Homework Set # 12 - Not Due – Solutions Available on November 25

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1) Let F(ω) be the Fourier (exponential) transform of f(x) and G(ω) be the Fourier transform of g(x)=f(x+a). Show that G(ω)=e^-iaωF(ω).

Solution (pdf format)

2) 20.2.2

3) 20.2.4

4) 20.2.7

5) 20.2.8

6) 20.3.6

7) For a point source at the origin, the three-dimensional neutron diffusion equation becomes

-D∆²φ(r)+K²Dφ(r)=Qδ(r),

∆² stands for the Laplacian. Apply a three-dimensional Fourier transform. Solve the transformed equation. Transform the solution back into r-space.

Solution (pdf format)


8)  a) Consider the function f(x)=1 for |x|<1 and 0 otherwise.

         1) Can you expand f(x) in terms of a sine Fourier transform? Why?

         2) What Fourier transform should you use? Calculate this FT for f(x).

     b) Now consider f(x)=1 for a<x<b with a>0 and b>0 and f(x)=0 otherwise.

         1) Can you expand f(x) in terms of a sine FT? Why?

         2) If the answer to the previous question is yes perform the FT.

         3) For what values of x is your sine transform appropriated?

         4) What assumption are you making about f(-x)?

         5) Can you expand f(x) as a cosine FT? What assumption do you need to make about f(-x)?

         6) If you wanted to obtain a FT of f(x) valid for all x what FT should you use? Why? Find the coefficients .




Solution (pdf format)