Math 623, W 2007: Homework 1.
For full credit, your solutions must be clearly presented and all code included.
(1) Consider the following initial value problem for the function
u
=
u
(
x
) deﬁned for 0
≤
x
≤
1.
u
xx
+
xu
x
+
u
= 0 and
u
(0) = 1
,u
x
(0) = 0
.
(a) What is the exact solution
u
(
x
)?
Hint
: it is of the form
u
(
x
) =
e
φ
(
x
)
for a polynomial
φ
(
x
).
(b) Write down the ﬁnite diﬀerence scheme for the ODE above, using a forward diﬀerence for
u
x
and a symmetric diﬀerence for
u
xx
.
(c) Same question as in (b) but use a backward diﬀerence for
u
x
.
(d) Same question as in (b) but use a central diﬀerence for
u
x
.
(e) Let
±
n
be the error at grid point
n
, i.e.
±
n
=
u
n

u
(
n
Δ
x
). Using your answers to (a)(d),
compute the values of
±
N
(
N
= 1
/
Δ
x
) for Δ
x
= 2

1
,
2

2
,
2

3
,...
(until the computations
become too slow for your computer). Do this for all three schemes in (b)(d). Plot

log

±
N

as a function of

log Δ
x
. What do you observe?
(2) Consider the following PDE:
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 Fall '08
 CONLON
 Math, Boundary value problem, finite difference scheme

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