Homework # 1
Due: Monday, 9/9
1. (10pts ) a. One way to derive difference methods for y
0
(t) = f (t, y) is to integrate
the equation from t
i
to t
i+1
, i.e.,
Z
t
i+1
t
i
y
0
(t) dt =
Z
t
i+1
t
i
f(t, y) dt
and then use a numerical quadrature rule to approximate the integral of f . Derive the
backward Euler method by using an appropriate numerical quadrature rule. Be sure and
state which quadrature rule you are using.
b. (Graduates only) Use the quadrature rule
Z
b
a
g(x) dx =
b − a
2
h
g(a) + g(b)
i
to approximate
R
t
i+1
t
i
f(t, y) dt and derive a difference equation for y
0
(t) = f(t, y). Is it
explicit or implicit?
2. (10 pts) Identify each method as explicit or implicit. Justify your answer.
a. Y
i+1
= Y
i
+
∆t
2
f(t
i
, Y
i
) + f
t
i
+
∆t
2
, Y
i
+
∆t
2
f(t
i
, Y
i
)
b. Y
i+1
= Y
i
+
∆t
2
f(t
i
, Y
i
) + f
t
i+1
, Y
i+1
c. Y
i+1
= Y
i
+
∆t
2
f(t
i
, Y
i
) + f
t
i+1
, Y
i
+ ∆tf(t
i
, Y
i
)
d. Y
i+1
= Y
i−3
+
4∆t
3
2f(t
i
, Y
i
) − f(t
i−1
, Y
i−1
) + 2f(t
i−2
, Y
i−2
)
e. Y
i+1
= Y
i−1
+
∆t
3
f(t
i−1
, Y
i−1
) + 4f(t
i
, Y
i
) + f
t
i+1
, Y
i+1
3. (10 its) Consider the specific IVP
y
0
(t) = t
3
+ 5t + 1 0 < t < 4 y(0) = 1
a. Perform one step of the forward Euler method with ∆t = .1 and compute the actual
error there (use exact arithmetic). Write the equation of the tangent line to the solution
at the origin. Plot the exact solution on [0, .1], the tangent line at the origin and your
approximation. Is your approximation on the exact solution curve, the tangent line or
neither? Why?
b. We know that the local truncation error in taking a step of the forward Euler method
is given by the remaining terms in the series
(∆t)
2
2!
y
00
(t) +
(∆t)
3
3!
y
000
(t) +
(∆t)
4
4!
y
[iv]
(t) + · · ·
1
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