Math 3260 - Mathematical Modelling II
Homework #3 due November 13th
1. Consider a pendulum released with a very high initial velocity V , so that it rotates with nearly
constant initial velocity (negl
Math 3260 - Mathematical Modelling I
Homework #1 Due Friday Oct. 4
1. Consider the species competition model we discussed
k1 N1 12 N2
dN1
= r1 N 1
,
dt
k1
k2 N2 21 N1
dN2
= r2 N 2
.
dt
k2
(a) The four
Math 3260 - Mathematical Modelling II
Homework #3 Solutions
1. Consider a pendulum released with a very high initial velocity V , so that it rotates with nearly
constant initial velocity (neglect fric
Math 3260 - Mathematical Modelling II
Solutions
1. The steady-state problem for a the sh population model we studied in class is
2 Pxx + P (1 P ) = 0 .
(a) Express this second order equation as 2 rst
Math 3260 - Mathematical Modelling II
Homework #2 Due October 25
1. The steady-state problem for a the sh population model we studied in class is
2 Pxx + P (1 P ) = 0 .
(a) Express this second order
Spatial Ecological Model
1
The Model
We will now consider the eect of a no shing zone on a population of sh.
Specically, is there a minimum size which will ensure a continued population.
Such a model
Math 3260 - Mathematical Modelling I
Solutions
1. Consider the species competition model we discussed
dN1
k1 N1 12 N2
= r1 N 1
,
dt
k1
k2 N2 21 N1
dN2
= r2 N 2
.
dt
k2
(a) The four cases given corresp
Math 3260 - Applied Dierential Equations
Homework #2 due Feb 10th
1. Consider a pendulum released with a very high initial velocity V , so that it rotates with nearly
constant initial velocity (neglec
Perturbation Methods
The main idea of perturbation methods is to consider a small change from
an easily solved problem. We wish to quantify how the solution changes when
include additional terms which
Systems of Differential Equations
Equilibria and Stability
1
Preliminaries - Stability
We will consider are nonlinear systems of ordinary differential equations. In
other words they will take the form
Math 3260 Practice Exam
1. Consider the system of dierential equations,
x = x(1 x y ) ,
y = y ( y x/2) ,
where is a constant (not necessarily positive).
(a) Find all equilibria. We solve the nonlinear
Math 3260 Practice Exam
1. Consider the system of dierential equations,
x = x(1 x y ) ,
y = y ( y x/2) ,
where is a constant (not necessarily positive).
(a) Find all equilibria.
(b) For each equilibri
Math 3260 - Mathematical Modelling II
Homework #4 Solutions
1. Consider the reaction diusion system
a
2a
= Da 2 + a + a2 1 ah ,
t
x
h
2h
= Dh 2 h2 + 2 ah ,
t
x
h
h
(0) =
( L) = 0 ,
x
x
a
a
(0) =
( L
Math 3260 - Mathematical Modelling II
Homework #4 due Dec 4
1. Consider the reaction diusion system
a
2a
= Da 2 + a + a2 1 ah ,
t
x
h
2h
= Dh 2 h2 + 2 ah ,
t
x
h
h
(0) =
(L) = 0 ,
x
x
a
a
(0) =
(L)
Math 3260 Practice Exam
1. Consider the system of dierential equations,
x = x(1 x y) ,
y = y( y x/2) ,
where is a constant (not necessarily positive).
(a) Find all equilibria.
(b) For each equilibrium
Epidemic Models
1
Preliminaries - Stability
All the epidemic models we will consider are nonlinear systems of ordinary
dierential equations. In other words they will take the form,
x = F (x) ,
(1)
whe
Estimate of the error of linear approximation
So far, we have studied how using some information around a point a we can construct a linear approximation
to approximate nearby values of the function.
Math 3260 - Applied Differential Equations
Homework #2 due Friday Feb 10th
1. Consider a pendulum released with a very high initial velocity V , so that it rotates with nearly
constant initial velocit
Quasi-Linear First Order PDEs
A quasi-linear pde in two variables has general form
A(x, y, u)ux + B(x, y, u)uy = C(x, y, u) ,
where A, B and C are given functions. We say it is quasi-linear because A,