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### Topics - Victor Ivrii

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31
##### TT1 / TT1-P4
« on: October 19, 2016, 10:29:21 PM »
Consider the PDE  with boundary conditions:
\begin{align}
&(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

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
does not depend on $t$.

32
##### TT1 / TT1-P3
« on: October 19, 2016, 10:27:53 PM »
Find  solution to
\begin{align}
&u_{tt}-9u_{xx}=0, \qquad&& t>0, \ \ 0<x< t,\label{eq-1}\\
&u|_{t=0}=\sin (x), && x>0,\label{eq-2}\\
&u_t|_{t=0}=3\cos (x), && x>0,\label{eq-3}\\
&u|_{x=t}= 0, &&t>0.\label{eq-4}
\end{align}

33
##### TT1 / TT1-P2
« on: October 19, 2016, 10:26:33 PM »
(a) Find solution $u(x,t)$ to
\begin{align}
&u_{tt}-u_{xx}= (x^2-1)e^{-\frac{x^2}{2}},\label{eq-1}\\
\end{align}
(b) (1 pts--bonus) Find $\lim _{t\to +\infty} u(x,t)$.

34
##### TT1 / TT1-P1
« on: October 19, 2016, 10:24:30 PM »
Consider the first order equation:

u_t +  xt u_x = - u.
\label{eq-1-1}

(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.

35
##### APM346--Announcements / Term Test 1
« on: October 10, 2016, 02:55:13 PM »
I recall that test will be Wednesday, October 19, 19:10-21:00 at  (SF 3202).

If you have not done yet please signâ€“in to forum and confirm that time works for you or select an early sitting from the list.  Please do it ASAP as it could be problematic to accommodate you at the last moment.

All early sittings will be in my office HU1008 (if there will be a change warranted by a number of students it wil be announced). If nobody selects some slot it will be dropped and I could be out of my office at that time.

Please do not select any option 2--6  if option 1 works for you. Please select  option 6 only if none of options 1--5 works for you--and in this case send me email so we will discuss yet another timeslot.

36
##### APM346--Announcements / Your background: other
« on: October 10, 2016, 02:44:25 PM »

37
##### APM346--Announcements / Your background: Calculus II
« on: October 10, 2016, 02:37:52 PM »
Please answer these question. If you think that your knowledge of Calculus II is inadequate talk to me urgently (we will discuss the remedy)

38
##### APM346--Announcements / Your background: ODE
« on: October 10, 2016, 02:35:12 PM »
Please answer these question. If you think that your knowledge of ODE  is inadequate talk to me urgently (we will discuss the remedy).

39
##### APM346--Announcements / Q1--7 graded
« on: October 03, 2016, 12:53:29 PM »
Q1
 Count 81 Average 2.73 Median 3.5 Standard Deviation 1.26

 90 - 100 14 80 - 89 27 70 - 79 9 60 - 69 11 50 - 59 6 40 - 49 0 30 - 39 2 20 - 29 1 10 - 19 1 0 - 9 10

40
##### APM346--Announcements / Bonuses
« on: October 02, 2016, 06:08:43 PM »
September Web bonuses
Roro Sihui Yap = .6
Shentao Yang = .2
XiuYu Zheng = .6

October Web bonuses
Shentao Yang = 2
XinYu Zheng = .6+.6+.2+1+1+1+1+.6 =6
Roro Sihui Yap=1.2+1+.6+.6+1+1 =5.4
Tianyi Zhang =.6

October Class bonuses
Hu Ziyao = .6
Shaghayegh Atashi = .6
Menghan Chen = 2

November Web bonuses
Tianyi Zhang =.6
Roro Sihui Yap= 3 +.5 + 4 (maxed her bonuses)
XinYu Zheng = .5 +.5+.5+4 =5.5 (maxed his bonuses)
Shentao Yang=.5 +1 +.5= 2
Luyu Cen = .5

November Class bonuses
Shentao Yang=.5
Luyu Cen = .5

41
##### Q1 / Q1-P3
« on: September 29, 2016, 09:30:39 PM »
Find the solution of

\left\{\begin{aligned}
&u_x+3u_y=u,\label{eq-1}\\
&u|_{x=0}=y.\label{eq-2}
\end{aligned} \right.

42
##### Q1 / Q1-P2
« on: September 29, 2016, 09:29:38 PM »
Find the general solutions to the following equation:

u_{xyz}=\sin (x)+\sin (y)+\sin(z)
\label{eq-1}

43
##### Q1 / Q1-P1
« on: September 29, 2016, 09:29:03 PM »
Consider first order equations and determine if they are linear homogeneous, linear inhomogeneous, quasilinear or non-linear ($u$ is an unknown function):
\begin{align}
&u_t+xu_x-u= 0,\label{eq-1}\\[5pt]
&u_x^2+u_y^2-1= 0. \label{eq-2}
\end{align}

44
##### APM346--Misc / Bonus Topic 3 (telegrapher's equation)
« on: September 22, 2016, 08:15:17 AM »
Consider a wire. Denote by $i=i(x,t)$ a current,  and by $v(x,t)$ a potential.
a. Explain
\begin{align}
&v_x = -Li_t,\\
&i_x=-Cv_t
\end{align}
where $L$ and $C$ are inductivity and capacity of the segment of unit length. Ignore resistance and leaks.
b. Prove that $i$ (and $v$) satisfy

i_{tt}-c^2i_{xx}=0

with $c=1/\sqrt{CL}$.

45
##### APM346--Misc / Bonus Topic 2 (air pipe)
« on: September 22, 2016, 08:03:52 AM »
Consider an air pipe and denote by $\rho =\rho(x,t)$ the density and by $u=u(x,t)$ a velocity of the air.
a.  Explain
\begin{align}
&\rho u_t = - p_x ,\\
&\rho_t  + (\rho u)_x=0
\end{align}
where $p=p(\rho)$ is a pressure.
b. Assuming that $\rho-\mu$ ($\mu$ is a constant) and $u$ are small linearize these equations to
\begin{align}
&\mu u_t = - k\rho_x ,\\
&\rho_t  + (\mu u)_x=0
\end{align}
where $k=p'(\mu)$.
c. Prove that then $u$ (and $\rho$) satisfy equation

u_{tt}-ku_{xx}=0.

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