n,f>²_r = ²_r |
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inhomogeneous Sturm-Louisville equation |
takes the form Lu + λu r(x) = f(x) plus BCs |
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Fredholm Alternative |
If Lu + λur = f, then (i) if λ≠λ_n, this has a unique solution given by u(x) = ∑[( ∫ f(x)θ_n(x) dx) / (λ - λ_n)] θ_n(x) sum from n=0 to ∞, integral from a to b (ii) if ∃n s.t. λ=λ_n, a necessary condition fo a solution is ∫ f(x)θ_n(x) dx = 0 so that the solution is u(x) = ∑[( ∫ f(x)θ_m(x) dx) / (λ - λ_m)] θ_m(x) + γθ_n(x) for arbitrary γ sum for m≠n, integral from a to b |
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