• Journal of applied mathematics & informatics
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• v.29 no.3_4
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• pp.763-779
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• 2011

In this paper we apply the Exp-function method to obtain traveling wave solutions of three nonlinear partial differential equations, namely, generalized sinh-Gordon equation, generalized form of the famous sinh-Gordon equation, and double combined sinh-cosh-Gordon equation. These equations play a very important role in mathematical physics and engineering sciences. The Exp-Function method changes the problem from solving nonlinear partial differential equations to solving a ordinary differential equation. Mainly we try to present an application of Exp-function method taking to consideration rectifying a commonly occurring errors during some of recent works.

• Bulletin of the Korean Mathematical Society
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• v.40 no.2
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• pp.243-251
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• 2003

This note Presents sufficient conditions for Lyapunov's stability and instability of a class of nonlinear nonautonomous second-order ordinary differential equations. Such a class includes as particular cases a remarkably large number of differential equations with specific physical applications. Two successive nonlinear transformations are applied to the original differential equation in order to convert it into a more convenient form for stability analysis purposes. The obtained stability / instability conditions depend closely on the parameterization of the original differential equation.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.5 no.2
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• pp.1-6
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• 2001

Multiplicity of nonlinear second order nonlinear ordinary differential equations will be discussed.

• Steel and Composite Structures
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• v.46 no.5
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• pp.637-647
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• 2023

Nonlinear free vibration and stability responses of a carbon nanotube reinforced composite beam under temperature rising are investigated in this paper. The material of the beam is considered as a polymeric matrix by reinforced the single-walled carbon nanotubes according to different distributions with temperature-dependent physical properties. With using the Hamilton's principle, the governing nonlinear partial differential equation is derived based on the Euler-Bernoulli beam theory. In the nonlinear kinematic assumption, the Von Kármán nonlinearity is used. The Galerkin's decomposition technique is utilized to discretize the governing nonlinear partial differential equation to nonlinear ordinary differential equation and then is solved by using of multiple time scale method. The critical buckling temperatures, the nonlinear natural frequencies and the nonlinear free response of the system is obtained. The effect of different patterns of reinforcement on the critical buckling temperature, nonlinear natural frequency, nonlinear free response and phase plane trajectory of the carbon nanotube reinforced composite beam investigated with temperature-dependent physical property.

• Journal of applied mathematics & informatics
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• v.11 no.1_2
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• pp.243-253
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• 2003

In this paper, oscillation and stability of nonlinear neutral impulsive delay differential equation are studied. The main result of this paper is that oscillation and stability of nonlinear impulsive neutral delay differential equations are equivalent to oscillation and stability of corresponding nonimpulsive neutral delay differential equations. At last, two examples are given to illustrate the importance of this study.

• Journal of applied mathematics & informatics
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• v.27 no.5_6
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• pp.1165-1176
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• 2009

This paper is concerned with the interval oscillation of the second order nonlinear ordinary differential equation (r(t)|y'(t)|$^{{\alpha}-1}$ y'(t))'+p(t)|y'(t)|$^{{\alpha}-1}$ y'(t)+q(t)f(y(t))g(y'(t))=0. By constructing ageneralized Riccati transformation and using the method of averaging techniques, we establish some interval oscillation criteria when f(y) is not differetiable but satisfies the condition $\frac{f(y)}{|y|^{{\alpha}-1}y}$ ${\geq}{\mu}_0$ > 0 for $y{\neq}0$.

• Bulletin of the Korean Mathematical Society
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• v.54 no.4
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• pp.1185-1194
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• 2017

In this paper, we present nonlinear differential equations arising from the generating function of the Eulerian polynomials. In addition, we give explicit formulae for the Eulerian polynomials which are derived from our nonlinear differential equations.

• Kyungpook Mathematical Journal
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• v.46 no.1
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• pp.65-77
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• 2006

By using general means, some oscillation criteria for second order nonlinear elliptic differential equation with damping $$\sum_{i,j=1}^{N}D_i[a_{ij}(x)D_iy]+\sum_{i=1}^{N}b_i(x)D_iy+p(x)f(y)=0$$ are obtained. These criteria are of a high degree of generality and extend the oscillation theorems for second order linear ordinary differential equations due to Kamenev, Philos and Wong.

• Structural Engineering and Mechanics
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• v.81 no.6
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• pp.705-714
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• 2022

The aim of this paper is to investigate nonlinear dynamic responses of functionally graded composite beam resting on the nonlinear viscoelastic foundation subjected to moving mass with temperature rising. The non-linear strain-displacement relationship is considered in the finite strain theory and the governing nonlinear dynamic equation is obtained by using the Hamilton's principle. The Galerkin's decomposition technique is utilized to discretize the governing nonlinear partial differential equation to nonlinear ordinary differential equation and then the governing equation is solved by using of multiple time scale method. The influences of temperature rising, material distribution parameter, nonlinear viscoelastic foundation parameters, magnitude and velocity of the moving mass on the nonlinear dynamic responses are investigated. Also, the buckling temperatures of the functionally graded beams based on the finite strain theory are obtained.

• Journal of the Chungcheong Mathematical Society
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• v.19 no.4
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• pp.393-402
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• 2006

We investigate relations between multiplicity of solutions and source terms in a elliptic equation. We have a concerne with a sharp result for multiplicity of a nonlinear Laplacian differential equation.