scholarly journals ON ASYMPTOTIC PROPERTIES OF SOLUTIONS, DEFINED ON THE HALF OF AXIS OF ONE SEMILINEAR ODE

2017 ◽  
Vol 21 (6) ◽  
pp. 130-134
Author(s):  
I.V. Filimonova ◽  
T.S. Khachlaev
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Unknown Function ◽  
Linear Equations ◽  
Unbounded Domains ◽  
Elliptic Partial Differential Equations ◽  
Order Derivative ◽  
Partial Differential ◽  
First Order ◽  
Fowler Equation

The paper deals with the solutions of ordinary differential semi-linear equa- tion, the coefficients of which depend on several real parameters. If the coefficient is chosen so that the equation does not contain the first-order derivative of the unknown function, it will be the case of Emden - Fowler equation. Asymp- totic behavior of Emden - Fowler equation solutions at infinity is described in the book of Richard Bellman. The equations with the first-order derivative, considered in this work, erase in some problems for elliptic partial differential equations in unbounded domains. The sign of the coefficient in first-order deriva- tive term essentially influences on the description of solutions. Partly the result of this paper can be obtained from the works of I.T. Kiguradze. In present work we use lemmas about the behavior of solutions of the linear equations with a strongly (weakly) increasing potential. The paper deals with the solutions of ordinary differential semi-linear equa- tion, the coefficients of which depend on several real parameters. If the coefficient is chosen so that the equation does not contain the first-order derivative of the unknown function, it will be the case of Emden - Fowler equation. Asymp- totic behavior of Emden - Fowler equation solutions at infinity is described in the book of Richard Bellman. The equations with the first-order derivative, considered in this work, erase in some problems for elliptic partial differential equations in unbounded domains. The sign of the coefficient in first-order deriva- tive term essentially influences on the description of solutions. Partly the result of this paper can be obtained from the works of I.T. Kiguradze. In present work we use lemmas about the behavior of solutions of the linear equations with a strongly (weakly) increasing potential.

scholarly journals On the integrability problem for systems of partial differential equations in one unknown function, I

2019 ◽  
Vol 11 (4) ◽  
pp. 35-71 ◽  
Author(s):  
Antonio Kumpera
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Unknown Function ◽  
Contact Structures ◽  
Partial Differential ◽  
Practical Applications ◽  
First Order ◽  
Integrability Problem ◽  
Integration Problem ◽  
In The Beginning

We discuss the integration problem for systems of partial differential equations in one unknown function and special attention is given to the first order systems. The Grassmannian contact structures are the basic setting for our discussion and the major part of our considerations inquires on the nature of the Cauchy characteristics in view of obtaining the necessary criteria that assure the existence of solutions. In all the practical applications of partial differential equations, what is mostly needed and what in fact is hardest to obtains are the solutions of the system or, occasionally, some specific solutions. This work is based on four most enlightening Mémoires written by Élie Cartan in the beginning of the last century.

scholarly journals On Ulam–Hyers Stability for a System of Partial Differential Equations of First Order

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1060
Author(s):  
Daniela Marian ◽  
Sorina Anamaria Ciplea ◽  
Nicolaie Lungu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Compatibility Condition ◽  
Unknown Function ◽  
Differential Inequalities ◽  
Partial Differential ◽  
First Order ◽  
Independent Variables ◽  
Rassias Stability ◽  
Two Independent Variables

The aim of this paper is to investigate generalized Ulam–Hyers stability and generalized Ulam–Hyers–Rassias stability for a system of partial differential equations of first order. More precisely, we consider a system of two nonlinear equations of first order with an unknown function of two independent variables, which satisfy the corresponding compatibility condition. The study method is that of differential inequalities of the Gronwall type.

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