Diff and Taylor to return expressions in terms of partial derivatives?

I have not read your stuff in detail or the wikipedia article but
(a) you might want to use gradef
(b) a variety of people have tried to
implement stuff like the method of multiple scales
using Macsyma or Maxima. e.g. by following the
book by Nayfeh on perturbation analysis.
One of the earliest was in the MIT thesis of
William Martin, c. 1967, using a predecessor system
to Macsyma...see http://hdl.handle.net/1721.1/6907
(c) eliminating secular terms is usually done by "hand"
or some ad hoc method.
(d) these techniques do not seem to be, for the most
part, algorithmic to the extent that you can expect them to
work on any problem which is thrown at them.  You can
probably get the methods sufficiently automated so that
you can solve the very same set of standard problems,
but you can get more terms in the asymptotic series.
automatically.
Unfortunately, the series are asymptotic and thus additional
terms are held in suspicion by the people interested in
using the series.

If you have a particular "in" with people truly interested in
new applications of multiple-scale analysis, that would be
great in keeping you on a path of doing something useful
(to at least one person...)

Good luck.



On 3/5/2013 1:01 PM, Mike Valenzuela wrote:
> Hello,
>
> I will start by providing the context of my question. I was playing around
> with Multiple-scale analysis (
> http://en.wikipedia.org/wiki/Multiple-scale_analysis), and a common trick I
> find in a couple of text books involve the following. See (C. C. Lin and L.
> A. Segel, *Mathematics Applied to Deterministic Problems in the Natural
> Sciences*, Chapter 11: Three Techniques Applied to the Pendulum) for the
> below example:
>
>
> (1) Introduce a slow variable: tau=epsilon*t
>
> (2) Introduce the assumption that: f(t, tau, epsilon) = f[0](t, tau) +
> epsilon*f[1](t, tau) + epsilon^2*f[2](t,tau) + ...
> In the textbook f[i] --> f superscript i,
> This is like a Taylor series expansion, except that the variable t is never
> multiplied on the outside to avoid secular terms
>
> (3) df[0]/dt = partial(f[0], t) * partial(t,t) + partial(f[0], tau) *
> partial(tau, t) = f[0]_1 + epsilon*f[0]_2
> Where the underscore denotes partial with respect to the ith argument
>
> (4) Thus
> df/dt = f[0]_1 + epsilon*(f[0]_2 + f[1]_1) + epsilon^2*(f[1]_2 + f[2]_1) +
> epsilon^3*(f[2]_2 + f[3]_1) + ...
> d^2f/dt^2 = f[0]_11 + epsilon*(f[0]_12 + f[0]_21 + f[1]_11) +
> epsilon^2*(f[0]_22 + f[1]_12 + f[1]_21 + f[2]_11) + ...
>
> So the question. How can I tell Maxima to replicate the above results
> easily in Maxima?
>
> Is there some method like def_diff, which would allow me to specify the
> first derivative and would produce the correct higher order derivatives via
> the chain rule?
> I think the basic approach would be to configure Maxima to provide
> derivatives and Taylor expansions using partial derivatives.
> I tried deftaylor, but it does not appear to like multivariate definitions.
> I tried using depends(tau, [t, epsilon]), but I couldn't get the same
> results as above.
> I tried using taylor( f(t,tau,epsilon), [t, tau], 0, 2), but I couldn't
> figure out how to make Maxima avoid the secular terms.
>
> I think it would be useful to have partial derivative expansions as an
> option in Taylor and Diff as these would be nice in other applications as
> well. A few that come to mind include Calculus of Variations and the
> derivation of the Runge Kutta methods.
>
> Thanks in advance for any advice in simplifying this process.
>
>
>
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