Caloric polynomial

In differential equations, the mth-degree caloric polynomial (or heat polynomial) is a "parabolically m-homogeneous" polynomial P_m(x, t) that satisfies the heat equation

\frac{\partial P}{\partial t} = \frac{\partial^2 P}{\partial x^2}.

"Parabolically m-homogeneous" means

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

The polynomial is given by

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

It is unique up to a factor.

With t = −1, this polynomial reduces to the mth-degree Hermite polynomial in x.

References

Cannon, John Rozier (1984), The One-Dimensional Heat Equation, Encyclopedia of Mathematics and Its Applications, 23 (1st ed.), Reading/Cambridge: Addison-Wesley Publishing Company/Cambridge University Press, pp. XXV+483, ISBN 978-0-521-30243-2, MR 0747979, Zbl 0567.35001 . Contains an extensive bibliography on various topics related to the heat equation.

External links

Zeroes of complex caloric functions and singularities of complex viscous Burgers equation

This article is issued from Wikipedia - version of the 9/23/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.