Math 401-501/502 Current Assignment — Spring 2013

Assignment 9 — Due Thursday, 4/18/13.

• Problems
  1. Find the Fourier transform of the funtion $f(x)$. You may use table A.2 (p. 314) in the text.
     1. $f(x) = \left\{\begin{array}{cc} e^{-x}& x \ge 0\\ 0 & x<0 \end{array} \right.$
     2. $f(x) = \left\{\begin{array}{cc} xe^{-x}& x \ge 0\\ 0 & x<0 \end{array} \right.$
  2. 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 $f(x) = \frac{\sin{2x}}{x}$.
  3. Use the convolution theorem and A.2, #7, to find $f*g$, if $f(x) = e^{-4x^2}$ and $g(x) =e^{-12x^2}$.
  4. Find the inverse Fourier transform of \[ \hat f(\omega) = \sin(\omega) e^{-2| \omega|} \]
  5. Use the Fourier transform method for the infinite bar to solve this convective heat flow problem: \[ \frac{1}{k}\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}+ \alpha \frac{\partial u}{\partial x}, \ u(x,0)=f(x),\ -\infty < x < \infty, \ \alpha>0. \]
  6. 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).
  7. Solve this wave equation for an infinite string: \[ \frac{\partial^2 u}{\partial t^2} = 9\frac{\partial^2 u}{\partial x^2}, \ u(x,0) =0,\ \frac{\partial u}{\partial t}\!(x,0) = xe^{-x^2},\ - \infty < x < \infty, \ t\ge 0. \]