406
Decompose into full Fourier series on interval $[-\pi,\pi]$ and sketch the graph of the sum of such Fourier series:
\begin{equation}
f(x)=|x|.
\end{equation}
\begin{equation}
f(x)=|x|.
\end{equation}
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407
Technical Questions / \erf etc
« on: October 21, 2016, 08:34:58 PM »
Definitely $erf$ should be printed upright, like $\sin$ and so on. Unfortunately LaTeX and standard packages do not know this command (macro) \erf. Neither does MathJax with standard extensions. But there is mechanism: I place* in the preamble of the LaTeX document
$\erf\bigl(\frac{x}{\sqrt{4kt}}\bigr)$
*) Only once
Code: [Select]
\newcommand{\erf}{\operatorname{erf}}and now I can use \erf. In Mathjax I place* on the page the same command, surrounded by dollar signs Code: [Select]
$\newcommand{\erf}{\operatorname{erf}}$and again! $\newcommand{\erf}{\operatorname{erf}}$ $\erf\bigl(\frac{x}{\sqrt{4kt}}\bigr)$
*) Only once
408
Find the solution $u(x,t)$ to
\begin{align}
&u_t=u_{xx} && -\infty<x<\infty, \ t>0,\label{eq-5-1}\\[2pt]
&u|_{t=0}=\left\{\begin{aligned}
&1-x^2\qquad && |x|<1,\\
&0 && |x|>1,
\end{aligned}\right.\label{eq-5-2}\\
&\max |u|<\infty. \label{eq-5-3}
\end{align}
Calculate the integral.
\begin{align}
&u_t=u_{xx} && -\infty<x<\infty, \ t>0,\label{eq-5-1}\\[2pt]
&u|_{t=0}=\left\{\begin{aligned}
&1-x^2\qquad && |x|<1,\\
&0 && |x|>1,
\end{aligned}\right.\label{eq-5-2}\\
&\max |u|<\infty. \label{eq-5-3}
\end{align}
Calculate the integral.
409
Consider the PDE with boundary conditions:
\begin{align}
&u_{tt}-c^2u_{xx} =0,\qquad&&0<x<L,\label{eq-4-1}\\
&(u_x -\alpha u_{tt})|_{x=0}=0,\label{eq-4-2}\\
&(u_x +\beta u_{tt})|_{x=L}=0\label{eq-4-3}
\end{align}
where $c>0$ and $\alpha>0$ are constant. Prove that the energy $E(t)$ defined as
\begin{equation}
E(t)= \frac{1}{2}\int_0^L \bigl( u_t^2 + c^2u_{x}^2 \bigr)\,dx +c^2\frac{\alpha}{2}u_t(0,t)^2+
c^2\frac{\beta}{2}u_t(L,t)^2\end{equation}
does not depend on $t$.
\begin{align}
&u_{tt}-c^2u_{xx} =0,\qquad&&0<x<L,\label{eq-4-1}\\
&(u_x -\alpha u_{tt})|_{x=0}=0,\label{eq-4-2}\\
&(u_x +\beta u_{tt})|_{x=L}=0\label{eq-4-3}
\end{align}
where $c>0$ and $\alpha>0$ are constant. Prove that the energy $E(t)$ defined as
\begin{equation}
E(t)= \frac{1}{2}\int_0^L \bigl( u_t^2 + c^2u_{x}^2 \bigr)\,dx +c^2\frac{\alpha}{2}u_t(0,t)^2+
c^2\frac{\beta}{2}u_t(L,t)^2\end{equation}
does not depend on $t$.
412
Consider the first order equation:
\begin{equation}
u_t + xt u_x = - u.
\label{eq-1-1}
\end{equation}
(a) Find the characteristic curves and sketch them in the $(x,t)$ plane.
(b) Write the general solution.
(c) Solve equation (\ref{eq-1-1}) with the initial condition $u(x,0)= (x^2+1)^{-1}$.
Explain why the solution is fully determined by the initial condition.
\begin{equation}
u_t + xt u_x = - u.
\label{eq-1-1}
\end{equation}
(a) Find the characteristic curves and sketch them in the $(x,t)$ plane.
(b) Write the general solution.
(c) Solve equation (\ref{eq-1-1}) with the initial condition $u(x,0)= (x^2+1)^{-1}$.
Explain why the solution is fully determined by the initial condition.
416
Final Exam / Official solutions
« on: December 21, 2015, 08:17:15 AM »
Basically all solutions posted where either correct from the start or stabilized to correct ones.
417
Final Exam / FE-7
« on: December 18, 2015, 07:52:03 PM »
Solve
\begin{align}
&\Delta u=0 && x^2+y^2+z^2<1,\label{7-1}\\
&u=g(x,y,z) && x^2+y^2+z^2=1\label{7-2}
\end{align}
with $g(x,y,z)=z(x^2+y^2)$.
Hint. If $g$ is a polynomial of degree $m$ look for
\begin{equation}
u=g - P(x,y,z)(x^2+y^2+z^2-R^2)
\label{7-3}
\end{equation}
with $P$ a polynomial of degree $(m-2)$. Here $R$ is the radius of the ball. If $g$ has some rotational symmetry, so $P$ has.
Bonus
Represent $u$ as a sum of homogeneous harmonic polynomials.
\begin{align}
&\Delta u=0 && x^2+y^2+z^2<1,\label{7-1}\\
&u=g(x,y,z) && x^2+y^2+z^2=1\label{7-2}
\end{align}
with $g(x,y,z)=z(x^2+y^2)$.
Hint. If $g$ is a polynomial of degree $m$ look for
\begin{equation}
u=g - P(x,y,z)(x^2+y^2+z^2-R^2)
\label{7-3}
\end{equation}
with $P$ a polynomial of degree $(m-2)$. Here $R$ is the radius of the ball. If $g$ has some rotational symmetry, so $P$ has.
Bonus
Represent $u$ as a sum of homogeneous harmonic polynomials.
418
Final Exam / FE-6
« on: December 18, 2015, 07:49:02 PM »
Consider spherically symmetric solutions of 3D-wave equation
\begin{align}
&&&u_{tt}-u_{rr}-\frac{2}{r} u_r=0, \qquad 0<r<1,\label{6-1}\\
&\text{with boundary conditions } \notag\\
&&&|u(0,t)|<\infty, \qquad u(1,t)=0\label{6-2}\\
&\text{and initial conditions}\notag\\
&&&u(r,0)=1,\qquad u_t(r,0)=0\label{6-3}
\end{align}
and solve by a separation of variables.
Hint. $r R''+2 R'= (rR)''$.
\begin{align}
&&&u_{tt}-u_{rr}-\frac{2}{r} u_r=0, \qquad 0<r<1,\label{6-1}\\
&\text{with boundary conditions } \notag\\
&&&|u(0,t)|<\infty, \qquad u(1,t)=0\label{6-2}\\
&\text{and initial conditions}\notag\\
&&&u(r,0)=1,\qquad u_t(r,0)=0\label{6-3}
\end{align}
and solve by a separation of variables.
Hint. $r R''+2 R'= (rR)''$.
419
Final Exam / FE-5
« on: December 18, 2015, 07:46:35 PM »
Consider Laplace equation in the half-strip
\begin{align}
&&&u_{xx} +u_{yy}=0 \qquad y>0, \ 0 <x< \pi/2 \label{5-1}\\
&\text{with the boundary conditions}\notag\\
&&&u (0,y)=0,\qquad u(\pi/2,y)=0,\label{5-2}\\
&&&u(x,0)=g(x)\label{5-3}
\end{align}
with $g(x)=\cos(x)$\; and condition $\max |u|<\infty$.
(a) Write the associated eigenvalue problem.
(b) Find all eigenvalues and corresponding eigenfunctions.
(c) Write the solution in the form of a series expansion.
\begin{align}
&&&u_{xx} +u_{yy}=0 \qquad y>0, \ 0 <x< \pi/2 \label{5-1}\\
&\text{with the boundary conditions}\notag\\
&&&u (0,y)=0,\qquad u(\pi/2,y)=0,\label{5-2}\\
&&&u(x,0)=g(x)\label{5-3}
\end{align}
with $g(x)=\cos(x)$\; and condition $\max |u|<\infty$.
(a) Write the associated eigenvalue problem.
(b) Find all eigenvalues and corresponding eigenfunctions.
(c) Write the solution in the form of a series expansion.
420
Final Exam / FE-4
« on: December 18, 2015, 07:44:29 PM »
Consider the Laplace equation in the ring
\begin{align}
&&&u_{xx} + u_{yy} =0\qquad &&\text{in }\ 1< r = \sqrt{x^2+y^2} < 2,
\label{4-1} \\
&\text{with the boundary conditions}\notag\\
&&& u =\sin(\theta)\qquad &&\text{for }\ r=1,\label{4-2}\\
&&& u= 3 \sin(\theta)\qquad &&\text{for }\ r=2.\label{4-3}\end{align}
(a) Look for solutions $u$ in the form of $u(r,\theta)= R(r) P(\theta)$ (in polar coordinates) and derive a set of ordinary differential equations for $R$ and $P$. Write the correct boundary conditions for $P$.
(b) Solve the eigenvalue problem for $P$ and find all eigenvalues.
(c) Solve the differential equation for $R$.
(d) Find the solution $u$ of (\ref{4-1})--(\ref{4-3}).
\begin{align}
&&&u_{xx} + u_{yy} =0\qquad &&\text{in }\ 1< r = \sqrt{x^2+y^2} < 2,
\label{4-1} \\
&\text{with the boundary conditions}\notag\\
&&& u =\sin(\theta)\qquad &&\text{for }\ r=1,\label{4-2}\\
&&& u= 3 \sin(\theta)\qquad &&\text{for }\ r=2.\label{4-3}\end{align}
(a) Look for solutions $u$ in the form of $u(r,\theta)= R(r) P(\theta)$ (in polar coordinates) and derive a set of ordinary differential equations for $R$ and $P$. Write the correct boundary conditions for $P$.
(b) Solve the eigenvalue problem for $P$ and find all eigenvalues.
(c) Solve the differential equation for $R$.
(d) Find the solution $u$ of (\ref{4-1})--(\ref{4-3}).