Английская Википедия:Caloric polynomial

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Шаблон:No footnotes 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

<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 mth-degree Hermite polynomial in x.

References

• Шаблон:Citation. 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

Шаблон:Polynomial-stub