1.1 Dimensional Analysis
1.1.1
The Program of Applied Mathematics
1) Applied mathematics is a broad subject area in the mathematical sciences dealing with
those topics, problems, and techniques that have been useful in analyzing real-world
phenomena.
2) I
Line Integrals of Vector Fields
In lecture, Professor Auroux discussed the non-conservative vector eld
F = h y, xi.
For this eld:
1. Compute the line integral along the path that goes from (0, 0) to (1, 1) by rst going
along the x-axis to (1, 0) and then
Problems: Vector Fields
1. Sketch the following vector elds. Pay attention to their names because we will be
encountering these elds frequently.
(a) Force, constant gravitational eld F(x, y) =
gj.
x
y
i+ 2
j = hx, yi/r2 . (This is a shrinking radial eld
2
I. (25 pts) Find the A for Which the integral equation
IE)
006) = /\ /0°° dy 6 (y + 6) W)
has nontrivial solutions. Give the corresponding
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-b
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fa
14094)
18.307: Integral Equations
M.I.T. Department of Mathematics
Spring 2006
Due: Wednesday, 03/22/06
Homework 4
11. (Similar to Prob. 4.1 in text by M. Masujima.) Show the following correspondence between
the kernel K(x, y) of the Fredholm equation and the de
4.4 Generalizations
4.4.1 Higher Derivatives
In the simplest variational problem the Lagrangian depends on x, y, and y'. An obvious
generalization is to include higher derivatives in the Lagrangian. Thus we consider the
second-order problem
b
J ( y ) = L(
5 Boundary Value Problems and Integral Equations
Topics in this chapter form the traditional core of applied mathematics boundary
value problems, orthogonal expansions, integral equations, Green's functions, and
distributions.
All of these subjects are in
3.4 Initial Layers
3.4.1 Damped Spring-Mass System
Initial value problems containing a small parameter can have a layer near t = 0. For
evolution problems these layers are called initial layers. They are zones in which there
are rapid temporal changes.
We
2 Two-Dimensional Dynamical Systems
2.1 Phase Plane Phenomena
We now extend the analysis to a system of two simultaneous equations
x = P( x, y ),
y = Q( x, y ),
(1.1)
in two unknowns x = x(t) and y = y(t). The functions P and Q are assumed to possess
cont
7 Wave Phenomena
Two of the fundamental processes in nature are diffusion and wave propagation. In the
present chapter we investigate wave phenomena and obtain equations that govern the
propagation of waves first in simple model settings.
The evolution eq
2 Two-Dimensional Dynamical Systems
2.2 Linear Systems
Linear systems are important for stability analysis near a critical point.
Linear systems also appear in physics, engineering, mechanics, etc.
Consider a two-dimensional linear system
x = ax + by,
y =
8.2 Momentum and Energy
8.2.1 Momentum Conservation
In classical mechanics, a particle of mass m having velocity v has linear momentum mv.
Newton's second law asserts that the time rate of change of momentum of the particle is
equal to the net external fo
Solutions Manual
Applied Mathematics, 3rd Edition
J. David Logan
Willa Cather Professor of Mathematics
University of Nebraska Lincoln
November 8, 2010
ii
Contents
Preface
v
1 Scaling, Dimensional Analysis (Secs. 1 & 2)
1.1 Dimensional Analysis . . . . . .
7.3 Quasi-linear Equations
A quasi-linear partial differential equation is an equation that is linear in its
derivatives.
Many of the equations that occur in applications (for example, the nonlinear equations of
fluid mechanics) are quasi-linear.
In this
5.4 Integral Equations
An integral equation is an equation where the unknown function occurs under an
integral sign.
Integral equations arise in the analysis of differential equations, and, in fact, many initial
and boundary value problems for differentia
Soums \To Set 19- i
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0
Sec oLé Sub; 1105) J, 13:33.3 «31103)
D
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l
1 Sd[ WWW] = 3 (53B) => {Cl-%)dAB=1
7
P=545£3+At~52p3>1= bug-fizz.) =>
0
The Aejtevmivxant 0: Htis sdem is
1-11/ -A 2 Z A A
'1» 2 (HimW3 =(\
18.307: Integral Equations
M.I.T. Department of Mathematics
Spring 2006
Due: Wednesday, 03/15/06
Homework 3
7. (Similar to Prob. 3.9 in text by M. Masujima.) Solve the integral equation
1, where
xy + (x y)1/2 ,
x>y
K(x, y) =
xy,
x < y.
1
0
dy K(x, y) u(y)