Applied Mathematics II
PROBLEM SET 5
Due March 6, 2003
1. In class we considered t + (1 2)x = 0 with 0 = 2 (1 + x0 ), 1 < x0 < 0 and otherwise zero.
3
From the solution we found,
(a) sketch the density of the pack of cars as a function of x when t = 1.5.
F UNCTIONS OF S EVERAL VARIABLES
M AXIMA AND M INIMA
Functions of one variable (review).
Interpret the function:
as a graph y = f (x)
as the position of a particle y = g(t ) at time t .
The derivative: slope of tangent line or velocity.
At a local maximum
Applied Math II
Spring 2003
The Riemann function for the wave equation in two space dimensions
We have established that the Riemann function in three dimensions is given by
R
(3)
(x, t) =
4c2
x 2 +y 2 +z 2
)
c
2 + y2 + z 2
x
(1) (t
We use the 3 superscr
Applied Mathematics II
PROBLEM SET 4
Due February 27, 2003
1. Solve, using characteristics, the following problem for a nonlinear wave equation:
ut + u2 ux = 0, u(x0 , 0) =
0,
if x0 < 0,
x0 , if x0 0.
Show that u is constant on a characteristic and that t
Some Applications of Linear Algebra
1. Given n linear equations in n unknowns how can you tell
a) when a solution exists?
b) if that solution is unique?
2. Linear maps F (X ) = AX , where A is a matrix, have the property
that F (0) = A0 = 0, so they neces
Applied Mathematics II
PROBLEM SET 3
Due February 20, 2003
1. Problem 1.4, page 34 of text.
2. Problem 2.7, page 41 of text. Gie the integral curves in parametric form. Show in part (b), for
example, that the curves are given by
3
3
x = t + x0 , y = [y0 +
Applied Mathematics II
PROBLEM SET 2
Due February 13, 2003
1. Consider the telegraphers system
ix + Cvt + Gv = 0, vx + Lit + Ri = 0
on the domain 0 x l, t > 0. Multiply the rst equation by v , the second by i, and add to get
(iv )x + C (v 2 /2)t + L(i2 /2
Applied Mathematics
PROBLEM SET 7
April 3, 2003
1. For t 0 let T (x, t) be a bounded solution of the heat equation Tt DTxx = 0 satisfying the initial
condition T (x, 0) = f (x) on the innite interval (, +). Let
T1 (x, t) = T (x, t) +
x
t3/2
x2
e 4Dt .
(a)
Applied Mathematics II
PROBLEM SET 1
Due February 6, 2003
1. The telegraphers equation is intended to model a coaxial electrical transmission line. The currentcarrying wire core is coupled by resistance and capacitance to the grounded covering. The contin
Applied Mathematics
PROBLEM SET 6
Due March 13, 2003
1. Let f (x) have Fourier transform f (k ), that is f (k ) = F (f ). Show that, if f (k ) 0 as k , that
1
F (f (k ) = ixF (f ). From this deduce that xf (x) has the Fourier transform if (k ). Use this
Applied math II
Spring 2003
The bursting balloon reconsidered
Kirchos solution of the IVP for the 3D wave equation is
u(x, t) =
1
4c2 t
S (x,t)
ut (x, t)dS +
1
t 4c2 t
u(x , 0)dS .
Here S is the spherical survace with center at x and radius ct and dS indi