8. a. Use the method of characteristics to solve
u
t
+
uu
x
+
cu
=0
with
u
(
x,
0) =
f
(
x
)where
c
is a positive constant.
Let’s rewrite the equation as
u
t
+
uu
x
=
−
cu
Now the characteristic equation is
dx
dt
=
u
x
(0) =
ξ
In order to solve this, we have to Frst Fnd
u
from the second ODE, which is
du
dt
=
−
cu
u
(
ξ,
0) =
f
(
ξ
)
This equation is separable
du
u
=
−
cdt
ln
u
=
−
ct
+ln
f
(
ξ
)
or
u
(
x
(
t
)
,t
)=
f
(
ξ
)
e
−
ct
Taking this to the right hand side of the characteristic equation we have
dx
dt
=
f
(
ξ
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This note was uploaded on 12/22/2011 for the course MAP 2302 taught by Professor Bell,d during the Fall '08 term at UNF.
 Fall '08
 BELL,D

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