MAT 5220 HOMEWORK
Due on Wed. Sep. 15
1.2.2. Derive the heat equation for a rod assuming constant thermal properties and no
sources.
(a) Consider the total thermal energy between x and x + x.
(b) Cons
Lecture 7
Equilibrium or steady-state temperature distributions (contd)
Equilibrium temperature distributions for other cases
We may investigate the existence of steady state distributions for other s
Lecture 4
Derivation of Heat Equation in higher dimensions
Section 1.5 of text by Haberman
We now derive the heat equation in higher dimensions, i.e., R2 (to model a thin solid plate) and
R3 (a solid
Appendix to Set 2 of AMATH 353 Lecture Notes
(taken from MATH 227 Lecture Notes by instructor)
Lecture 34
Surface integrals of vector-valued functions ux integrals
From the previous lecture, we can, a
University of Waterloo
Department of Applied Mathematics
AMATH 353: Partial Dierential Equations I
Winter 2010
Lecture Notes
E.R. Vrscay
Department of Applied Mathematics
c
E.R. Vrscay 2010
1
Lecture
Lecture 13
The wave equation - nal comments
Sections 4.2-4.6 of text by Haberman
In the previous lecture, we studied the so-called wave equation in one-dimension, i.e., for a function
u(x, t),
u
2u
=
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AMATH 353 Midterm Exam
University of Waterloo
February 12, 2007
One hour and thirty minutes
RCH 305 6:00-7:30 p.m.
MC 4044 5:00-6:30 p.m.
Instructions:
1.
Lecture 10
Heat equation with zero-ux endpoint conditions (contd)
Section 2.4 of text by Haberman
In the last lecture, we considered the 1D heat equation
2u
u
= k 2,
t
x
(1)
with zero-ux boundary cond
Lecture 16
Laplaces equation - nal comments
To summarize, we have investigated Laplaces equation, 2 = 0, for a few simple cases, namely,
1. On the line, i.e., on R, i.e., 0 x L, in general, a x b. Her
Lecture 34
Method of characteristics (contd)
Relevant section of textbook:
12.6
Let us now review the main idea behind the method of characteristics. Suppose that the function
u(x, t) satises the foll
Lecture 31
Fourier transforms and the Dirac delta function
In the previous section, great care was taken to restrict our attention to particular spaces of functions
for which Fourier transforms are we
Lecture 28
Solution of Heat Equation via Fourier Transforms and Convolution Theorem
Relvant sections of text:
10.4.2, 10.4.3
In the previous lecture, we derived the unique solution to the heat/diusion
Lecture 25
Vibrating circular membrane (conclusion)
Some normal modes of vibration
Relevant section of text:
7.8.3 (p.
320-1)
We now examine a few normal modes of vibration of the circular drum. Recal
Lecture 22
Higher-dimensional PDEs
Relevant section of text:
Chapter 7
We now examine some PDEs in higher dimensions, i.e., R2 and R3 . In general, the heat and wave
equations in higher dimensions are
Lecture 19
The Rayleigh quotient for approximating eigenvalues and eigenfunctions
Relevant section of text:
5.6
The Rayleigh quotient is named in honour of John Strutt, Lord Rayleigh (1842-1919), who
Some useful facts
Fourier sine and cosine series:
An sin
nx
,
L
nx
,
L
A0 =
f (x) =
n=1
f (x) = A0 +
n=1
An cos
An =
1
L
L
2
L
L
0
f (x) dx,
0
nx
dx.
L
f (x) sin
An1 =
2
L
L
0
f (x) cos
nx
dx
L
Laplac
AMATH 353
Assignment 1
Due : Monday, January 16, 2012
Instructor: K. D. Papoulia
1. Derive the heat equation for a rod with variable cross-sectional area A(x) and no sources by considering the
total t