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Homework 6: Fourier transforms

 

Math 456/556

 


 

For the following, express convolutions as explicit integrals.

 


 

Question 1 (An integral equation) Perform a Fourier transform of

 

?

 


 

e?|y| u(x ? y)dy = g(x).

 


 

??

 


 

to solve for u(x). Check your answer with g(x) = x2 .

 


 

Question 2 Use the Fourier transform to solve ?u = 0 on the lower half plane subject to u(x, 0) = ?(x)

 

and limy??? u = 0.

 


 

Question 3 (Convective diffusion equation) Suppose that in addition to diffusion, there is something (i.e.

 

wind or water currents) pushing around a conserved quantity u(x, t). The ?ux of material is the sum of

 

diffusive and convective ?uxes

 

J = ?Dux + V u

 

Plugging into the general form for conservation laws gives the convection - diffusion equation

 

ut = Duxx ? V ux ,

 


 

?? < x < ?.

 


 

(1)

 


 

Suppose that u(x, 0) = f (x) initially.

 


 

A. Solve this equation by Fourier transform.

 

B. Show that w(y, t) = u(y + V t, t) satis?es wt = Dwyy . Therefore, the solution to (1) is just a translation

 

u(x, t) = w(x ? V t, t) of the solution to the usual diffusion equation.

 


 

3

 


 

2

 


 

1.5

 


 

1

 


 

(2 ?)

 


 

?1

 


 

4

 


 

? exp( ikx ? k ) dk

 


 

2.5

 


 

0.5

 


 

0

 


 

?0.5

 

?15

 


 

?10

 


 

?5

 


 

0

 

x

 


 

5

 


 

10

 


 

15

 


 

Figure 1: Inverse transform computed numerically.

 

Question 4 Consider the fourth order diffusion equation

 

ut = ?uxxxx ,

 


 

?? < x < ?.

 


 

A. Find the transform of the fundamental solution S(x, x0 , t) solving the equation with initial condition

 

S(x, 0) = ?(x ? x0 ) (that is to say, don?t perform the inverse transform yet).

 

B. The inverse transform of part A cannot be written in terms of elementary functions, but one can numerically evaluate the Fourier integral (for various x); its graph is shown above. Call h(x) the function whose

 

Fourier transform is

 

?

 

h(k) = exp(?k 4 ).

 

Use the dilation property for Fourier transforms to write S in terms of h. C. Write the solution for a general

 

initial condition u(x, 0) = f (x) as an integral involving h.

 


 

Question 5 Solve explicitly by transform in x,

 

1

 

ut = uxx + cos(bx) = uxx + (eibx + e?ibx ),

 

2

 


 

?? < x < ?,

 


 

u(x, 0) = 0.

 


 

Your answer should not involve any integrals. (Hint: it might be easier to take the inverse transform directly

 

rather than using table formulas!)

 


 

 


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