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 (neglect friction).
(a) Rescale the problems, so there is a
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 equation as 2 rst order ones and locate all the equilib
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 friction).
(a) Rescale the problems, so there is a small pa
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 must involve both space and time, so we will construct
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 order ones and locate all the equilibria
points. (The
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 cases given correspond to four possible sets of inequa
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 correspond to four possible sets of inequalities satised by th
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 (neglect friction).
(a) Rescale the problems, so there is a sm
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 can make the problem unsolvable. We will consider a fe
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,
~x = f~(~x) ,
(1)
where ~x Rn and f~ : Rn Rn . For su
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 system
x(1 x y ) = 0 ,
y ( y x/2) = 0 .
Clearly one so
Math 3260 - Applied Differential Equations
Homework #1 Due Wednesday Jan 25
1. Consider the species competition model
k1 N1 12 N2
dN1
= r1 N1
,
dt
k1
dN2
k2 N2 21 N1
= r2 N2
.
dt
k2
For the questions below, consider the four cases:
case 1
case 2
case 3
ca
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, give values of for which it is stable.
2. Consider
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) = 0 .
x
x
(a) Find all spatially homogeneous steady-s
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) = 0 .
x
x
(a) Find all spatially homogeneous steady-sta
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, give values of for which it is stable.
2. Consider th
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)
where x Rn and F : Rn Rn . For such systems, an equilibriu
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.
An interesting question is now to estimate how good a
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 velocity (neglect friction).
(a) Rescale the problems, so ther
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,
B and C depend on u, but not on its derivatives. If in