Re: Problem dealing with Cases and MemberQ
- To: mathgroup at smc.vnet.net
- Subject: [mg68221] Re: Problem dealing with Cases and MemberQ
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Sat, 29 Jul 2006 01:00:55 -0400 (EDT)
- Organization: The University of Western Australia
- References: <ea72js$k6a$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
In article <ea72js$k6a$1 at smc.vnet.net>,
"Charlie Brummitt" <cbrummitt at wisc.edu> wrote:
> I am trying to write code to detect whether a given (partial
> differential) equation is dissipative, i.e. whether it contains
> -u[x,t] or +D[u[x,t],{x,2}] or -D[u[x,t],{x,4}] or +D[u[x,t],{x,6}].
I assume that you are talking about the dissipative properties of the
finite-difference scheme associated with the PDE, rather than of the PDE
itself? So I would guess that the application is to automatically
construct stable finite-difference schemes?
> In other words, given an expression (for example, -0.1 u[x,t] +
> u[x,t]^2 + u[x,t] D[u[x,t],{x,2}] ) I would like to determine
> whether the expression contains any of the following: -U, +Uxx,
> -Uxxxx, +Uxxxxxx --- with the specified signs. (Note that here I used
> the shorthand U = u[x,t], Uxx = D[u[x,t],{x,2}], etc.)
>
> I am having trouble determining whether a given variable (say U) has a
> negative sign in front of it. It does not matter whether that U is
> also multiplied by a real number (say 0.1) or by another variable (say
> Ux). Could anyone please help me write the code to do this?
But what if -U is multiplied by -U (just U^2)? And how should you handle
-U Uxx ?
> Here is a method I have tried, to moderate success:
>
> This method successfully determines whether a particular term has a
> plus or minus sign on it when the term has a real coefficient, but it
> fails when the term in the equation is multiplied by another variable
> (e.g., Ux).
>
> Here is essentially what the code does: I do Cases[equation, term] to
> find the cases of the term I'm looking for, then I divide the
> resulting list by the term to just get the coefficient that's in front
> of each case. Then I map the appropriate sign function (Positive or
> Negative, depending on which term I'm searching for -- e.g. Negative
> for -U, Positive for +Uxx, etc.). Then I use MemberQ on the result to
> find out whether any of the values of the resulting list is True.
>
> However, this method fails when the term in the equation is multiplied
> by a different variable. This is because when I divide the list
> generated by Cases by the term I'm searching for, I am still left with
> any other variables that were multiplied by the term. For example, if
> - U*Ux is in the equation, after dividing by U I am left with -Ux, and
> when I apply Negative to this quantity it does not evaluate to True
> (even though I want it to!) because it doesn't know whether Ux is
> positive or negative. How might I fix this?
>
> Here is the actual code:
>
> containsDissipative[eq_, term_, sign_] := MemberQ[Map[sign, Cases[eq,
> _*term]/term], True];
>
>
> If you are so inclined, here is a good equation to use as a test equation:
> Plus[Times[-0.001`, u[x, t]], Times[10, Power[u[x, t], 2]], Times[-1,
> u[x, t], Derivative[1, 0][u][x, t]]]
> Then flipping the signs on the leading U term and the U*Ux term is a
> good way to test the code.
>
> Might there be another way using Select?
Here's a slightly more complicated equation:
de = 10 u[x, t]^2 - Derivative[1, 0][u][x, t]*u[x, t] -
0.001 u[x, t] - Derivative[2, 0][u][x, t] Derivative[3, 0][u][x, t]^2
Turn it into a list,
ld = List @@ de
Extract the overall sign,
sg = Sign[First /@ ld]
and locate the even derivatives:
Union[
Cases[#,u[x,t] | Derivative[n_ /; EvenQ[n],_][u][x,t],Infinity]]& /@ ld
This should allow you to select the terms that you want.
Also, perhaps the following links are of interest?
http://library.wolfram.com/infocenter/Conferences/5775/
http://library.wolfram.com/infocenter/Conferences/4960/
Cheers,
Paul
_______________________________________________________________________
Paul Abbott Phone: 61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 1014
The University of Western Australia (CRICOS Provider No 00126G)
AUSTRALIA http://physics.uwa.edu.au/~paul