Problem #5
UCI. MAE200B, winter 2006. by nasser Abbasi.
Problem: Solve the following problems by transform methods
(a)
Consider the special case when
.
Sketch the corresponding
for various values of
.
Solution for part (a)
Some definitions first. The Fourier transform is defined as (using textbook definition)
and the inverse Fourier transform is
And the convolution is defined as
The shift property:
If
has Fourier transform
then
has the Fourier transform
The delay property:
If
has fourier transform
then
has fourier transform
If
has fourier transform
then
has fourier transform
and
And the property of differentiation
Now start the solution by assuming that
Start by taking the fourier transform of each term in the PDE w.r.t
,
and assume that
and assume that the fourier transform of
is
Hence
and
and
Now take the fourier transform of the PDE and using the above relations
Hence this is now a first order ODE, the solution is
To find the solution
then take the inverse fourier transform of the above
Using the delay property we get
Hence the solution is
When
,
hence
Hence the solution is
The effect of the term
is
to introduce the term
in the exponential as shown above.
has units of distance, so the term
acts as a shift in distance. So it is a transport phenomena. I.e. diffusion
with mass transport.
This is a plot of the solution, I picked
where the dirac delta function is to show how
changes with time at that location.
Problem: Solve the following problems by transform methods
(b)
Solution for part (b)
Start by taking the fourier transform of each term in the PDE w.r.t
,
and assume that
and assume that the fourier transform of
is
Hence
and
Now take the fourier transform of the PDE w.r.t
and using the above relations
Hence this is now a first order ODE, nonhomogeneous. Star by solving the homogeneous ODE. The solution is
To find the solution
then take the inverse fourier transform of the above
I am not sure if we should assume that
as well as part(a) as the problem was not clear. To precessed, I assume so,
else not knowing what
is one can not do anything more here.
When
,
hence
Hence the homogeneous solution is
Not knowing what
is, assume that the particular solution is
which can be found by using method of finding integrating factors, hence the
solution is
Problem: Solve the following problems by transform methods
(c)
Solution for part (c)
Take fourier transform w.r.t
,
assume that
Take fourier transform of the PDE we obtain
This is a second order ODE, the characteristic equation is
solution is
Using the second B.C. that
,
this implies that
must be zero, else
will not vanish as
goes to
.
Hence we obtain
Hence
But
Hence (1) becomes
at
,
hence
Hence
So the solution is
Problem: Solve the following problems by transform methods
(d)
Solution for part (d)
Taking Fourier transform w.r.t
assume that
Hence the PDE becomes
This is a second order ODE. The characteristic equation is
Hence
Hence the solution is
Apply BC, at
Apply second BC
at
,
hence
Hence we have 2 equations
Solve for A,B
Hence
and
Hence solution is
Hence
But
and
Where
is the sign function which gives -1, 0 or 1 depending on whether
is negative, zero, or positive
and
and
Hence