Reactiondiffusion equations and bifurcation theory
MATH XXX

Spring 2012
Homework 2
This problem set is due on March 12 (Tuesday) unless with my special permission.
1. Consider the growth equation y 0 =
y
, with y(0) = h/2.
h2 + y 2
(a) Find the (implicit) solution of the equation.
(b) Show that the solution is a convexconcav
Reactiondiffusion equations and bifurcation theory
MATH XXX

Spring 2012
Homework 4
This problem set is due on March 26 (Tuesday) unless with my special permission.
Rule of extra problem: If you solve one extra problem, you can get a homework waiver for another homework assignment.
1. Consider the chemical in river problem: (d
Reactiondiffusion equations and bifurcation theory
MATH XXX

Spring 2012
Homework 1
This problem set is due on March 5 (Tuesday) unless with my special permission. Maximum possible stars
earned from this assignment: 10.
1. Find the bifurcation points for y 0 = y(1 y)2 , and plot the bifurcation diagram using Matlab.
y
y
)
(y 0
Reactiondiffusion equations and bifurcation theory
MATH XXX

Spring 2012
Homework 9
This problem set is due on April 30th (Tuesday) unless with my special permission.
1. Consider the following epidemic with vaccination model (SIRV model):
S 0 = (1 )d dS SI S + (1 S I R) + R,
I 0 = SI + (1 S I R)I (d + )I,
R0 = I (d + )R,
V =
Reactiondiffusion equations and bifurcation theory
MATH XXX

Spring 2012
Homework 5
This problem set is due on April 2nd (Tuesday) unless with my special permission.
Rule of extra problem: If you solve one extra problem, you can get a homework waiver for another homework assignment.
Problem 14 are for the FisherKPP equation
Reactiondiffusion equations and bifurcation theory
MATH XXX

Spring 2012
Homework 7
This problem set is due on April 16th (Tuesday) unless with my special permission.
1. Let M be a nonempty set. Suppose that f : M M is a mapping and for all x, y M , if x 6= y
then d(f (x), f (y) < d(x, y). Such an f is called a weak contracti
Reactiondiffusion equations and bifurcation theory
MATH XXX

Spring 2012
Homework 5
This problem set is due on April 9th (Tuesday) unless with my special permission.
Rule of extra problem: If you solve one extra problem, you can get a homework waiver for another homework assignment.
1. Suppose that (u , v ) is a constant equil
Reactiondiffusion equations and bifurcation theory
MATH XXX

Spring 2012
Homework 3
This problem set is due on March 19 (Tuesday) unless with my special permission.
1. Find all polynomial solutions with degree no more than 3 of the Helmholtzs equation uxx +uyy +u = 0
and the modified Helmholtzs equation uxx + uyy u = 0.
2. Fin
Reactiondiffusion equations and bifurcation theory
MATH XXX

Spring 2012
Homework 8
This problem set is due on April 23rd (Tuesday) unless with my special permission.
1. Use the degree argument to show that the equation f (x, y) = (2x2 y 3y, y 2 4xy) = (0, 0) has a
positive solution. (Hint: you have to exclude the trivial solu
Reactiondiffusion equations and bifurcation theory
MATH XXX

Spring 2012
Homework 10 (last one!)
This problem set is due on May 7th (Tuesday) unless with my special permission.
1. (see Homework 1 problem 4) Schnakenberg model is a basic differential equation model to describe
an autocatalytic chemical reaction with possible os