scholarly journals Generalized Weierstrass Integrability of a Class of Second-order Nonlinear Differential Equations

2021 ◽  
pp. 119-133
Author(s):  
Xin Zhao ◽  
Yanxia Hu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Traveling Wave ◽  
Nonlinear Differential Equations ◽  
Traveling Wave Solutions ◽  
Second Order ◽  
Wave Solutions ◽  
The Relationship ◽  
Integrating Factors

The generalized Weierstrass integrability of a class of second-order nonlinear differential equations is considered. The conditions of existence and the corresponding expressions of generalized Weierstrass inverse integrating factors of the second-order nonlinear differential equation are presented. The relationship between the generalized Weierstrass inverse integrating factors and the Weierstrass inverse integrating factors is given. Finally, as an application of the main results, a Kudryashov-Sinelshchikov equation for obtaining traveling wave solutions is considered.

scholarly journals Integrability via Functional Expansion for the KMN Model

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1819
Author(s):  
Radu Constantinescu ◽  
Aurelia Florian
Keyword(s):  
Differential Equations ◽  
Traveling Wave ◽  
Nonlinear Differential Equations ◽  
Traveling Wave Solutions ◽  
Auxiliary Equation ◽  
First Order ◽  
Second Order Differential Equations ◽  
Functional Expansion ◽  
Wave Solutions ◽  
First Time

This paper considers issues such as integrability and how to get specific classes of solutions for nonlinear differential equations. The nonlinear Kundu–Mukherjee–Naskar (KMN) equation is chosen as a model, and its traveling wave solutions are investigated by using a direct solving method. It is a quite recent proposed approach called the functional expansion and it is based on the use of auxiliary equations. The main objectives are to provide arguments that the functional expansion offers more general solutions, and to point out how these solutions depend on the choice of the auxiliary equation. To see that, two different equations are considered, one first order and one second order differential equations. A large variety of KMN solutions are generated, part of them listed for the first time. Comments and remarks on the dependence of these solutions on the solving method and on form of the auxiliary equation, are included.

scholarly journals Exact Traveling Wave Solutions to the (2 + 1)-Dimensional Jaulent-Miodek Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Yongyi Gu ◽  
Bingmao Deng ◽  
Jianming Lin
Keyword(s):  
Differential Equations ◽  
Computer Simulations ◽  
Traveling Wave ◽  
Nonlinear Differential Equations ◽  
Traveling Wave Solutions ◽  
Mathematical Physics ◽  
Complex Method ◽  
Exact Traveling Wave Solutions ◽  
Applied Method ◽  
Wave Solutions

We derive exact traveling wave solutions to the (2 + 1)-dimensional Jaulent-Miodek equation by means of the complex method, and then we illustrate our main result by some computer simulations. It has presented that the applied method is very efficient and is practically well suited for the nonlinear differential equations that arise in mathematical physics.

Oscillation of Second-Order Nonlinear Differential Equations

2014 ◽  
Vol 548-549 ◽  
pp. 1007-1010
Author(s):  
Qing Zhu ◽  
Zhi Bin Ma
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Second Order ◽  
Oscillation Criterion ◽  
Order Nonlinear Differential Equation

A new oscillation criterion is established for a certain class of second-order nonlinear differential equation x"(t)-b(t)x'(t)+c(t)g(x)=0, x"(t)+c(t)g(x)=0 that is different from most known ones. Some applications of the result obtained are also presented. Our results are sharper than some previous ones.

scholarly journals Oscillation and nonoscillation theorems for second order nonlinear differential equations

2001 ◽  
Vol 32 (2) ◽  
pp. 95-102
Author(s):  
Jiang Jianchu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Second Order ◽  
Order Nonlinear Differential Equation ◽  
Oscillation And Nonoscillation

New oscillation and nonoscillation theorems are obtained for the second order nonlinear differential equation $$ (|u'(t)|^{\alpha -1} u'(t))' + p(t)|u(t)|^{\alpha -1} u(t) = 0 $$ where $ p(t) \in C [0, \infty) $ and $ p(t) \ge 0 $. Conditions only about the integrals of $ p(t) $ on every interval $ [2^n t_0, 2^{n+1} t_0] $ ($ n = 1, 2, \ldots $) for some fixed $ t_0 >0 $ are used in the results.

Oscillation Criteria for Second Order Nonlinear Differential Equations Involving Integral Averages

1993 ◽  
Vol 45 (5) ◽  
pp. 1094-1103 ◽  
Author(s):  
James S. W. Wong
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Equation ◽  
Nonlinear Differential Equation ◽  
General Equation ◽  
Nonlinear Differential Equations ◽  
Nonlinear Function ◽  
Second Order ◽  
Oscillation Criteria ◽  
Fowler Equation

AbstractConsider the second order nonlinear differential equationy" + a(t)f(y) = 0where a(t) ∈ C[0,∞),f(y) GC1 (-∞, ∞),ƒ'(y) ≥ 0 and yf(y) > 0 for y ≠ 0. Furthermore, f(y) also satisfies either a superlinear or a sublinear condition, which covers the prototype nonlinear function f(y) = |γ|γ sgny with γ > 1 and 0 < γ < 1 known as the Emden-Fowler case. The coefficient a(t) is allowed to be negative for arbitrarily large values of t. Oscillation criteria involving integral averages of a(t) due to Wintner, Hartman, and recently Butler, Erbe and Mingarelli for the linear equation are shown to remain valid for the general equation, subject to certain nonlinear conditions on f(y). In particular, these results are therefore valid for the Emden-Fowler equation.

scholarly journals The First Integrals of a Second Order Ordinary Differential Equation and Application

2019 ◽  
pp. 1-15
Author(s):  
Yanxia Hu ◽  
Xiaofei Du
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
First Integrals ◽  
Second Order ◽  
Order Ordinary Differential Equation ◽  
Partial Differential ◽  
Wave Solutions ◽  
Symmetry Method

The first integrals of second order ordinary differential equations are considered. The necessary conditions of the existence of analytical first integrals for the equation are presented. Then, the first integrals of the equation are obtained using Lie symmetry method. The results of the first integrals are applied to certain classes of partial differential equations, the conditions of nonexistence of the traveling wave solutions of the partial differential equations are obtained, and traveling wave solutions of the equations under the certain parametric conditions are also obtained.

scholarly journals Some Study Methods for Ordinary Differential Equation Integrability of the Second Order of a Certain Type

2019 ◽  
pp. 48-62
Author(s):  
O. V. Zadorozhnaya ◽  
V. K. Kochetkov
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Practical Interest ◽  
Second Order ◽  
Parameter Method ◽  
Study Methods ◽  
Special Cases ◽  
General Conditions

The paper deals with treating some study methods of the equation integrability of a certain type that are little studied in the theory of differential equations. It is known that a significant part of the differential equations cannot be integrated. Then, to develop methods for their study is, certainly, of scientific interest. The obtained results, formulated as theorems and statements, are of scientific and practical interest because of their importance for applications in modern science.In the paper we present an alternative method for studying the integrability of both linear and nonlinear differential equations of the second order. An introduction of parameters allowed us to develop a study method for the integrability of ordinary differential equations of the second order. We also formulate the theorems describing some General conditions for the integrability of the second-order linear equation and consider special cases of integrability, which arise out of the above facts.Based on the obtained parameter method, some General conditions for the integrability of the nonlinear differential equation of the second order are given, and the consequences of these General conditions are indicated.We have obtained new results related to the construction and development of methods for studying the differential equation to which some types of differential equations are reduced and laid the foundations for a rigorous and systematic study of the introduced special nonlinear differential equation of the second order.

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