• Kyungpook Mathematical Journal
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• v.27 no.1
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• pp.7-13
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• 1987

This paper is a study of the oscillatory and asymptotic behaviour of solutions of the second order nonhomogeneous linear differential equation y"+P(x)y=f(x), and the associated homogeneous equation. Conditions are determined, under which the nonhomogeneous equation is oscillatory if and only if the homogeneous equation is oscillatory.

• Journal of the Chungcheong Mathematical Society
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• v.1 no.1
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• pp.19-26
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• 1988

In this paper, we calculate the precise estimation of range of a Gateaux differentiable operator, and apply to the existence of a periodic solution of the second order nonlinear differential equation $$z^{{\prime}{\prime}}+Az^{\prime}+G(z)=e(t)=e(t+2{\pi})$$.

• Kyungpook Mathematical Journal
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• v.46 no.4
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• pp.527-541
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• 2006

In this paper, an existence theorem for a nonlinear two point boundary value problem of second order differential equations in Banach algebras is proved using a nonlinear alternative based on Leray-Schauder alternative.

• The Mathematical Education
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• v.22 no.1
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• pp.67-68
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• 1983

In this note we consider properties for the discreteness of the spectrum of second-order differential operators. If we give some conditions, then the spectrum of any self-adjoint extension of $A_{0}$ , $A_{0}$ u=$\alpha$[u], D( $A_{0}$ )= $C_{0}$ $^{\infty}$(0.1) is discrete.1) is discrete.

• Journal of computational fluids engineering
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• v.12 no.3
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• pp.55-61
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• 2007

A computational study on a strongly stratified natural convection is performed with the elliptic blending second-moment closure. The turbulent heat flux is treated by both the algebraic flux model (AFM) and the differential flux model (DFM). Calculations are performed for a turbulent natural convection in a square cavity with conducting top and bottom walls and the calculated results are compared with the available experimental data. The results show that both the AFM and DFM models produce very accurate solutions with the elliptic-blending second-moment closure without invoking any numerical stability problems. These results show that the AFM and DFM models for treating the turbulent heat flux are sufficient for this strongly stratified flow. However, a slight difference between two models is observed for some variables.

• Journal of applied mathematics & informatics
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• v.30 no.1_2
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• pp.219-234
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• 2012

By using the generalized Riccati transformation and the inequality technique, we establish some new oscillation criterion for the second-order nonlinear delay dynamic equations $$(a(t)(x^{\Delta}(t))^{\gamma})^{\Delta}+q(t)f(x({\tau}(t)))=0$$ on a time scale $\mathbb{T}$, here ${\gamma}{\geq}1$ is the ratio of two positive odd integers with $a$ and $q$ real-valued positive right-dense continuous functions defined on $\mathbb{T}$. Our results not only extend and improve some known results, but also unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation.

• Bulletin of the Korean Chemical Society
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• v.31 no.2
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• pp.429-434
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• 2010

A novel T-type polyester 7 containing 1-(2,5-dioxyphenyl)-2-{5-(1,2,2-tricyanovinyl)-2-thienyl}ethenes as nonlinear optical (NLO) chromophores, which are part of the polymer backbone, was prepared and characterized. Polyester 7 is soluble in common organic solvents such as dimethylsulfoxide and N,N-dimethylformamide. It showed a thermal stability up to $300^{\circ}C$ in thermogravimetric analysis thermogram and the glass-transition temperature ($T_g$) obtained from differential scanning calorimetry thermogram was around $113^{\circ}C$. The second harmonic generation (SHG) coefficient ($d_{33}$) of poled polymer films at 1,560 nm fundamental wavelength was around 1.85 pm/V. The dipole alignment exhibits a greater thermal stability even at $10^{\circ}C$ higher than $T_g$, and there is no SHG decay below $125^{\circ}C$ due to the partial mainchain character of the polymer structure, which is acceptable for nonlinear optical device applications.

• Wind and Structures
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• v.25 no.5
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• pp.415-432
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• 2017

The nonlinear aerodynamic instability of a tensioned plane orthotropic membrane structure is theoretically investigated in this paper. The interaction governing equation of wind-structure coupling is established by the Von $K\acute{a}rm\acute{a}n's$ large amplitude theory and the D'Alembert's principle. The aerodynamic force is determined by the potential flow theory of fluid mechanics and the thin airfoil theory of aerodynamics. Then the interaction governing equation is transformed into a second order nonlinear differential equation with constant coefficients by the Bubnov-Galerkin method. The critical wind velocity is obtained by judging the stability of the second order nonlinear differential equation. From the analysis of examples, we can conclude that it's of great significance to consider the orthotropy and geometrical nonlinearity to prevent the aerodynamic instability of plane membrane structures; we should comprehensively consider the effects of various factors on the design of plane membrane structures; and the formula of critical wind velocity obtained in this paper provides a more accurate theoretical solution for the aerodynamic stability of the plane membrane structures than the previous studies.

• Convergence Security Journal
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• v.5 no.3
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• pp.93-97
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• 2005

MacCormack's method is an explicit, second order finite difference scheme that is widely used in the solution of hyperbolic partial differential equations. Apparently, however, it has shown entropy violations under small discontinuity. This non-physical shock grows fast and eventually all the meaningful information of the solution disappears. Some modifications of MacCormack's methods follow ideas of central schemes with an advantage of second order accuracy for space and conserve the high order accuracy for time step also. Numerical results are shown to perform well for the one-dimensional Burgers' equation and Euler equations gas dynamic.

• Macromolecular Research
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• v.15 no.3
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• pp.234-237
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• 2007

3,4-Bis-(3,4-dicarboxyphenylcarboxyethoxy)-4'-nitrostilbene dianhydride was prepared and reacted with 4,4'-(hexafluoroisopropylidene)dianiline to yield a novel Y-type polyimide containing the 3,4-dioxynitrostilbenyl group as an NLO-chromophore, which constituted part of the polymer backbone. The resulting polyimide was soluble in polar solvents such as acetone and N,N-dimethylformamide. The polymer exhibited good thermal stability up to $370^{\circ}C$ in the thermogravimetric analysis. The glass-transition temperature ($T_g$) obtained from the differential scanning calorimetry thermogram was near to $153^{\circ}C$. The second harmonic generation (SHG) coefficient ($d_{33}$) of the poled polymer film at the fundamental wavelength of $1064\;cm^{-1}$ was around $2.15\;{\times}\;10^{-8}\;esu$ (9.01 pm/V). The dipole alignment exhibited exceptionally high thermal stability even at a temperature $30^{\circ}C$ above the $T_g$, and there was no SHG decay below $180^{\circ}C$ because of the partial main chain character of the polymer structure.