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Unformatted text preview: 2
u(η (0), 0) = 1 + cos α.
5
The solution of the ﬁrst ODE, gives the characteristics in the transformed domain:
η= 1
ξ+α
2 The solution of the second ODE:
1
ξ
u(η (ξ ), ξ ) = Ke 2
Using the initial condition
2
1 + cos α = K
5
Thus 1
But α = η − ξ thus
2 Now substitute back: Thus 1
ξ
2
u(η (ξ ), ξ ) = (1 + cos α)e 2
5
1
ξ
1
2
u(η (ξ ), ξ ) = (1 + cos (η − ξ ))e 2
5
2
1
1
ξ = x+t
2
2
1
5
1
η − ξ = (t − 2x) − ( x + t) = x
2
2
2
1
x+t
u(x, t) = (1 + cos x)e 2
. 249 ...
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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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