MATH 340
LAB 14 Assignment
ODEs:
Problem 1: System of ODEs
Use Explicit (forward) Euler to solve the following system of ODEs
(
u
0 = 9u + 24v + 5 cos(t) − (1/3) sin(t), 0 ≤ t ≤ 1
v
0 = −24u − 51v − 9 cos(t) + (1/3) sin(t)
(1)
with initial conditions
(
u(0) = 4/3
v(0) = 2/3
(2)
whose actual solution is given by
(
U(t) = 2e
−3t − e
−39t + (1/3) cos(t)
V (t) = −e
−3t + 2e
−39t − (1/3) cos(t)
(3)
Find the approximated solutions at the point t = 1 for h = 20
, 2
−1
, . . . , 2
−9
. Write
your results in a table with columns in order: hi
, the approximated solutions ui(t =
1), vi(t = 1), the errors eui = |U(t = 1) − ui(t = 1)|, evi = |V (t = 1) − vi(t = 1)|.
Check the order of convergence of the method (how does the error depend on h?)
and confirm with our previous results on Explicit Euler for one single ODE.
Problem 2:
Solve the same system above with R-K4 method and print out the corresponding
table. Comment about your results and the order of convergence.
1
