This preview shows page 1. Sign up to view the full content.
Unformatted text preview: s 36ˆ + 3 (⇠ ⌘ )ˆ + 4 (⇠ ⌘ )ˆ + sin( 4 ⇠ 1 ⌘ )ˆ = 0.
u
3
3u
1
1
1
4
1
Canonical form: u
ˆ
(⇠ ⌘ )u
(⇠ ⌘ )ˆ
u
sin( 3 ⇠ 3 ⌘ )ˆ = 0.
u
108
27
36 6. a) Given uxx +Dux +Euy +F u = g (x, y ), (where E 6= 0), we let u (x, y ) = eax+by v (x, y ),
as instructed. Then
ux = aeax+by v + eax+by vx
uy = beax+by v + eax+by vy
uxx = a2 eax+by v + 2aeax+by vx + eax+by vxx .
Substituting into the DE gives
eax+by a2 v + 2avx + vxx +Deax+by (av + vx )+Eeax+by (bv + vy )+F eax+by v = g (x, y ) .
Collecting terms, this is
vxx + (2a + D) vx + Evy + a2 + Da + Eb + F v = e ax by g (x, y ) . b) Choosing a = D/2 and b =
a2 Da F /E = D2 /4 F /E will
eliminate the second and fourth terms, giving the following DE for v (x, y ):
vxx + Evy = e
c) Here we have D =
1
(
a=
and b =
2
the PDE will be ax by g (x, y ) . a, E = 1, F =
, and g (x, y ) = 0, so we use
2
1) /4 +
= (1 + )2 . The simpliﬁed form of
1
vxx vy = 0. AMATH 350 Page 6 Assignment #8 Solutions  Fall 2013 7. Solve the IVP
ut = xux , u(x, 0) = 5
,
x3 x>0 by the method of separation of variables.
Solution:
Assume that u(x, y ) = F (x)G(t). Then ut = F (x)G0 (t) and ux = F 0 (x)G(t),
so
ut = xux =) F (x)G0 (t) = xF 0 (x)G(t).
Isolating functions of x and t, we have xF 0 (x)
G0 ( t )
=
=.
F ( x)
G( t )
Here we have also drawn the conclusion that the expressions in each variable
must equal a constant, since they are equal for every value of x and every value
of t. We have thus separated the problem into two ODEs:
xF 0 (x) = F (x),
Z
Z
dF
=)
=
dx,
F
x =) ln F  = =) =) l n x + C1 ,  F  = e C1 e G 0 ( t ) = G( t )
Z
Z
dG
=
dt
G ln G = t + C2 ln x  G  = e C2 e F = C3 x ,
=) G( t) = C4 e u(x, t) = C5 x e t t t Applying the initial condition, we have
5
= C5 x ,
x3
so we must have C5 = 5 and = 3; t...
View
Full
Document
This note was uploaded on 02/08/2014 for the course AMATH 350 taught by Professor Davidhamsworth during the Fall '12 term at University of Waterloo, Waterloo.
 Fall '12
 DavidHamsworth

Click to edit the document details