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Homework 6: Fourier transforms
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 < ?.
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.
? exp( ikx ? k ) dk
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,
ut = uxx + cos(bx) = uxx + (eibx + e?ibx ),
?? < 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!)
This question was answered on: Sep 18, 2020
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