Assignment schedule:

THESE WEBPAGES DESCRIBE HOW THIS COURSE WILL BE RUN.
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Homework: Typically there will be one homework per week. The HW will be posted here every Wednesday (at the latest) and it will due the following Wednesday (3am; so really Tuesday night) in Gradescope (through Canvas). HW is posted 7 days before due date so you are encouraged to start on it early! Late HW will NOT be accepted.

Homework: Typically there will be one homework per week. The HW will be posted here every Wednesday (at the latest) and is due the following Wednesday and must be handed in as soon as you enter the classroom before the lecture starts . In some occasions the HW will be posted a few days before Wednesday so you are encouraged to start on it earlier. Late HW will not be accepted.

Assignments:

Week #
 Topics: Sec.: Exercises: Due:
01
 19/01
Introduction
Heat equation
Derivation of heat equation in 1D
1.1
Notes: #1
    HW#00:
  • Go over the Matlab tutorial (HTML, PDF) [For students without previous knowledge on Matlab (if you are an "expert" in Matlab just send me an email indicating so)]:
    Type/explore all instructions in Matlab (follow the whole tutorial) and submit to me by email a diary of it.
    To do the diary:
    • Write "diary LastName_M531.txt" as the first instruction.
    • Type/explore the tutorial. Everything you key will be saved.
    • Type "diary OFF". This will close the diary and save the file in your current directory (nothing is saved to the file until you close the diary).
    • If you want to do the diary in several seatings, just open different diaries every time using "diary LastName_M531_A.txt", "diary LastName_M531_B.txt", etc...
    • Email me your diary(ies).
We 26 Jan
02
 24-26/01
1. Heat Equation:
1.1 Intro
1.2 Derivation of heat equation in 1D
1.3 Boundary Conditions
1.4 Steady State
1.4.1 Prescribed BCs
1.4.2 Insulated BCs
1.5 Derivation in 3D
1.1 Laplacian, polar and cylindrical coord.

2. Method of Speration of Variables:
2.1 Introduction
2.2 Linearity
2.3 Heat eq with zero BCs
2.3.1 Introduction
2.3.2 Separation of Variables
2.3.3 Time-dependent ODEs
1.2-1.5 + 2.1-2.3
Notes: #2, #3
     HW#01:
  • 1.4.1.(a) + 1.4.1.(e) + 1.4.1.(f)
  • 1.4.3 + 1.4.7
  • 1.5.13 + 1.5.22
  • 2.2.4
We 02 Feb
03
 31/01-02/02
2. Method of Speration of Variables:
2.3 Heat eq with zero BCs
2.3.4 Boundary value (eigenvalue) problem
2.3.5 Product solution and superposition principle
2.3.6 Orthogonality of sines
2.3.7 Full solution
2.3.8 Summary

3. Fourier Series: (watch Video in Canvas)
3.1 Introduction
3.2 Convergence
3.3 Sine-cosine FS
3.3.1 Sine FS
3.3.2 Cosine FS
3.3.3 Sine-Cosine FS
3.3.4 Parity of f(x)
3.4 Term-by-term differentiation
3.5 Term-by-term integration
3.6 Complex FS
2.3 + Chap 3
Notes: #4, #5
   HW#02:
  • 2.3.1.(a) + 2.3.1.(b) + 2.3.1.(f)
  • 2.3.2.(b) + 2.3.2.(c)
  • 2.3.3.(b)
  • 2.3.8 + 2.3.11
We 09 Feb
04
 07-09/02
2. Method of Speration of Variables:
2.4 Examples with heat equation (other BVPs)
2.4.1 Rod with iinsulated ends
2.4.1 Thin insulated circular ring
2.4.3 Summary of BVPs
2.5 Laplace equation
2.5.1 Laplace equation in a rectangle
2.5.2 Laplace equation in a disk
2.5.4 Qualitative properties of Laplace eq.
2.4-2.5
    HW#03:
  • 2.4.1 + 2.4.3
  • 2.5.2 + 2.5.5.(a) + 2.5.9
We 16 Feb
05
 14-16/02
2.5.4 Qualitative properties of Laplace eq. (cont)

4. Wave equation: Vibrating Strings and Membranes.
4.1 Introduction
4.2 Vertically vibrating string
4.3 Boundary conditions
4.4 Vibrating string with fixed ends
4.5 Vibrating membrane
Planes Waves
d'Alambert solution

5 Sturm-Liouville (SL) eigenvalue problems
5.1 SL: Introduction
4.1-4.5 + 5.1
    HW#04:
  • 4.4.1 + 4.4.3 + 4.4.8 + 4.4.9 + 4.4.10
    [typo in 4.4.10.(a): u(0,T)=0 🡢 u(0,t)=0]
  • Read+Study Chap#3: turn in an extensive summary of the most important results and formulae
We 23 Feb
06
 21-23/02
5 Sturm-Liouville (SL) eigenvalue problems
SL: Theorems/properties
SL: Self-adjointness
SL: Lagrange's identity
SL: Green's formula
SL: Orthogonal/Real/Unique/Non-unique eigenfunctions
SL: Rayleigh quotient
SL: Vibrations on non-uniform string
SL: Boundary conditions of third kind
5.3-5.8
    HW#05:
  • 5.3.2 + 5.3.3 + 5.3.9
  • 5.5.5
We 02 Mar
07
 28/02-02/03
SL: Boundary conditions of third kind
SL: Approximation properties

7. Higher dimensional PDEs:
7.2 Introduction
7.2 Separation of variables
    Vibrating membrane of any shape
    Rectangular membrane
    Helmholtz equation
    Laplace equation
7.3 Vibrating Rectangular Membrane
5.8-5.10 + 7.1-7.3
    HW#06:
  • 5.6.1
  • 5.7.1
  • 5.8.3 [only (a)+(b)]
  • 5.8.8 [only (a)+(b)+(d)]
Fr 11 Mar
MT#1
 09/03
MIDTERM#1, Wednesday 9 Mar
Chaps 1-5
    Chaps 1 through 5 (all)
 
08
 07-09/03
7. Higher dimensional PDEs:
7.7 Vibrating circular membrane
  Bessel's equation

MT#1
7.4-7.7
    HW#07:
  • 7.3.1 (a, b and c only) + for 7.3.1.a plot a few (5x5) of the spatial templates + [for extra credit] do a movie of solution for f(x)=sum over n and m of Φnm = sin(n*pi*x/L)*sin(m*pi*y/H) for L=10, H=5 and choose a "good" k and time span with the coefficients:
    cn = (4*L*p2*sin((1/2)*n*pi)*sin((1/2)*n*pi/p))/(n*pi*(-n2+4*p2));
    cm = (4*H*q2*sin((1/2)*m*pi)*sin((1/2)*m*pi/q))/(m*pi*(-m2+4*q2));
    Anm = cn*cm*pi2/(L*H);
    with p=5.1 and q=5.1.
  • 7.3.4 (a AND b) + plot a few (5x5) of the spatial templates + [for extra credit] do a movie of solution for f(x) and L and H the same as above (you might need to change sines -> cosines) and choose a "good" c and time span.
  • You might want to check the matlab movie command and the command movie2avi (built in) or mpgwrite [click here to download].
We 16 Mar
09
 14-16/03
7.7 Vibrating circular membrane [VIDEO]
    Bessel's equation
    Singular points of Bessel's equation
    Bessel functions
    Asymptotics of Bessel functions
    The Bessel Eigenvalue problem
    IVP with Bessel functions
    Circularly symmetric case

7.8 More on Bessel functions
    Qualitative properties
    Asymptoptics for eigenvalues
    Zeros
    Series representation
7.7-7.8
    HW#08:
  • 7.7.1 + 7.7.5
  • 7.9.1.b
  • 7.10.1.a
Fr 25 Mar
10
 21-23/03
7.9 Laplace eq. in a cylinder [VIDEO]
    Modified Bessel functions

Wave equation on a Sphere [VIDEO]

7.10 Spherical problems and Legendre Polynomials [VIDEO]
    Spherical problems
    Legendre polynomials
    Associated Legendre function
    Legendre polynomials
    Radial eigenvalue problems
    Spherical cavity
7.9-7.10
     
 
--
28-30/03

SPRING BREAK
 
NO CLASSES
 
11
 04-06/04
8. Non-homogeneous (NH) problems
8.1 Introduction
8.2 Heat flow with sources and NH
    Time independent BCs
    Time dependent BCs
8.3+8.4 Eigenfunction expansion [VIDEO]
8.1-8.4
MT#2
 13/04
MIDTERM#2, Wednesday 13 Apr
Chap 7
    Chap 7 [+ all the SL theory/concepts of Chap 5]
 
12
 11-13/04
MT#2 Review +
MT#2 		     
13
 18-20/04
8.5 Forced vibrating membranes
8.6 Poisson equation
    1D eigenfunctions
    2D eigenfunctions
    Arbitrary geometry

9. Green's function for time-independent problems
Green's function (just intro) [VIDEO]

10. Infinite domain problems
Fourier Transform
Heat eq on an infinite domain [VIDEO]
FT of Gaussian
Heat equation and FT [VIDEO]
8.5, 8.6, 9.1-9.2, 10.1-10.3
    
14
 25-27/04
Dirac delta
Dirac delta and heat eq.
Step function
Convolution integral
Fourier sine & cosine transforms [VIDEO]
Examples of PDEs using FT
10.4-10.6
    HW#09:
  • 8.2.2.a + 8.2.3
  • 8.3.3
  • 8.5.2.b
  • 10.2.2
  • 10.3.5 + 10.3.11
Sa 30 Apr
15
 02-04/05
Intro to Laplace transforms
Dissipative waves
Dispersive waves
Dispersion relation
   HW#10:
  • 10.4 : 10.4.3 + 10.4.7
  • 10.5 : 10.5.2
  • 10.6 : Do an exhaustive summary of Sec. 10.6.5
We 04 May
FINAL
 06/05
FINAL, Friday 6 May
All Chaps
    All Chaps