LECTURE 3
Characteristics and First Order Equations
We shall now generalize the methods developed in the preceding lecture.
Definition 3.1. A partial di erential equation in n variables, xi is said to be quasi-linear if it is linear
in the partial derivat
Math 4263
Homework Set 5
1. Show that the following ODEs are of the Sturm-Liouville type
d
dy
p (x)
q (x) y + r (x) y = 0
,
p (x) > 0 , r (x) > 0
dx
dx
and if so identify the functions p (x) , q (x) and r (x).
(a) y + k 2 y = 0
(b) x2 y + xy + x2 2 y = 0
Math 4263
Homework Set 6
1. Show that a function f (z ) = u (z ) + iv (z ) of a complex variable z = x + iy that satises the CauchyRiemann equations
v
u
v
u
=
,
=
x
y
y
x
also has the property that both its real part u (z ) and its imaginary part v (z ) s
Math 4263
Homework Set 4
1. Prove the Maximum Principle for solutions of the homogeneous Laplace equation: i.e., show that any
solution of
uxx + uyy = 0
;
0 x a;0 y b
attains its maximal value on one of the four boundary lines
`1
`2
`3
`4
=
=
=
=
f(x; 0)
Math 4263
Homework Set 3
1. Solve utt c2 uxx = 0, u (x, 0) = ex , ut (x, 0) = sin (x)
2. Solve uxx 3uxt 4utt = 0, u (x, 0) = x2 , ut (x, 0) = ex . (Hint: nd a change of variables that factors
the dierential operator as we did for the wave equation.).
3. S
Math 4263
Homework Set 1
1. Solve the following PDE/BVP
2ut + 3ux
=
0
u (x, 0)
=
sin (x)
2. Solve the following PDE/BVP
ux + ex uy
u (0, y )
=
0
= y2
3.
(a) Find the curves : t (x (t) , y (t) such that
dx
=x
,
dt
that cross the line y = 1 at t = 0.
dy
=y
Math 4263
Homework Set 2
1. Use the Maximum Principle for the Heat Equation to demonstrate that there is a unique solution to
ut k 2 uxx
= f (x, t)
0xL
,
,
t>0
t>0
(1a)
u (0, t)
= g (t)
,
(1b)
u (L, t)
= h (t)
,
t>0
(1c)
u (x, 0)
= (x)
,
0xL
(1d)
0
(2a)
2
Math 4263
Homework Set 7
1.
(a) Use the Taylor expansion formula
1 3f
f
1 2f
2
3
4
(x, y ) (x) +
(x, y ) (x) + O (x)
(x, y ) x +
2
x
2 x
6 x3
to derive the following approximations
u
u (x, t + t) u (x, t)
(x, t)
+ O (t)
t
t
u (x + x, t) 2u (x, t) + u (x
LECTURE 1
Introduction
1. Partial Dierential Equations: Basics
A partial dierential equation (or PDE, for short) is an equation relating a function of n variables
x1 ; : : : ; xn , its partial derivatives with respect to the variables x1 ; : : : ; xn , an
LECTURE 6
Explicit Solutions of the Heat Equation
Recall the 1-dimensional (homogeneous) Heat Equation:
ut a2 uxx = 0 .
(1)
In this lecture our goal is to construct an explicit solution to the Heat Equation (1) on the real line, satisfying
a given initial
LECTURE 7
The Wave Equation
A wave equation (in 1 + 1 dimensions) is a partial dierential equation of the form
2
2
c2 2 = f (x, t) .
2
t
x
Such equations crop up in a variety of physical contexts; vibrating strings, electrical circuits, electromagnetism,
LECTURE 5
The Heat Equation
We now turn our attention to the Heat (or Diusion) Equation:
ut k2 uxx = 0 .
This PDE is used to model systems in which heat or some other property (e.g. the concentration of a
solvent in a solution) distributes itself througho
LECTURE 4
Second Order Linear Equations
1. The Basic Types of 2nd Order Linear PDEs:
1.1. Generic and Standard Forms of 2nd Order Linear PDEs. The generic form of a second
order linear PDE in two variables is
(1)
A(x; y )
@2
@2
@2
@
@
+ B (x; y )
+ C (x;
LECTURE 2
First Order Equations, Contd
Example 2.1. Solve the following rst order linear PDE
(1)
x + 2xy = y
subject to the boundary condition
(0, y ) = 1 + y 2
(2)
,
for1 < y < 2
.
Suppose that (x, y ) is a solution of this PDE/BVP. If we look along the
Math 4263
FINAL EXAM
2:-00 3:50 pm, May 6, 2009
Name:
1. State the Maximum Principle for solutions of the homogeneous heat equation and then use it to prove
the uniqueness of any solution of
ut k 2 uxx
0xL
= f (x, t)
u(0, t)
= (t)
t>0
u(L, t)
= (t)
t>0
u(