A Picone-type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial differential equations of second order

2000 ◽  
Vol 40 (1-8) ◽  
pp. 381-395 ◽  
Author(s):  
Takaŝi Kusano ◽  
Jaroslav Jaroš ◽  
Norio Yoshida
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Second Order ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Type Identity ◽  
Oscillation Theorems

XVI.—Simultaneous Linear Partial Differential Equations of the Second Order

1943 ◽  
Vol 61 (3) ◽  
pp. 195-209
Author(s):  
E. L. Ince
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Hypergeometric Function ◽  
Second Order ◽  
Partial Differential ◽  
Compatible System ◽  
Linear Partial Differential Equations ◽  
Polynomial Coefficients ◽  
Linearly Independent ◽  
Image Position

SummaryIn the system of two linear partial differential equations of the second ordera,…,f were supposed to be polynomials in x, and a1…, f1 polynomials in y. These polynomial coefficients were subjected to certain restrictions, including conditions for the system having exactly four linearly independent solutions, and conditions for preserving the symmetrical aspect, in x and y, of the system. It has been proved that any compatible system of the contemplated form whose coefficients satisfy the stipulated conditions is equivalent with, i.e. transformable into, a hypergeometric system. More particularly it has been shown that the hypergeometric systems involved are the system of partial differential equations associated with Appell's hypergeometric function in two variables F2 and the confluent systems arising herefrom.

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