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) Consider the total thermal energy between x = a and x = b.
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 situations, including:
1. Mixed boundary conditions: For
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 region). In what follows, the derivation will be in R3
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, at least in principle, compute ux integrals of vector el
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 1
1. Introduction
A partial dierential equation to be a
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
= c2 2 .
t
x
(1)
It was derived from the modelling of a v
Family Name:
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Signature:
I.D. No.:
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. Put your name, signature and I.D. No. in the blanks ab
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 conditions
u
(0, t) = 0,
x
u
u(L, t) = 0,
x
(2)
and initial
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. Here , Laplaces equation assumes
the simple form
d2 u
= 0,
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 made a
great number of contributions to the study of so
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 following quasilinear PDE:
u
u
+ c(u, x, t)
= Q(u, x, t).
t
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 well-dened. That being said, it is often necessary to ext
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 equation on R,
2u
u
= k 2,
t
x
< x < ,
(1)
with initi
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. Recall that they are given
by
umn (r, , t) = fmn (r )gm ( )h
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 given by
u
= k 2 u
t
(heat equation),
(wave equation).
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
Laplacian in planar polar coordinates:
2 u(r, ) =
1
u
1 2u
r
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 thermal energy within an arbitrary interval [a, b]. Cons