Difference between revisions of "Parabolic partial differential equation"
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N.V. Krylov, "Nonlinear elliptic and parabolic equations of the second order" , Reidel (1987) (Translated from Russian)</TD></TR></table> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964) {{MR|0181836}} {{ZBL|0144.34903}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N.V. Krylov, "Nonlinear elliptic and parabolic equations of the second order" , Reidel (1987) (Translated from Russian) {{MR|0901759}} {{ZBL|0619.35004}} </TD></TR></table> |
Revision as of 12:12, 27 September 2012
An equation (cf. Differential equation, partial) of the form
 |
 |
where 
is a positive-definite quadratic form. The variable 
is singled out and plays the role of time. A typical example of a parabolic partial differential equation is the heat equation
 |
Comments
The above defines second-order linear parabolic differential equations. There also exist notions of non-linear parabolic equations. For instance, in [a2] equations are studied of the form 
, where 
is a function of variables 
such that for a certain 
one has 
on the domain under consideration.
A semi-linear partial differential equation of the second order, i.e. one of the form 
, is said to be of parabolic type if 
at each point of the domain under consideration.
References
| [a1] | A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964) MR0181836 Zbl 0144.34903 |
| [a2] | N.V. Krylov, "Nonlinear elliptic and parabolic equations of the second order" , Reidel (1987) (Translated from Russian) MR0901759 Zbl 0619.35004 |
How to Cite This Entry:
Parabolic partial differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parabolic_partial_differential_equation&oldid=12983
Parabolic partial differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parabolic_partial_differential_equation&oldid=12983
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article