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