Laplace type invariants for parabolic equations

Document type: Journal Articles
Article type: Original article
Peer reviewed: Yes
Author(s): Nail H. Ibragimov
Title: Laplace type invariants for parabolic equations
Year: 2002
Pagination: 125-133
ISSN: 0924-090X
ISI number: 000174933300003
Organization: Blekinge Institute of Technology
Department: School of Engineering - Dept. Mathematics and Science (Sektionen för teknik – avd. för matematik och naturvetenskap)
School of Engineering S- 371 79 Karlskrona
+46 455 38 50 00
Language: English
Abstract: The Laplace invariants pertain to linear hyperbolic differential equations with two independent variables. They were discovered by Laplace in 1773 and used in his integration theory of hyperbolic equations. Cotton extended the Laplace invariants to elliptic equations in 1900. Cotton's invariants can be obtained from the Laplace invariants merely by the complex change of variables relating the elliptic and hyperbolic equations. To the best of my knowledge, the invariants for parabolic equations were not found thus far. The purpose of this paper is to fill this gap by considering what will be called Laplace type invariants (or seminvariants), i.e. the quantities that remain unaltered under the linear transformation of the dependent variable. Laplace type invariants are calculated here for all hyperbolic, elliptic and parabolic equations using the unified infinitesimal method. A new invariant is found for parabolic equations.
Subject: Mathematics\General
Keywords: hyperbolic, elliptic and parabolic equations, equivalence transformation, seminvariants, Laplace type invariants