Hellenica World

.

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

Mathematics Encyclopedia

Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License

Home - Hellenica World