On a class of systems of second order elliptic type differential equations with two independent variables

1968 ◽  
Vol 9 (1) ◽  
pp. 39-50
Author(s):  
A. Dzhuraev
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
Elliptic Type ◽  
Independent Variables ◽  
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.

scholarly journals On a method for constructing Bergman kernels

1980 ◽  
Vol 29 (4) ◽  
pp. 454-461
Author(s):  
A. Azzam ◽  
E. Kreyszig
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Second Order ◽  
Partial Differential ◽  
Bergman Kernels ◽  
Order Linear ◽  
Independent Variables ◽  
New Class ◽  
Linear Partial Differential Equations ◽  
Two Independent Variables

AbstractWe establish a method of constructing kernels of Bergman operators for second-order linear partial differential equations in two independent variables, and use the method for obtaining a new class of Bergman kernels, which we call modified class E kernels since they include certain class E kernals. They also include other kernels which are suitable for global representations of solutions (whereas Bergman operators generally yield only local representations).

scholarly journals On the Solution of Non-linear Partial Differential Equations of the Second Order

1891 ◽  
Vol 10 ◽  
pp. 63-70
Author(s):  
John M'Cowan
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Second Order ◽  
Partial Differential ◽  
Independent Variables ◽  
Linear Partial Differential Equations ◽  
General Utility ◽  
Non Linear ◽  
Two Independent Variables

§ 1. It is proposed to discuss in this paper partial differential equations involving two independent variables x and y, and a dependent variable z. The method of reduction which is explained can be applied to certain equations involving more than two independent variables, but such application is subject to too many restrictions to be of much general utility.

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

1898 ◽  
Vol 191 ◽  
pp. 1-86 ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Second Order ◽  
Original Equation ◽  
Partial Differential ◽  
First Order ◽  
Independent Variables ◽  
Dependent Variables ◽  
Independent Variable ◽  
Two Independent Variables

The general feature of most methods for the integration of partial differential equations in two independent variables is, in some form or other, the construction of a set of subsidiary equations in only a single independent variable; and this applies to all orders. In particular, for the first order in any number of variables (not merely in two), the subsidiary system is a set of ordinary equations in a single independent variable, containing as many equations as dependent variables to be determined by that subsidiary system. For equations of the second order which possess an intermediary integral, the best methods (that is, the most effective as giving tests of existence) are those of Boole, modified and developed by Imschenetsky, and that of Goursat, initially based upon the theory of characteristics, but subsequently brought into the form of Jacobian systems of simultaneous partial equations of the first order. These methods are exceptions to the foregoing general statement. But for equations of the second order or of higher orders, which involve two independent variables and in no case possess an intermediary integral, the most general methods are that of Ampere and that of Darboux, with such modifications and reconstruction as have been introduced by other writers; and though in these developments partial differential equations of the first order are introduced, still initially the subsidiary system is in effect a system with one independent variable expressed and the other, suppressed during the integration, playing a parametric part. In oilier words, the subsidiary system practically has one independent variable fewer than the original equation. In another paper I have given a method for dealing with partial differential equations of the second order in three variables when they possess an intermediary integral; and references will there be found to other writers upon the subject. My aim in the present paper has been to obtain a method for partial differential equations of the second order in three variables when, in general, they possess no intermediary integral. The natural generalisation of the idea in Darboux’s method has been adopted, viz., the construction of subsidiary equations in which the number of expressed independent variables is less by unity than the number in the original equation; consequently the number is two. The subsidiary equations thus are a set of simultaneous partial differential equations in two independent variables and a number of dependent variables.

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