Unformatted text preview: ntradiction.
y
Apply second BC: uy (0,, y ) = y ) 0 = y ,, which is a contradiction.
Apply second BC: uy (0 y ) = y ) 0 = y which is a contradiction.
y
So the BVP has no solution.
So the BVP has no solution.
So the BVP has no solution.
So the BVP has no solution.
2
x
2
(ii) u(x, 0) = x22,, uy (x, y )y=x = exx.. As above, H (x) = x22 G(x).
(ii) u(x, 0) = x22 uy (x, y ) y=x = ex As above, H (x) = x22 G(x).
x
x
(ii) u(x, 0) = x2,, uy (x, y )y=x = ex .. As above, H (x) = x2 G(x).
(ii) u(x, 0) = x
uy (x, y ) y=x = ex As above, H (x) = x
x
G(x).
y
3
y
y=x
=x
2 x3
Apply second BC: uy (x, y )yy=x = e x ) x22e xx33G(x) = e x.. Thus
Apply second BC: uy (x, y ) y=x = ex ) x22e x 33G(x) = ex Thus
y
2x
x
x
x
y =x = e x )
Apply second3 BC: uy (x, y )y=xx+=3 e ) x e x G(x) = e x.. Thus
Apply second BC: uy (x, y ) y=xx+x3
x e G(x) = e Thus
y
3
3
y =x x3
e x +x 3
e
e x +x 3
2
ex+x 3, H (x) = x2 + e x+x 3 , and the BVP has the unique solution:
x+x , H (x) = x2 + e x+x 3 , and the BVP has the unique solution:
x+x 3
+x
x+x
+x
G(x) = e x 22
G(x) = ex 2
ex
2
x
G(x) = ex 22 ,, H (x) = x2 + e xx222 ,, and the BVP has the unique solution:
H (x) = x2 + exx22 and the BVP has the unique solution:
G(x) =
x2
x2
x
x
x
3
x+x3
exx+xx33
ex+ x 3
2
2
+
2
2
x2 y
x+x (1
2 + ex+x 3(1
u(x, y ) = x 2 + e 2
e xx22yy)..
u(x, y ) = x 2
e x 2yy)
xy
2
x ).
2 (1
2 (1
u(x, y ) = x + x 2
e
u(x, y ) = x + x 2
e
).
x2
x
2
2y
2
2y
y
2
(iii) u(x, 0) = x 22,, uy (1,, y ) = e 22y.. As above H (x) = x22 G(x).
(iii) u(x, 0) = x2 uy (1 y ) = e2 y As above H (x) = x22 G(x).
(iii) u(x, 0) = x 2,, uy (1,, y ) = ey y.. As above H (x) =y x2 G(x).
(iii) u(x, 0) = x
uy (1 y ) = 2 y As above H (x) = x
e2
G(x).
y
y
2
3y
Apply second BC: uy (1,, y ) = e 22y ) e yyG(1) = e22yy ) G(1) = e33yy..
Apply second BC: uy y y ) = e2y ) e yyG(1) = e22yy ) G(1) = e33yy
y (1
2y
y
2y
y
Apply second BC: uy (1,, y ) = e 2 ) e G(1) = e ) G(1) = e3y ..
Apply second BC: uy (1 y ) = e ) e G(1) = e ) G(1) = e
y
This is impossible, so the BVP has no solutions.
This is impossible, so the BVP has no solutions.
This is impossib...
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 Fall '12
 DavidHamsworth
 Trigraph, Emoticon, ASCII art, BVP

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