differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...

, the ''m''th-degree caloric polynomial (or heat polynomial) is a "parabolically ''m''-homogeneous" polynomial ''P''_''m''(''x'', ''t'') that satisfies the heat equation :

\frac = \frac.

"Parabolically ''m''-homogeneous" means :

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

The polynomial is given by :

P_m(x,t) = \sum_^ \frac x^ t^\ell.

It is unique

up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...

a factor. With ''t'' = −1, this polynomial reduces to the ''m''th-degree

Hermite polynomial In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well as ...

in ''x''.

References

*{{Citation , last = Cannon , first = John Rozier , author-link = John Rozier Cannon , title = The One-Dimensional Heat Equation , place =

Reading Reading is the process of taking in the sense or meaning of Letter (alphabet), letters, symbols, etc., especially by Visual perception, sight or Somatosensory system, touch. For educators and researchers, reading is a multifaceted process invo ...

Cambridge Cambridge ( ) is a university city and the county town in Cambridgeshire, England. It is located on the River Cam approximately north of London. As of the 2021 United Kingdom census, the population of Cambridge was 145,700. Cambridge beca ...

, publisher =

Addison-Wesley Publishing Company Addison-Wesley is an American publisher of textbooks and computer literature. It is an imprint of Pearson PLC, a global publishing and education company. In addition to publishing books, Addison-Wesley also distributes its technical titles throug ...

Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambr ...

, 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

Zeroes of complex caloric functions and singularities of complex viscous Burgers equation
Differential equations Polynomials Partial differential equations