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Zdenek Hurak
science forum beginner
Joined: 10 May 2005
Posts: 10
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 Posted: Tue May 17, 2005 5:33 am Post subject:
MuPAD: Differentiation w.r.t. functions.
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Hello,
I would like to "linearize" a nonlinear differential equation arouns some
"operating point". For example:
| Quote: | deq := diff(x(t),t)^2+x(t)*cos(y(t))=u(t)
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To do that, I need to find the first terms of multivariate Taylor expansion,
that is, I need to find diff with respect to "variables" dot(x)(t), x(t),
y(t).
How can I do that? Of course, a working way is to substitute "expressions"
for "functions" like:
| Quote: | subs(deq,diff(x(t),t)=x_dot)
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and then to use diff or even linalg::jacobian w.r.t. variables x, x_dot, y
But is there a more elegant way? The thing is that I have many variables in
the equation and I don't feel like doing the above described substitution
by hand.
Thanks,
Zdenek Hurak |
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Stefan Wehmeier
science forum beginner
Joined: 09 May 2005
Posts: 17
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| Posted: Tue May 17, 2005 9:27 am Post subject:
Re: MuPAD: Differentiation w.r.t. functions.
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Zdenek Hurak wrote:
| Quote: | Hello,
I would like to "linearize" a nonlinear differential equation arouns some
"operating point". For example:
deq := diff(x(t),t)^2+x(t)*cos(y(t))=u(t)
To do that, I need to find the first terms of multivariate Taylor
expansion, that is, I need to find diff with respect to "variables"
dot(x)(t), x(t), y(t).
How can I do that? Of course, a working way is to substitute "expressions"
for "functions" like:
subs(deq,diff(x(t),t)=x_dot)
and then to use diff or even linalg::jacobian w.r.t. variables x, x_dot, y
But is there a more elegant way? The thing is that I have many variables
in the equation and I don't feel like doing the above described
substitution by hand.
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you will have to. This is not a bug, it is to make you think about what you
are doing :-)
For example, you in your situation want to see x(t) and y(t) as independent,
such that diff(x(t), y(t)) = 0. But dx/dy = (dx/dt)*(dt/dy), such that from
another point of view the result should rather be diff(x(t), t)*
diff((y@@-1)(t), t). In other words, differentiating a function with
respect to another function has no common meaning so we won't allow it.
Best regards,
--
Stefan Wehmeier
stefanw@math.uni-paderborn.de |
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Jerzy Karczmarczuk
science forum addict
Joined: 09 May 2005
Posts: 50
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| Posted: Thu May 19, 2005 8:57 am Post subject:
Re: MuPAD: Differentiation w.r.t. functions.
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Stefan Wehmeier on the subject raised by Zdenek Hurak, (about
differentiating wrt. function) :
| Quote: | For example, you in your situation want to see x(t) and y(t) as independent,
such that diff(x(t), y(t)) = 0. But dx/dy = (dx/dt)*(dt/dy), such that from
another point of view the result should rather be diff(x(t), t)*
diff((y@@-1)(t), t). In other words, differentiating a function with
respect to another function has no common meaning so we won't allow it.
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Perhaps no 'common meaning' with *some* meaning of 'common meaning'.
I think nevertheless that it is useful in variational calculus,in
manipulating distributions, etc. The "functional differentiation" is the
everyday bread for many theoretical physicists...
Jerzy Karczmarczuk
--
Posted via Mailgate.ORG Server - http://www.Mailgate.ORG |
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Stefan Wehmeier
science forum beginner
Joined: 09 May 2005
Posts: 17
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| Posted: Thu May 19, 2005 9:19 am Post subject:
Re: MuPAD: Differentiation w.r.t. functions.
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Jerzy Karczmarczuk wrote:
| Quote: | Stefan Wehmeier on the subject raised by Zdenek Hurak, (about
differentiating wrt. function) :
For example, you in your situation want to see x(t) and y(t) as
independent, such that diff(x(t), y(t)) = 0. But dx/dy = (dx/dt)*(dt/dy),
such that from another point of view the result should rather be
diff(x(t), t)* diff((y@@-1)(t), t). In other words, differentiating a
function with respect to another function has no common meaning so we
won't allow it.
Perhaps no 'common meaning' with *some* meaning of 'common meaning'.
I think nevertheless that it is useful in variational calculus,in
manipulating distributions, etc. The "functional differentiation" is the
everyday bread for many theoretical physicists...
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ok, you will have to write your own package then ... It should not be too
difficult if you just proceed by substitution as Zdenek suggested. Sorry
for the inconvenience. If there are more theoretical physicists around
here, maybe you share your work and write a MuPAD package for theoretical
physics?
--
Stefan Wehmeier
stefanw@mupad.de |
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