Fixed end moment

2014-01-07T02:00:28Z

219.92.27.238: /* Examples */

In [[differential equations]], the ''m''th-degree '''caloric polynomial''' (or '''heat polynomial''') is a "parabolically ''m''-homogeneous" polynomial ''P''<sub>''m''</sub>(''x'', ''t'') that satisfies the [[heat equation]]

: <math> \frac{\partial P}{\partial t} = \frac{\partial^2 P}{\partial x^2}. </math>

"Parabolically ''m''-homogeneous" means

: <math> P(\lambda x, \lambda^2 t) = \lambda^m P(x,t)\text{ for }\lambda > 0.\, </math>

The polynomial is given by

: <math> P_m(x,t) = \sum_{\ell=0}^{\lfloor m/2 \rfloor} \frac{m!}{\ell!(m - 2\ell)!} x^{m - 2\ell} t^\ell. </math>

It is unique [[up to]] a factor.

With ''t'' = −1, this polynomial reduces to the ''m''th-degree [[Hermite polynomial]] in ''x''.

==References==
*{{Citation
| last = Cannon
| first = John Rozier
| author-link = John Rozier Cannon
| title = The One-Dimensional Heat Equation
| place = [[Reading, Massachusetts|Reading]]/[[Cambridge]]
| publisher = [[Addison–Wesley|Addison-Wesley Publishing Company]]/[[Cambridge University Press]]
| year = 1984
| series = Encyclopedia of Mathematics and Its Applications
| volume = 23
| edition = 1st
| pages = XXV+483
| url = http://books.google.com/?id=XWSnBZxbz2oC&printsec=frontcover#v=onepage&q&f=true
| id =
| mr = 0747979
| zbl = 0567.35001
| isbn =978-0-521-30243-2 }}. Contains an extensive bibliography on various topics related to the [[heat equation]].

== External links ==

* [http://arxiv.org/pdf/math.AP/0612506.pdf Zeroes of complex caloric functions and singularities of complex viscous Burgers equation]

[[Category:Differential equations]]
[[Category:Polynomials]]
[[Category:Partial differential equations]]

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