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Messages - Victor Ivrii

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2431
APM346 Misc / Re: Where to buy the Strauss textbook w/ solutions?
« on: September 30, 2012, 11:35:00 PM »
The U of T bookstore doesn't seem to have it in stock. All they had was a textbook by Lawrence Evans which seemed much too advanced.

Evans book is a graduate textbook and very popular but I am not too excited and find it boring.

Neither I am very excited about Strauss book. One can buy from Amazon.ca


2432
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 $u_{tt}-u_{xx}-u_{yy}=0$.

2433
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

2434
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.

2435
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.

2436
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

2437
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.

2438
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.

2439
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.

2440
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.

2441
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)

2442
Home Assignment 2 / Re: Problem 3
« on: September 29, 2012, 03:58:54 PM »
thanks, notes are more clear! However, my understanding for the problem is that I can define odd function to solve the Dirichlet boundary problem and define even function to solve Neumann boundary problem, isn't it?
You are right. However remember that method of continuation has certain preconditions (satisfied in this case)

2443
Home Assignment 2 / Re: Problem 3
« on: September 29, 2012, 03:13:13 PM »
What is meaning of "method of continuation"? I read the textbook and find it define the odd function to solve the problem, can I use that method?

See also Lecture 8  and note that it could be either odd or even

2444
Home Assignment 2 / Re: problem 1 typo?
« on: September 29, 2012, 03:10:20 PM »
Professor I have a question for part (c). Does the solution have to be continuous? For example I have f(x) on x>2t, g(x) on -2t<x<2t and h(x) on -3t<x<-2t. Should f(2t) = g(2t) and g(-2t)=h(-2t) (so the overall solution is continuous)?

My problem is that some of the f, g, h involve a constant K and I was wondering if I should use continuity to specify what K is.

Thanks a lot!

You should be able to find constants from initial and boundary conditions. Solutions may be discontinuous along lines you indicated

2445
APM346 Announcements / Re: Please register ASAP
« on: September 29, 2012, 02:42:53 PM »
Also again: select username whatever you want (make it short enough to save yourself a time while logging in) but change DisplayName (open Profile > Modify Profile > Account Settings and there will be Username and the next Name) matching to one on BlackBoard.

Credits for active and useful forum participation will be given only to those who are easily identified. Well, for identifying those who behaves badly I would go an extra mile :D


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