Topics

421

Final Exam / FE-3

« on: December 18, 2015, 07:42:21 PM »

\begin{align}
&u_{tt}-  9u_{xx}=0,\qquad 0<x<2, \; t>0,\label{3-1}\\[2pt]
& u (0,t)= u (2,t)=0,\label{3-2}\\[2pt]
& u(x,0)=f(x),\label{3-3}\\[2pt]
& u_t(x,0)=g(x)\label{3-4}\end{align}
with $f(x)=0$, $g(x)=x(2-x)$.  Write the answer in terms of  Fourier series.

422

Final Exam / FE-2

« on: December 18, 2015, 07:41:43 PM »

$\newcommand{\erf}{\operatorname{erf}}$
Solve  IVP for the heat equation
\begin{align}
&u_t - u_{xx}=0,\qquad &&-\infty <x<\infty,\; t>0,\label{2-1}\\[2pt]
&u|_{t=0}= f(x)\label{2-2}
\end{align}
with $f(x)=|x|$.

Solution should be expressed  through $\displaystyle{\erf(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-z^2}\,dz}$.

423

Final Exam / FE-1

« on: December 18, 2015, 07:39:28 PM »

Solve by the method of characteristics the BVP for a wave equation
\begin{align}
&4 u_{tt}-  u_{xx}=0,\qquad 0<x<\infty , \; t>0\label{1-1}\\[2pt]
& u(x,0)=f(x),\label{1-2}\\[2pt]
& u_t(x,0)=g(x),\label{1-3}\\[2pt]
& u (0,t)= h(t)\label{1-4}
\end{align}
with $f(x)=2xe^{-2x^2}$,   $g(x)=0$ and  $h(t)=te^{-t^2/2}$.

424

HA10 / HA10-P5

« on: November 28, 2015, 11:38:33 AM »

Problem 5 http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter10/S10.P.html#problem-10.P.5

425

HA10 / HA10-P4

« on: November 28, 2015, 11:38:14 AM »

Problem 4 http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter10/S10.P.html#problem-10.P.4

426

HA10 / HA10-P3

« on: November 28, 2015, 11:37:50 AM »

Problem 3 http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter10/S10.P.html#problem-10.P.3

427

HA10 / HA10-P2

« on: November 28, 2015, 11:36:58 AM »

Problem 2 http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter10/S10.P.html#problem-10.P.2

428

HA10 / HA10-P1

« on: November 28, 2015, 11:36:40 AM »

Problem 1 http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter10/S10.P.html#problem-10.P.1

429

Test 2 / Solutions to TT2

« on: November 20, 2015, 08:12:37 AM »

There are my solutions

430

Test 2 / TT2-P5

« on: November 18, 2015, 08:43:06 PM »

Find Fourier transforms of the  function
\begin{equation*}
f(x)= \left\{\begin{aligned}
&\cos (x) &&|x|<\frac{\pi}{2},\\
&0 &&|x|>\frac{\pi}{2}
\end{aligned}\right.
\end{equation*}
and write this function as a Fourier integral.

431

Test 2 / TT2-P4

« on: November 18, 2015, 08:42:30 PM »

  Consider Laplace equation in the half-disk
  \begin{equation}
u_{rr} +\frac{1}{r}u_r +\frac{1}{r^2}u_{\theta\theta}=0 \qquad  r<1,0<\theta<\pi \label{4-1}
 \end{equation}
with the Dirichlet boundary conditions as $\theta=0$ and $\theta=\pi$
 \begin{equation}
u|_{\theta=0}=u|_{\theta=\pi}=0\label{4-2}
 \end{equation}
and the Robin boundary condition as $r=1$
 \begin{equation}
(u_r + u)|_{r=1}=1.\label{4-3}
\end{equation}
Using separation of variables find solution as a series.

432

Test 2 / TT2-P3

« on: November 18, 2015, 08:39:01 PM »

Using Fourier method find eigenvalues and eigenfunctions of Laplacian in the rectangle $\{0<x<a, <y<b\}$ with Dirichlet boundary conditions:
\begin{align}
&u_{xx}+u_{yy}=-\lambda u\qquad 0<x<a,\ 0<y<b,\label{3-1}\\[3pt]
&u_{x=0}=u_{x=a}=u_{y=0}=u_{y=b}=0.\label{3-2}
\end{align}

433

Test 2 / TT2-P2

« on: November 18, 2015, 08:38:04 PM »

Solve
\begin{align}
&u_{xx}+u_{yy}=0\qquad -\infty<x<\infty, \ 0<y<\infty,\label{2-1}\\
&u_{y=0}=\frac{1}{x^2+1},\label{2-2}\\
&\max |u |<\infty.\label{2-3}
\end{align}

Hint: Use partial Fourier transform with respect to $x$, and formula
\begin{equation}
F (x^2+a^2)^{-1}=\frac{1}{2a}e^{-|k|a}\qquad \text{as\ \ } a>0.
\label{2-4}
\end{equation}

434

Test 2 / TT2-P1

« on: November 18, 2015, 08:36:52 PM »

Solve by Fourier method
\begin{align}
& u_{tt}-u_{xx}=0\qquad -\frac{\pi}{2}<x<\frac{\pi}{2},\label{1-1}\\
& u_x|_{x=-\pi/2}=u_x|_{x=\pi/2}=0,\label{1-2}\\
&u| _{t=0}=x^2,\qquad u_t|_{t=0}=0.\label{1-3}
\end{align}

435

HA9 / HA9-P6

« on: November 14, 2015, 11:43:38 AM »

Problem 6
http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter7/S7.P.html#problem-7.P.6