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 = −1/2, this polynomial reduces to the mth-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/Cambridge

| publisher = Addison-Wesley Publishing Company/Cambridge University Press

| year = 1984

| series = Encyclopedia of Mathematics and Its Applications

| volume = 23

| edition = 1st

| pages = XXV+483

| url = https://books.google.com/books?id=XWSnBZxbz2oC

| 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

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

Category:Polynomials

Category:Partial differential equations