search

# partial differential equation

Exam Practice

### Rating

Showing Page:
1/1
5577-10.2-9E AID: 595|18/09/2012
RID: 175| 19/10/12
Consider the initial – boundary value problem
( ) ( )
( )
, 0 1, 0
0, 20, 1, 10, 0
,0 0, 0 1
xx t
u u x t
u t u t t
u x x
= < < >
= = >
= < <
...... (1)
For the initial - boundary value problem
( ) ( )
( ) ( )
2
0 1
, 0 , 0
0, , , , 0
,0 , 0
t xx
u c u x p t
u t T u p t T t
u x f x x p
= < < >
= = >
= < <
...... (2)
the solution is
( ) ( ) ( )
, ,u x t v x t S x= +
...... (3)
Where
( )
2
2
1
, sin
n
c t
p
n
n
n
v x t b x e
p
π
π
÷
=
=
÷
...... (4)
( )
1 0
0
T T
S x T x
p
= +
( ) ( )
( )
0
2
sin , 1, 2,3
p
n
n
b f x S x x dx n
p p
π
= =
÷
L
...... (5)
Observe that
( )
0f x =
Compare (1) with (2),
1c =
( )
0f x =
0
0T =
1
10T =
So,
( )
10 20
20
1
20 10
S x x
x
= +
=
Substitute
1c
=
,
,
( )
0f x =
,
( )
20 10S x x=
in (5), it follows
( )
( ) ( )
( ) ( )
( )
( )
1
0
1 1
0 0
1
1
2
0
0
2
0 20 10 sin
1 1
20 sin 40 sin
cos sin cos
20 40
cos cos 1
20 0 40
n
n
b x x dx
x n x dx n x dx
x n x n x n x
n n
n
n n
n n n
π
π π
π π π
π π
π
π π
π π π
= +
÷
=
= +
= + +
[ ]
( ) ( )
20
cos 2
20
1 2 Since cos 1 , 1, 2,
n n
n
n
n n
n
π
π
π
π
=
= = =
L
So,
( )
20
1 2
n
n
b
n
π
=
Also, use
1c =
,
,
( )
20
1 2
n
n
b
n
π
=
in (4), it follows that
( ) ( )
( ) ( )
2
2
2 2
1
1
1
1
20
, 1 2 sin
1
20
1 2 sin
n
t
n
n
n
n t
n
n
v x t x e
n
n x e
n
π
π
π
π
π
π
÷
=
=
=
÷ ÷
=
So,
( ) ( ) ( )
2 2
1
20
, 1 2 sin
n
n t
n
v x t n x e
n
π
π
π
=
=
Use this result and
( )
10S x x=
in (3), the solution is
( ) ( ) ( )
2 2
1
20
, 1 2 sin 10
n
n t
n
u x t n x e x
n
π
π
π
=
= +

### Unformatted Attachment Preview

5577-10.2-9E AID: 595|18/09/2012 RID: 175| 19/10/12Consider the initial - boundary value problem ...... (1)For the initial - boundary value problem ...... (2)the solution is ...
Purchase document to see full attachment
User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.
Review
Review

Anonymous
I was struggling with this subject, and this helped me a ton!

Studypool
4.7
Trustpilot
4.5
Sitejabber
4.4