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

APM 346 (2012) Home Assignment X

The purpose of this assignment is a survey of material in preparation to Term Test 1. Don't submit it--it will not be graded!

Problem 1

\begin{equation} (t^2+1) xu_t + (x^2+1)t u_x = 0. \label{eq-1} \end{equation}

  1. Find the characteristic curves and sketch them in the $(x,t)$ plane.
  2. Write the general solution.
  3. Where the solution is fully determined by the initial condition $u(x,0)=g(x)$?
  4. Solve equation (\ref{eq-1}) with the initial condition $u(x,0)= x^2$.
  5. Solve equation (\ref{eq-1}) with the initial condition $u(x,0)= x$.

Problem 2

Solve the wave equation with the following initial conditions

\begin{equation} \left\{\begin{aligned} & u_{tt}- 25 u_{xx}= e^{-x-t} ,\qquad&& -\infty <x< \infty\\ & u (x,0) = xe^{-x} ,\\ & u_t(x,0)=e^{-x}. \end{aligned}\right. \label{eq-2} \end{equation}

Problem 3

Consider wave equation with boundary conditions:

\begin{equation} \left\{ \begin{aligned} &u_{tt}-c^2 u_{xx} + \gamma u =0,\qquad &&0<x<L,\\ &(u_x-\alpha u_t)(0,t)=0,\\ &(u_x-\beta u_t)(L,t)=0,\\ \end{aligned} \right. \label{eq-3} \end{equation} with $\alpha,\beta \in \mathbb{C}$, $\gamma\in \mathbb{R}$ and $c>0$.

Therefore $u$ is a complex-valued function. Consider an energy $E(t)$ defined as

\begin{equation} E(t)= \frac{1}{2}\int_0^L \bigl( |u_t|^2 + c^2 |u_{x}|^2 + \gamma |u|^2)\,dx. \label{eq-4} \end{equation}

  1. Find conditions to $\alpha,\beta,\gamma$ such that $E(t)$ does not depend on $t$ for any $u$ satisfying (\ref{eq-3});
  2. Find conditions to $\alpha,\beta,\gamma$ such that $E(t)$ is non-increasing function of $t$ for any $u$ satisfying (\ref{eq-3});

Hint. Each end is independent and conditions are separate for $\alpha$, and for $\beta$.

Problem 4

Find solutions $u$ of IVP for a heat equation \begin{equation} \left\{\begin{aligned} & u_t-u_{xx}=0 \qquad -\infty < x < \infty ,\\ & u (0,x)=f(x) \end{aligned}\right. \label{eq-5} \end{equation} where

  1. $f(x)=\theta(x):=\left\{ \begin{aligned} & 0 \qquad &&x<0,\\ & 1 &&x>0; \end{aligned}\right.$
  2. $f(x)=\left\{ \begin{aligned} & 0 \qquad &&|x|<1,\\ & 1 &&|x|>1. \end{aligned}\right.$

Hint. In (b) represent $f(x)$ as $\theta (x+1)-\theta(x-1)$.

Remark. Solution must be expressed through

$$\erf(z)= \sqrt{\frac{2}{\pi}}\int_0^z e^{-\frac{s^2}{2}}\,ds.$$