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2236

Home Assignment 2 / Re: Problem 2

« on: October 02, 2012, 07:13:01 AM »

MZ solution is OK except (d), PW solution of (d) is correct

There are my comments in 1 y.a. forum
http://weyl.math.toronto.edu:8888/APM346-2011F-forum/index.php?topic=22.msg64#msg64

$\frac{1}{r}f (ct+r)$ is an expanding spherical wave


$\frac{1}{r}g (ct-r)$ is a collapsing spherical wave (note my slightly different notations)

however both of them violate an original 3D wave equation as $r=0$ as an expanding spherical wave requires a source and a collapsing spherical wave require a sink and only for

$\frac{1}{r}\bigl[f (ct+r)-f(ct-r0\bigr])$ both source and sink cancel one another at original 3D wave equation holds in the origin as well.

2237

Home Assignment 2 / Re: Problem 1 -- not done yet!

« on: October 02, 2012, 06:56:08 AM »

Posted by: Rouhollah Ramezani
« on: October 01, 2012, 09:00:03 pm »

A) is correct

B) definitely contains an error which is easy to fix. Why I know about error? --  solution of RR  is not $0$ as $x=t$

C) Contains a logical error in the domain $\{−2t<x<2t\}$ (`middle sector`) which should be found and fixed. Note that the solution of RR there is not in the form $\phi(x+2t)+\psi(x-2t)$

RR deserves a credit but there will be also a credit to one who fixes it



So, for a wave equation with a propagation speed $c$ and moving boundary (with a speed $v$) there are three cases (we exclude exact equalities $c=\pm v$ ) -- interpret them as a piston in the cylinder:

• $-c<v<c$ The piston moves with a subsonic speed: one condition as in the case of the staying wall
• $v>c$ The piston moves in with a supersonic speed: no conditions => shock waves etc
• $v<-c$ The piston moves out with a supersonic speed: two conditions.

3D analog: a plane moving in the air. If it is subsonic then everywhere on its surface one boundary condition should be given but for a supersonic flight no conditions on the front surface, one on the side surface and two on the rear (with $\vec{v}\cdot \vec{n} >c$, $-c< \vec{v}\cdot \vec{n} <c$ and $\vec{v}\cdot \vec{n} <-c$ respectively where $\vec{v}$ is the plane velocity and $\vec{n}$ is a unit outer normal at the given point to the plane surface. The real fun begins at transonic points where $\vec{v}\cdot \vec{n} =\pm c$).




PS MathJax is not a complete LaTeX and does not intend to be, so it commands like \bf do not work outside of math snippets (note \bf); MathJax has no idea about \newline as it is for text, not math. For formatting text use either html syntax (in plain html) or forum markup

PPS \bf is deprecated, use \mathbf instead

2238

Home Assignment 3 / Re: Problem5

« on: October 01, 2012, 10:42:19 PM »

I thought for a heat equation there is max principle in general. Is this happening because the coefficient is x but not another random variable? Thanks

The truth is that it is not a heat equation as coefficient at $u_{xx}$ is not everywhere positive

2239

Home Assignment 1 / Re: WTH?

« on: October 01, 2012, 10:39:39 PM »

BTW, has anyone any idea how much time it takes to prepare 1 hour of lecture notes or 1 home assignment? 2-3 hours :-)

I started using vim recently in an effort to type TeX commands faster. But it has a steep learning curve 

I am not sure if vim speeds things up. I am using Mou (MacOSX markdown editor with real time preview and html export), still slow

2240

Home Assignment 2 / Re: Problem4

« on: October 01, 2012, 02:58:36 PM »

hi y'all, a quick question,
would it be appropriate to assume x & t are independent variables in this question? as like, they are presumably not correlated in any function of each other.

Also, what does it mean by rho = T = 1 ? (is T the the kinetic energy or something?)

When modeling a physical string, $T$ represents the tension force and $\rho$ is the mass density. I think they are just asking us to consider $c = 1$ for this problem, since $c = \sqrt{\frac{T}{\rho}}$.

$\rho$ is the linear mass density. On the first question: yes, $x,t$ are independent variables ($t$ is a time and $x$ is a spatial coordinate)

2241

Misc Math / Re: Classification criteria for PDEs

« on: September 30, 2012, 11:31:47 PM »

Yes if matrix of the corresponding coefficients is non-degenerate, the l.o.t. are of no importance and classification depends only on the sign of discriminant $B^2-4AC$. However if discriminant is 0, l.o.t. play role. Your equation is hyperbolic and you can find characteristics.

Also there are profound differences between hyperbolic equations with 2 independent variables like $u_{tt}-u_{xx}=0$ and with $n\ge 3$  independent variables like $u_{tt}-u_{xx}-u_{yy}=0$.

2242

Home Assignment 2 / Re: Problem4

« on: September 30, 2012, 12:13:59 PM »

Are we allowed in part "a" to use initial wave equation for equality proof? ([d^2u/dt^2-d^2u/dx^2=0] in de/dt=dp/dx and de/dx=dp/dt (partials) )

Please use \partial instead of d

Yes, you are allowed

2243

Home Assignment 2 / Re: Problem 2

« on: September 30, 2012, 03:33:53 AM »

I suspect I gave too many hints and this discussion should be stopped.

2244

Home Assignment 2 / Re: Problem 2

« on: September 29, 2012, 11:53:34 PM »

You are solving problem from home assignment 2, not from Strauss book. While result will be the same I see no compelling reason to assume a'priory that initial data must be even or odd.

2245

Home Assignment 2 / Re: Problem 2

« on: September 29, 2012, 11:02:40 PM »

Sorry, I am still having trouble understanding how #2. b is asking to solve for v using v. Could someone help please?

We are looking for $u$, not $v$ -- but $v$ satisfies 1D wave equation and we know everything (well, almost everything) about it

2246

Home Assignment 2 / Re: Problem 2

« on: September 29, 2012, 07:14:54 PM »

In this question r is always positive right (since it's the distance to the origin)? Should u(r,0) and ut(r,0) be even functions of r? I guess we need additional information about u(r,0) and ut(r,0) so that v can be extended to negative values.

Thanks!

See reply #7.

2247

Home Assignment 2 / Re: Problem 2

« on: September 29, 2012, 06:34:05 PM »

I'm even more confused after your response to Thomas' comment - why is part b) the only part that has x as a variable now?

Could you make it clear which version you change first, so I know if I should always check the html version instead of the pdf version for changes?

Usually changes come first to html.

2248

Misc Math / Re: Classification criteria for PDEs

« on: September 29, 2012, 06:31:37 PM »

Good question. However an answer is more complicated: among 2-nd order equations there are elliptic, hyperbolic, parabolic but also a lot of equations which are neither (and some of them are rather important). Ditto for higher order equations and the systems.

There is no complete classifications of PDEs and cannot be because any reasonable classification should not be based on how equation looks like but on the reasonable analytic properties it exhibits (which IVP or BVP are well-posed etc).

2D If we consider only 2-nd order equations with constant real coefficients then in appropriate coordinates they will look like either
\begin{equation}
u_{xx}+u_{yy}+\text{l.o.t} =f
\label{ell-2}
\end{equation}
or
\begin{equation}
u_{xx}-u_{yy}+\text{l.o.t.} =f.
\label{hyp-2}
\end{equation}
Here l.o.t. means "lower order terms". (\ref{ell-2}) are elliptic,   (\ref{hyp-2}) are hyperbolic.

 What to do if one of the 2-nd derivatives is missing? We get
\begin{equation}
u_{xx}-cu_{y}+\text{l.o.t.} =f.
\label{par-2}
\end{equation}
with $c\ne 0$ and  IVP $u|_{y=0}=g$ is well-posed in the direction of $y>0$ if $c>0$ and in direction $y<0$ if $c<0$. We can dismiss $c=0$ as not-interesting.

However this classification leaves out very important Schrödinger equation
\begin{equation}
u_{xx} +i c u_y=0
\label{Schr-2}
\end{equation}
with real $c\ne 0$. For it IVP $u|_{y=0}=g$ is well-posed in both directions $y>0$ and $y<0$ but it lacks many properties of parabolic equations (like maximum principle or mollification).

3D If we consider only 2-nd order equations with constant real coefficients then in appropriate coordinates they will look like either
\begin{equation}
u_{xx}+u_{yy}+u_{zz}\text{l.o.t} =f
\label{ell-3}
\end{equation}
or
\begin{equation}
u_{xx}+u_{yy}-u_{zz}+\text{l.o.t.} =f.
\label{hyp-3}
\end{equation}
 (\ref{ell-3}) are elliptic,   (\ref{hyp-3}) are hyperbolic.

Also we get parabolic equations like
\begin{equation}
u_{xx}+u_{y}-cu_z+\text{l.o.t.} =f.
\label{par-3}
\end{equation}
What about
\begin{equation}
u_{xx}-u_{y}-cu_z+\text{l.o.t.} =f?
\label{crap-3}
\end{equation}
Algebraist-formalist would call them parabolic-hyperbolic but since this equation exhibits no interesting analytic properties (unless one considers lack of such properties interesting) it would be a perversion.

Yes, there will be Schrödinger equation
\begin{equation}
u_{xx} +u_{yy}+i c u_z=0
\label{Schr-3}
\end{equation}
with real $c\ne 0$ but $u_{xx} -u_{yy}+i c u_z=0$ would also have IVP $u|_{z=0}=g$ well posed in both directions.

4D Here we would get also elliptic
\begin{equation}
u_{xx}+u_{yy}+u_{zz}+u_{tt}+\text{l.o.t.} =f,
\label{ell-4}
\end{equation}
hyperbolic
\begin{equation}
u_{xx}+u_{yy}+u_{zz}-u_{tt}+\text{l.o.t.} =f,
\label{hyp-4}
\end{equation}
but also ultrahyperbolic
\begin{equation}
u_{xx}+u_{yy}-u_{zz}-u_{tt}+\text{l.o.t.} =f
\label{uhyp-4}
\end{equation}
which exhibits some interesting analytic properties but these equations are way less important than elliptic, hyperbolic or parabolic.

Parabolic and Schrödinger will be here as well.

The notions of elliptic, hyperbolic or parabolic equations are generalized to higher-order equations but most of the randomly written equations do not belong to any of these types and there is no reason to classify them.

To make things even more complicated there are equations changing types from point to point, f.e. Tricomi equation
\begin{equation}
u_{xx}+xu_{yy}=0
\label{Tric}
\end{equation}
which is elliptic as $x>0$ and hyperbolic as $x<0$ and at $x=0$ has a "parabolic degeneration". It is a toy-model describing stationary transsonic flow of gas.


My purpose was not to give exact definitions but to explain a situation.

2249

Home Assignment 2 / Re: problem 1 typo?

« on: September 29, 2012, 05:10:55 PM »

Consider this:
\begin{align*}
&u_{tt}-c^2u_{xx}=f(x,t)\qquad \text{as  } x>0,t>0,\\
&u|_{t=0}=g(x),\\
&u_t|_{t=0}=h(x),\\
&u|_{x=0}=p(t)
\end{align*}
has a continuous solution if and only if $p(0)=g(0)$ (compatibility condition) but with the Neumann BC solution would be always $C$ (albeit not necessarily $C^1$.

BTW  heat equation
\begin{align*}
&u_{t}-ku_{xx}=f(x,t)\qquad \text{as  } x>0,t>0,\\
&u|_{t=0}=g(x),\\
&u|_{x=0}=p(t)
\end{align*}
also has a continuous solution if and only if $p(0)=g(0)$ (compatibility condition) but the discontinuity stays in $(0,0)$ rather than propagating along characteristics as for wave equation.

2250

Home Assignment 2 / Re: Problem 2

« on: September 29, 2012, 04:01:39 PM »

should the $\phi$ in eq. 6 read $\phi(r+ct)$ rather than $\phi(x+ct)$?

Yes! I fixed pdf (a bit of hassle to maintain two versions)