Math 401-501/502 Current Assignment — Spring 2013
Assignment 9 Due Thursday, 4/18/13.
- Problems
- Find the Fourier transform of the funtion $f(x)$. You may use
table A.2 (p. 314) in the text.
- $f(x) = \left\{\begin{array}{cc} e^{-x}& x \ge 0\\ 0 & x<0
\end{array} \right.$
- $f(x) = \left\{\begin{array}{cc} xe^{-x}& x \ge 0\\ 0 & x<0
\end{array} \right.$
- Show that if $F(\omega)$ is the Fourier transform of $f(x)$, then
the Fourier transform of $F(x)$ is $f(-\omega)$. (Hint: this follows
straight from the definition of the Fourier transform and inverse
Fourier transform.) Use this and table A.2 to find the Fouier
transform of $f(x) = \frac{\sin{2x}}{x}$.
- Use the convolution theorem and A.2, #7, to find $f*g$, if $f(x)
= e^{-4x^2}$ and $g(x) =e^{-12x^2}$.
- Find the inverse Fourier transform of
\[
\hat f(\omega) =
\sin(\omega) e^{-2| \omega|}
\]
- Use the Fourier transform method for the infinite bar to solve
this convective heat flow problem:
\[
\frac{1}{k}\frac{\partial u}{\partial t} = \frac{\partial^2
u}{\partial x^2}+ \alpha \frac{\partial u}{\partial x}, \
u(x,0)=f(x),\ -\infty < x < \infty, \ \alpha>0.
\]
- For $f(x) = \left\{\begin{array}{cc} 1 &0 \le x \le 1\\ 0 &
x<0 \end{array}\right.$ and $\, k=\alpha =1$, find a solution to the
previous problem in terms of the error function, erf(x).
- Solve this wave equation for an infinite string:
\[
\frac{\partial^2 u}{\partial t^2} = 9\frac{\partial^2
u}{\partial x^2}, \ u(x,0) =0,\ \frac{\partial u}{\partial t}\!(x,0)
= xe^{-x^2},\ - \infty < x < \infty, \ t\ge 0.
\]
Updated 4/11/2013 (fjn)