THE PROBLEM I NEED HELP WITH,
Solve: BUT IT'S PREREQUISITE TO THE
d d
2 PROBLEM I NEED HELP WITH,
u ( X , t) = k u ( X , t) - u ( X , t)
dt dX
2 LOOK AT THE SECOND PAGE!!!
with boundry condition: - < X <
and initial condition u ( X , 0) = f ( X)
2
d d
U ( X , t) = k U ( X , t) - U ( X , t)
dt 2
dX
Taking the Fourier Transform of both sides we see that:
d ( 2
U ( , t) = k -i ) U ( , t) - U ( , t)
dt
d 2
U ( , t) = -k U ( , t) - U ( , t)
dt
Solving this simple O.D.E. we get:
(
- k 2 t ( - t)
U ( , t) = F ( ) e e
)
Applying the inverse Fourier Transform to both sides we see that:
U ( , t) e
- i X
d =
F ( ) e e
( e
)
- k 2 t ( - t) - i X
d
- -
u ( X , t) =
F ( ) e e
( e
)
- k 2 t ( - t) - i X
d
-
u ( X , t) = e
( - t)
F ( ) e
(
- k 2 t - i X
e d
)
-
And now we can either apply the convolution theorm, or notice
directly from our text (pg. 467) that the R.H.S. of this equation is:
So we now have (after simplifying):
2
- ( X)
f ( X)
2
e
( - t) 1
e
4k t
dX ( )
- X
2
-
k t ( - t)
f ( X) e
e 4k t
u ( X , t) = dX
4 kt -
d d2
And this is the solution to: u ( X , t) = k u ( X , t) - u ( X , t)
dt 2
dX
with boundry condition: -
Homework Set # 6 Page - 6 of 13
Problem - 2, 10.4.4.B
Does the solution sugest a simplifying transformation?
This is the previous problem and it's awnser below -
the current problem pre-suposes that you know this:
Solve:
2
d d
u ( X , t) = k u ( X , t) - u ( X , t)
dt 2
dX
with boundry condition: - < X <
and initial condition u ( X , 0) = f ( X)
The solution is:
2
( )
- X
f ( X)
( - t) 1 4k t
u ( X , t) = e e dX
2
-
kt
In this section we're working with Fourier transform and inverse
transform, and the Convolution Theoorm. - I'm not really sure what
kind of simplifying transformation this could be looking for?
Homework Set # 6 Page - 7 of 13
