MA 3457 / CS 4033 HW #6

MA 3457 / CS 4033
HW #6
1. (6 points) Unique Solutions to IVPs
Show that each of the following initial-value problems has a unique solution:
(a) y
0 =
4t
3y
1+t
4 , 0 ≤ t ≤ 1, y(0) = 1
(b) y
0 = cos2
(y), 0 ≤ t ≤ 1, y(0) = π/4
2. (14 points total) Epidemic Model.
A community with L members contains I(t) infected individuals at time t and S(t) is the uninfected/
susceptible individuals. Assuming L is a constant, we have S(t) = L − I(t). For a mild illness, such as
the common cold, everyone continues to be active and the epidemic spreads from those who are infected
to those who are susceptible. Since there are I(t)S(t) possible contacts between these two groups, the
rate of change of I(t) is proportional to I(t)S(t). The model can be stated as the following IVP:
I
0 = kI(L − I), I(0) = Io, 0 ≤ t ≤ 60
where t is in days.
(a) (8 points) Write your own Euler’s method code with h = 0.2 to approximate the solution to above
IVP on [0,60] using the parameters L = 25, 000, k = 0.00003, with initial condition Io = 250.
(b) (4 points) Create a plot that includes the approximations at each mesh point using h = 0.2, 0.1,
0.05, and 0.025. Looking at I(30), what order of convergence are you seeing?
Note: There should be a different value for the infected population at t = 30 for the different
values of h. When we did the integration error check for Trapezoid and Simpsons rule, we looked
at abs(NumericalSolution−Exact)/hp where h
p was the expected theoretical error. The error is
approximately chp
, so when we observed a constant value c in the previous homework for different
h values, we could say that the error was as we expected. Do the same analysis for Euler method
using the solution to h = 0.0125 as the ‘exact’ solution in the calculation. Often, if we do not have
an analytical solution, we can use a finer grid as a good estimate (as long as the method converges
and approaches the true solution as the grid is refined).
(c) (2 points) Estimate the average number of individuals infected by finding the average solution
value. Compare this to fitting a curve to the Euler solution and using mean value theorem for
integrals.
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