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| Table 1.1: Class lectures review
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date |
book section |
note |
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1 |
Sept 3, 2019 |
Chapter 1 |
Order of ODE, On Laplacian, why it shows
up so frequently everywhere, review |
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2 |
Sept 5, 2019 |
Chapter 2 |
Transport PDE \(u_t+c u_x=0\), characteristic lines.
Transport with decay \(u_t+c u_x+ au=0\) |
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3 |
Sept 10, 2019 |
Chapter 2 |
Continue with Transport PDE \(u_t+c u_x=0\), examples
\(u_t+(x^2-1)u_x=0, u(0,x)=e^{-x^2}\) |
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4 |
Sept 12, 2019 |
Chapter 2.4 |
Wave equation \(u_{tt}=c^2 u_{xx}\), derivation of d’Alembert
solution on infinite line. Example. Domain
of influence. Also with external force.
Resonance |
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5 |
Sept 17, 2019 |
Chapter 3 |
Starting
Fourier series. Heat PDE \(u_t= k u_{xx}\). Separation
of variables. Periodic boundary conditions
(ring). Obtain Fourier series solution. How
to find coefficients, convergence, etc... |
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6 |
Sept 19, 2019 |
Chapter 3 |
More Fourier series. \(f(x) \in L_2\), definition of
norm of \(f(x)\), basis functions. How to
find Fourier coefficients. Example using
\(f(x)=x\). Definitions, jump discontinuity. Fourier
series convergence theorem. |
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7 |
Sept 24, 2019 |
Chapter 3 |
even and odd functions. Complex Fourier
series. Example. |
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8 |
Sept 26, 2019 |
Chapter 3 |
Integration of Fourier series. Find F.S.
of \(f(x)\) using integration of known F.S. for
\(g(x)\). Convergence of functions Uniform and
piecewise. M test. |
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9 |
Oct 1, 2019 |
Chapter 3 |
More on convergence. Convergence in
norm. Definitions
and examples. More theories on Fourier
series convergence. Bessel inequality. Proof
(long). Riemann-Lebesgue Lemma |
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10 |
Oct 3, 2019 |
Chapter 3 |
Decay and smoothness of Fourier series.
Proof of the Fourier series convergence
theorem. Dirichlet kernel. |
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11 |
Oct 8, 2019 |
N/A |
First exam |
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12 |
Oct 10, 2019 |
Chapter 4 |
Heat ODE \(u_t=k u_{xx}\), going over instantaneous
smoothness. Transport PDE we can go
back and forward in time, but not with
heat PDE. Heat PDE with non zero
boundary conditions |
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13 |
Oct 15, 2019 |
Chapter 4 |
Root cellar problem. Solving heat PDE
in complex domain example. Starting on
wave equation. Fourier series solution |
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14 |
Oct 17, 2019 |
Chapter 4 |
Solving wave PDE on finite domain
using d’Alembert. 2 cases. B.C. B.C. is
Neumann and B.C. is Dirichlet (Even and
Odd extension of initial position). Solving
Laplace PDE \(u_{xx}+u_{yy}=0\) on rectangle. |
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15 |
Oct 22, 2019 |
Chapter 4.3 |
Laplace in disk.
Polar coordinates. Separation of variables.
Converting back the solution from polar
to Cartesian coordinates. Closed form
integral formula. |
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16 |
Oct 24, 2019 |
Chapter 4.4 |
Closed form integral solution for Laplace
PDE inside disk. thm 4.6 and thm 4.9 (max
or min of solution at boundary), thm 4.11.
Classification of PDE’s. General formula to
find characteristic curves. |
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17 |
Oct 29, 2019 |
Chapter 6 |
Delta function. Definitions. Two cases,
using limits and using integral. Integration
of delta function, differentiation.
Introduction to Green function |
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18 |
Oct 31, 2019 |
Chapter 6.2 |
Green
function. Examples for \(-u''(x)=f(x)\) with Dirichlet and
Neumann B.C. Full derivation |
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19 |
Nov 5, 2019 |
Chapter 6.2 |
More Green function. Neumann B.C.
Higher dimensions Green function. Laplace
on square. Exam review |
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20 |
Nov 7, 2019 |
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Second exam |
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21 |
Tuesday Nov 12,
2019 |
Chapter 6 |
Green function in higher dimensions. On
whole plane. Green formula. Review of
multivariable calculus. Derivation of Green
function in 2D and 3D on whole space.
Exam 2 returned. |
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22 |
Thursday Nov 14,
2019 |
Chapter 6 |
Green function. Method of images. half
space and disk. Eigenfunctions |
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23 |
Tuesday Nov 19,
2019 |
Chapter 6 |
Eigenfunctions
and eigenvalues for Laplacian in 2 and
3D. Behaviour of eigenvalues, Weyl law for
eigenvalues. Solving PDE on 2D. |
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24 |
Thursday Nov 21,
2019 |
Chapter 6 |
Laplacian is energy minimizer.
Equivelance between \(E(u)=\int{ 1/2 | \triangle (u) |^2 -f u \, dx}\) and solution to \(-\triangle (u)=f\)
with Dirichlet B.C. Proofing that if \(u\) solves
Laplace PDE then it minimizes the energy.
And proofing that if \(u\) minimizes energy
then it solves Laplace PDE |
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25 |
Tuesday Nov 26,
2019 |
Chapter 6 |
Fourier transform. Derivations and two
examples using a box function and
Gaussian \(e^{-x^2}\) |
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