Fourier transform, Fourier integral

$\newcommand{\const}{\mathrm{const}}$ $\newcommand{\erf}{\operatorname{erf}}$

Deadline Thursday, April 2

It is 10-th Home assignment. Still only 7 best count

APM 346 (2015) Home Assignment 10

  1. Problem 1
  2. Problem 2
  3. Problem 3
  4. Problem 4
  5. Problem 5

Problem 1

Consider variational problem under constrain: \begin{align} &\Phi(u):=\frac{1}{2}\int_0^2 u_{xx}^2\,dx,\label{eq-10.1}\\ &\Psi(u)= \frac{1}{2}\int_0^2 u_{x}^2\,dx =1,\label{eq-10.2}\\ &u(0)=u'(0)=u(2)=u'(2)=0.\label{eq-10.3} \end{align}

  1. Write down Euler–Lagrange equation $\delta (\Phi-\lambda \Psi)=0$.
  2. Under boundary conditions (\ref{eq-10.3}) solve it and find out eigenvalues $\lambda$ for each solution exists.

Hint: a. it will be 4th order ODE.

Hint: b. Changing interval to $(-1,1)$ and observing that ODE and boundary conditions are symmetric (survive $x\mapsto -x$) one can consider separately even and odd eigenfunctions and corresponding eigenvalues.

Problem 2

Consider variational problem \begin{equation} \Phi(u):=\frac{1}{2}\int_\Omega (u_x^2 + y u_y^2- 2 yu)\,dxdy \label{eq-10.4} \end{equation} where $\Omega=\{ (x,y):\, 0 < x< 1, 0 < y < 1\}$.

  1. Write down Euler-Lagrange equation
  2. Write down all boundary conditions for it.

Problem 3

Consider distributions now. Calculate right from the definition

  1. $x^3\delta''(x)$;
  2. $x^2\delta''(x)$;
  3. $x^2\delta '''(x)$.

The results may be written through $\delta$-function or its derivatives of the smallest possible order.

Problem 4

Find Fourier transform of $\theta(x)$ (Heaviside function).

Problem 5

Consider wave equation \begin{equation} \rho u_{tt} - (k(x)u_x)_x=0 \label{eq-10.5} \end{equation} on $-\infty < x < \infty$ where $\rho(x)=\left\{\begin{aligned}\rho_-(x)& &&x<0,\\ \rho_+(x)& &&x>0\end{aligned}\right.$ and $k(x)=\left\{\begin{aligned}k_-(x)& &&x<0,\\ k_+(x)& &&x>0\end{aligned}\right.$.

  1. Write down equation (\ref{eq-10.5}) for $x< 0$ and $x > 0$ separately.
  2. Find out transmission conditions (there must be 2 of them) linking $u(-0,t)$, $u(+0,t)$, $u_x(-0,t)$, $u_x(+0,t)$.