LECTURE 2
Dimensional Analysis, Scaling, and Similarity
1. Systems of units
The numerical value of any quantity in a mathematical model is measured with
respect to a system of units (for example, meters in a mechanical model, or dollars
in a nancial model
Solutions: Problem set 3
Math 207C, Spring 2012
1. Consider the following IVP for y (t; ) that describes the logistic growth
of a population with a slowly-varying linear growth rate
y = a(t)y y 2 ,
y (0; ) = c,
where c > 0 and
1
sin t.
2
Find inner [t = O
Solutions: Problem set 4
Math 207C, Spring 2012
1. Consider the BVP
y + xy + y = 0,
y (0) = 2, y (1) = 1
0<x<1
where 0 <
1. Where do you expect a boundary layer? Use a dominant
balance argument to determine the appropriate inner variable, and nd leading o
Solutions: Problem set 6
Math 207C, Spring 2012
1. Consider the following scalar initial value problem for x(t; ):
x = x sin2 t,
x(0; ) = 1.
(a) Write down the averaged equation y = f (y ) and solve for y (t; ).
(b) Solve the full problem and verify expli
Solutions: Problem set 2
Math 207C, Spring 2012
1. Find the rst two terms in the asymptotic expansion as
root of the cubic equation
0 of each
23
x + x2 + 2x + = 0.
Solution
Introducing a scaled variable x = y/ , where y = O(1) as
get
2
1
2
y3 + 2 y2 + y
Solutions: Problem set 5
Math 207C, Spring 2012
1. Use the Poincar-Lindstedt method to nd an asymptotic approximation
e
for small-amplitude periodic solutions of the conservative nonlinear ODE
x + x + x2 = 0
Hint. Look for solutions of the form
x(t; ) = x
Solutions: Problem set 1
Math 207C, Spring 2012
1. A solute is injected at the left end of a pipe of length L and removed
from the right. Suppose that the solute is advected with velocity c0 > 0 and
diuses with disivity , when its density (x, t) satises t
128
LECTURE 5
Stochastic Processes
We may regard the present state of the universe as the eect
of its past and the cause of its future. An intellect which at a
certain moment would know all forces that set nature in motion,
and all positions of all items
94
LECTURE 4
Sturm-Liouville Eigenvalue Problems
Possibly one of the most useful facts in mathematics is that a symmetric matric
has real eigenvalues and a set of eigenvectors that form an orthonormal basis. This
property of symmetric matrices has a wide-
LECTURE 3
The Calculus of Variations
The variational principles of mechanics are rmly rooted in the
soil of that great century of Liberalism which starts with Descartes
and ends with the French Revolution and which has witnessed
the lives of Leibniz, Spin
LECTURE 1
Introduction
The source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance
of a concept of seemingly great generality is in essence the same
as a small and concrete special case.