scholarly journals On the Generalized Simplest Equations: Toward the Solution of Nonlinear Differential Equations with Variable Coefficients

2021 ◽  
Author(s):  
Gunawan Nugroho ◽  
Purwadi Agus Darwito ◽  
Ruri Agung Wahyuono ◽  
Murry Raditya
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Riccati Equation ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Variable Coefficients ◽  
Higher Order ◽  
First Order ◽  
Generalized Riccati Equation ◽  
Polynomial Differential Equation

The simplest equations with variable coefficients are considered in this research. The purpose of this study is to extend the procedure for solving the nonlinear differential equation with variable coefficients. In this case, the generalized Riccati equation is solved and becomes a basis to tackle the nonlinear differential equations with variable coefficients. The method shows that Jacobi and Weierstrass equations can be rearranged to become Riccati equation. It is also important to highlight that the solving procedure also involves the reduction of higher order polynomials with examples of Korteweg de Vries and elliptic-like equations. The generalization of the method is also explained for the case of first order polynomial differential equation.

scholarly journals Asymptotic Dichotomy in a Class of Third-Order Nonlinear Differential Equations with Impulses

2010 ◽  
Vol 2010 ◽  
pp. 1-20 ◽  
Author(s):  
Kun-Wen Wen ◽  
Gen-Qiang Wang ◽  
Sui Sun Cheng
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Functional Differential Equations ◽  
Higher Order ◽  
Third Order ◽  
Order Nonlinear Differential Equation ◽  
Functional Differential

Solutions of quite a few higher-order delay functional differential equations oscillate or converge to zero. In this paper, we obtain several such dichotomous criteria for a class of third-order nonlinear differential equation with impulses.

scholarly journals Existence for Nonoscillatory Solutions of Higher-Order Nonlinear Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
Yazhou Tian ◽  
Fanwei Meng
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Nonoscillatory Solution ◽  
Higher Order ◽  
Nonoscillatory Solutions ◽  
Order Nonlinear Differential Equation

The existence of nonoscillatory solutions of the higher-order nonlinear differential equation [r(t)(x(t)+P(t)x(t-τ))(n-1)]′+∑i=1mQi(t)fi(x(t-σi))=0,  t≥t0, where m≥1,n≥2 are integers, τ>0,  σi≥0,  r,P,Qi∈C([t0,∞),R),  fi∈C(R,R)  (i=1,2,…,m), is studied. Some new sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general Qi(t)  (i=1,2,…,m) which means that we allow oscillatory Qi(t)  (i=1,2,…,m). In particular, our results improve essentially and extend some known results in the recent references.

Asymptotics for third-order nonlinear differential equations: Non-oscillatory and oscillatory cases

2021 ◽  
pp. 1-19
Author(s):  
Calogero Vetro ◽  
Dariusz Wardowski
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Oscillatory Behavior ◽  
Third Order ◽  
First Order ◽  
Third Order Differential Equation ◽  
Comparison Technique ◽  
First Order Differential Equations

We discuss a third-order differential equation, involving a general form of nonlinearity. We obtain results describing how suitable coefficient functions determine the asymptotic and (non-)oscillatory behavior of solutions. We use comparison technique with first-order differential equations together with the Kusano–Naito’s and Philos’ approaches.

scholarly journals On the asymptotic behavior of solutions of certain third-order nonlinear differential equations

2005 ◽  
Vol 2005 (1) ◽  
pp. 29-35 ◽  
Author(s):  
Cemil Tunç
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Third Order ◽  
The Third ◽  
All Solutions

We establish sufficient conditions under which all solutions of the third-order nonlinear differential equation x ⃛+ψ(x,x˙,x¨)x¨+f(x,x˙)=p(t,x,x˙,x¨) are bounded and converge to zero as t→∞.

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 of even-order neutral differential equations via comparison principles

2014 ◽  
Vol 30 (3) ◽  
pp. 293-300
Author(s):  
J. DZURINA ◽  
◽  
B. BACULIKOVA ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Order Differential Equation ◽  
Higher Order ◽  
Comparison Principles ◽  
Neutral Differential Equations ◽  
First Order ◽  
Even Order ◽  
Delay Differential

In the paper we offer oscillation criteria for even-order neutral differential equations, where z(t) = x(t) + p(t)x(τ(t)). Establishing a generalization of Philos and Staikos lemma, we introduce new comparison principles for reducing the examination of the properties of the higher order differential equation onto oscillation of the first order delay differential equations. The results obtained are easily verifiable.

PHASE-PLANE STUDY USING WOLFRAM MATHEMATICA

2018 ◽  
pp. 149-153
Author(s):  
Krum Videnov
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Phase Plane ◽  
Plane Method ◽  
First Order ◽  
Specialized Software ◽  
Wolfram Mathematica ◽  
Phase Plane Method

In this paper, the capabilities of the specialized software Wolfram Mathematica for investigating processes described with differential equations are discussed. The aim is to create procedures and algorithms in Mathematica environment for study and analysis of systems and processes using the Phase-plane method. The proposed algorithm has been experimented to evaluate a nonlinear differential equation of first order.

scholarly journals THE NONLINEAR SUPERPOSITION PRINCIPLE AND THE WEI-NORMAN METHOD

1998 ◽  
Vol 13 (21) ◽  
pp. 3601-3627 ◽  
Author(s):  
J. F. CARIÑENA ◽  
G. MARMO ◽  
J. NASARRE
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Riccati Equation ◽  
Order Differential Equation ◽  
Superposition Principle ◽  
Theoretical Perspective ◽  
Nonlinear Superposition ◽  
Theoretical Methods ◽  
First Order ◽  
Second Order Differential Equations

Group theoretical methods are used to study some properties of the Riccati equation, which is the only differential equation admitting a nonlinear superposition principle. The Wei–Norman method is applied to obtain the associated differential equation in the group SL(2, ℝ). The superposition principle for first order differential equation systems and Lie–Scheffers theorem are also analyzed from this group theoretical perspective. Finally, the theory is applied in the solution of second order differential equations like time independent Schrödinger equation.

scholarly journals On the correct formulation of a nonlinear differential equations in Banach space

2001 ◽  
Vol 26 (7) ◽  
pp. 437-444
Author(s):  
Mahmoud M. El-Borai ◽  
Osama L. Moustafa ◽  
Fayez H. Michael
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Nonlinear Differential Equations ◽  
Initial Value ◽  
Correct Formulation

We study, the existence and uniqueness of the initial value problems in a Banach spaceEfor the abstract nonlinear differential equation(dn−1/dtn−1)(du/dt+Au)=B(t)u+f(t,W(t)), and consider the correct solution of this problem. We also give an application of the theory of partial differential equations.

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.

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