scholarly journals Analytic-hypoelliptic differential operators of first order in two independent variables

1964 ◽  
Vol 16 (4) ◽  
pp. 367-374 ◽  
Author(s):  
Humio SUZUKI
Keyword(s):  
Differential Operators ◽  
First Order ◽  
Independent Variables ◽  
Two Independent Variables

scholarly journals An integrable system of partial differential equations on the special linear group

2002 ◽  
Vol 44 (1) ◽  
pp. 83-93
Author(s):  
Peter J. Vassiliou
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Group ◽  
Special Linear Group ◽  
Partial Differential ◽  
Special Linear ◽  
First Order ◽  
Independent Variables ◽  
Two Independent Variables

AbstractWe give an intrinsic construction of a coupled nonlinear system consisting of two first-order partial differential equations in two dependent and two independent variables which is determined by a hyperbolic structure on the complex special linear group regarded as a real Lie groupG. Despite the fact that the system is not Darboux semi-integrable at first order, the construction of a family of solutions depending.upon two arbitrary functions, each of one variable, is reduced to a system of ordinary differential equations on the 1-jets. The ordinary differential equations in question are of Lie type and associated withG.

scholarly journals On the Cauchy problem for Dt2 − Dx(b(t)a(x))Dx

2020 ◽  
Vol 17 (01) ◽  
pp. 75-122
Author(s):  
Ferruccio Colombini ◽  
Tatsuo Nishitani
Keyword(s):  
Cauchy Problem ◽  
Differential Operators ◽  
Second Order ◽  
Gevrey Class ◽  
Independent Variables ◽  
The Cauchy Problem ◽  
Well Posed ◽  
Two Independent Variables

We consider the Cauchy problem for second-order differential operators with two independent variables [Formula: see text]. Assuming that [Formula: see text] is a nonnegative [Formula: see text] function and [Formula: see text] is a nonnegative Gevrey function of order [Formula: see text], we prove that the Cauchy problem for [Formula: see text] is well-posed in the Gevrey class of any order [Formula: see text] with [Formula: see text].

scholarly journals Memoir on the integration of partial differential equations of the second order in three independent variables, when an intermediary integral does not exist in general

1898 ◽  
Vol 62 (379-387) ◽  
pp. 283-285
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
General Feature ◽  
Partial Differential ◽  
First Order ◽  
Independent Variables ◽  
Dependent Variables ◽  
Two Independent Variables

The general feature of most of the methods of integration of any partial differential equation is the construction of an appropriate subsidiary system and the establishment of the proper relations between integrals of this system and the solution of the original equation. Methods, which in this sense may be called complete, are possessed for partial differential equations of the first order in one dependent variable and any number of independent variables; for certain classes of equations of the first order in two independent variables and a number of dependent variables; and for equations of the second (and higher) orders in one dependent and two independent variables.

scholarly journals IV. On simultaneous partial differential equations

1901 ◽  
Vol 195 (262-273) ◽  
pp. 151-191 ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Single Equation ◽  
Second Order ◽  
Partial Differential ◽  
First Order ◽  
Independent Variables ◽  
Two Independent Variables

In this paper, without touching on the question of the existence of integrals of systems of simultaneous partial differential equations, I have given a method by which the problem of finding their complete primitives may he attacked. The cases discussed are two: that of a pair of equations of the first order in two dependent and two independent variables, and that of a single equation of the second order, with one dependent and two independent variables.

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