• Communications of the Korean Mathematical Society
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• v.20 no.4
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• pp.765-774
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• 2005

The orthogonality of polynomials plays an important role in many areas and in many cases only finite orthogonalities are used. Concerning this fact we find characterizations of a finite orthogonal polynomial system satisfying a second order differential equation and then give several examples.

• Journal of applied mathematics & informatics
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• v.26 no.5_6
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• pp.1057-1069
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• 2008

We consider singularly perturbed Boundary Value Problems (BVPs) for third and fourth order Ordinary Differential Equations(ODEs) of convection-diffusion type with discontinuous source term and a small positive parameter multiplying the highest derivative. Because of the type of Boundary Conditions(BCs) imposed on these equations these problems can be transformed into weakly coupled systems. In this system, the first equation does not have the small parameter but the second contains it. In this paper a computational method named as 'An asymptotic finite element method' for solving these systems is presented. In this method we first find an zero order asymptotic approximation to the solution and then the system is decoupled by replacing the first component of the solution by this approximation in the second equation. Then the second equation is independently solved by a fitted mesh Finite Element Method (FEM). Numerical experiments support our theoritical results.

• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.265-277
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• 2004

This article deals with the number of periodic solutions of the second order polynomial differential equation using the Riccati equation, and applies the property of the solutions of the Riccati equation to study the property of the solutions of the more complicated differential equations. Many valuable criterions are obtained to determine the number of the periodic solutions of these complex differential equations.

• Bulletin of the Korean Mathematical Society
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• v.40 no.2
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• pp.235-241
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• 2003

It is proved that the product of any two linearly independent meromorphic solutions of second order linear differential equations with coefficients of small lower growth must have infinite exponent of convergence of its zero-sequences, under some suitable conditions.

• Kyungpook Mathematical Journal
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• v.56 no.1
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• pp.137-146
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• 2016

In this paper we study the oscillation criteria for the second order nonlinear differential equation with delay and advanced arguments of the form $$([x(t)+a(t)x(t-{\sigma}_1)+b(t)x(t+{\sigma}_2)]^{\alpha})^{{\prime}{\prime}}+q(t)x^{\beta}(t-{\tau}_1)+q(t)x^{\gamma}(t+{\tau}_2)=0,\;t{\geq}t_0$$ where ${\sigma}_1$, ${\sigma}_2$, ${\tau}_1$ and ${\tau}_2$ are nonnegative constants and ${\alpha}$, ${\beta}$ and ${\gamma}$ are the ratios of odd positive integers. Examples are provided to illustrate the main results.

• Journal of the Korean Mathematical Society
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• v.50 no.1
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• pp.173-187
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• 2013

This paper studies the properties of solutions of the Riccati equation arising from the quadratic optimal control problem of the general damped second order system. Using the semigroup theory, we establish the weak differential characterization of the Riccati equation for a general class of the second order distributed systems with arbitrary damping terms.

• Communications of the Korean Mathematical Society
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• v.19 no.3
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• pp.495-506
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• 2004

Some oscillation criteria are given for second order nonlinear differential equations by means of integral averaging technique.

• 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 applied mathematics & informatics
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• v.15 no.1_2
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• pp.173-184
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• 2004

In this paper, we find the approximate solution of a second order nonlinear partial differential equation on a simple connected region in $R^2$. We transfer this problem to a new problem of second order nonlinear partial differential equation on a rectangle. Then, we transformed the later one to an equivalent optimization problem. Then we consider the optimization problem as a distributed parameter system with artificial controls. Finally, by using the theory of measure, we obtain the approximate solution of the original problem. In this paper also the global error in $L_1$ is controlled.

• Journal of Korean Society for Atmospheric Environment
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• v.23 no.6
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• pp.708-717
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• 2007

In the present study, one of the widely applied equations for gas-solid cyclones, Leith and Licht model, was evaluated based on the 3-D CFD technique. The initial and boundary values of radial position and tangential velocity obtain-ed from the CFD simulation enabled complete calculation of the nonlinear second differential equation. This approach showed about 30% errors between calculations with and without the second order differential term. The calculation by using the simple first order equation presented shorter times to migrate up to the inner wall of the cyclone than by the second order, which theoretically implies higher separation efficiency. Further comparison is now under evaluation in terms of the detailed grade efficiency.