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Answered: - Directions: answer using complete sentences, and I need pr

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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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