 

Hakop A. Hakopian and Mariam G. Tonoyan
Partial differential analogs of ordinary differential equations and systems


Published: 
April 3, 2004

Keywords: 
Higher order and first order systems of PDEs, equivalence, consistency conditions, system of algebraic equations, multiple solutions, multivariate fundamental theorem of algebra, interpolation/intersection multiplicitiy described by partial differential operators. 
Subject: 
35N05, 35N10, 35G05, 35G20, 14C17, 13H15, 13F20 


Abstract
In this paper we develop a multivariate setting
which includes natural partial differential analogs of the
wellknown firstorder normal system of
ordinary differential
equations and the nth order normal differential equation.
The multivariate counterparts of both of the above examples
are overdetermined normal systems of PDEs:
a firstorder one, which is
the wellknown Pfaff system of PDEs and
a higher order (HO) system of PDEs, with a single unknown function,
which was introduced by the authors (see Hakopian and Tonoyan, 1998b and 2002b).
We generalize the wellknown results, including the connection
of equivalence which is rather unexpected for the theory of PDEs.
Among other generalized results are the method of variation of constants,
fundamental set of solutions,
Wronskian, Liouville's formula.
The equivalence, in view of the existence and uniqueness result of the
Pfaff system, yields a similar
result for the HO system.
Also the linear constant coefficient cases and an
algebraic system, arising from the multivariate
characteristic polynomials, are studied.
This algebraic system is a multivariate counterpart
of the univariate polynomial equation.
Interestingly a multivariate analog of the fundamental theorem of algebra
(FTA) holds for the algebraic system (see Hakopian and Tonoyan, 1998a, 2000 and 2002b,
and Hakopian, 2003a), which allows us
to find a fundamental set of solutions of the HO systems of PDEs,
similarly to the univariate case.


Author information
Hakop A. Hakopian:
Department of Mathematics, Yerevan State University, A. Manoukian Str. 1, Yerevan 375049, Armenia
Current Address: Department of Mathematics, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
hakop@ksu.edu.sa
Mariam G. Tonoyan:
Department of Mathematics, Yerevan State University, A. Manoukian Str. 1, Yerevan 375049, Armenia
Tgarnik@netsys.com

