• Kyungpook Mathematical Journal
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• v.47 no.2
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• pp.191-202
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• 2007

By using the weighted averaging techniques, we establish oscillation criteria for the second order damped quasilinear elliptic differential equation $$\sum_{i,j=1}^{N}D_i(a_{ij}(x){\parallel}Dy{\parallel}^{p-2}D_jy)+{\langle}b(x),\;{\parallel}Dy{\parallel}^{p-2}Dy{\rangle}+c(x)f(y)=0,\;p>1$$. The obtained theorems include and improve some existing ones for the undamped halflinear partial differential equation and the semilinear elliptic equation.

• Structural Engineering and Mechanics
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• v.24 no.2
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• pp.247-268
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• 2006

The parts of pile, above the soil and embedded in the soil are called the first region and second region, respectively. The forth order differential equations of both region for critical buckling load of partially embedded pile with shear deformation are obtained using the small-displacement theory and Winkler hypothesis. It is assumed that the behavior of material of the pile is linear-elastic and that axial force along the pile length and modulus of subgrade reaction for the second region to be constant. Shear effect is included in the differential equations by considering shear deformation in the second derivative of the elastic curve function. Critical buckling loads of the pile are calculated for by differential transform method (DTM) and analytical method, results are given in tables and variation of critical buckling loads corresponding to relative stiffness of the pile are presented in graphs.

• Nonlinear Functional Analysis and Applications
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• v.28 no.2
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• pp.449-471
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• 2023

In this paper, we study a linear system of homogeneous commensurate /incommensurate fractional-order differential equations by developing a new semi-analytical scheme. In particular, by decoupling the system into two fractional-order differential equations, so that the first equation of order (δ + γ), while the second equation depends on the solution for the first equation, we have solved the under consideration system, where 0 < δ, γ ≤ 1. With the help of using the Adomian decomposition method (ADM), we obtain the general solution. The efficiency of this method is verified by solving several numerical examples.

• Journal of applied mathematics & informatics
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• v.32 no.1_2
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• pp.39-53
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• 2014

In this paper, a non-asymptotic method is presented for solving singularly perturbed delay differential equations whose solution exhibits a boundary layer behavior. The second order singularly perturbed delay differential equation is replaced by an asymptotically equivalent first order neutral type delay differential equation. Then, Simpson's integration formula and linear interpolation are employed to get three term recurrence relation which is solved easily by Discrete Invariant Imbedding Algorithm. Some numerical examples are given to validate the computational efficiency of the proposed numerical scheme for various values of the delay and perturbation parameters.

• Journal of the Korean Mathematical Society
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• v.29 no.2
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• pp.225-238
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• 1992

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.25 no.4
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• pp.224-261
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• 2021

The interest in implicit-explicit (IMEX) integration methods has emerged as an alternative for dealing in a computationally cost-effective way with stiff ordinary differential equations arising from practical modeling problems. In this paper, we introduce implicit-explicit second derivative linear multi-step methods (IMEX SDLMM) with error control. The proposed IMEX SDLMM is based on second derivative backward differentiation formulas (SDBDF) and recursive SDBDF. The IMEX second derivative schemes are constructed with order p ranging from p = 1 to 8. The methods are numerically validated on well-known stiff equations.

• Korean Journal of Mathematics
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• v.19 no.3
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• pp.309-319
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• 2011

We give sufficient conditions that the homogeneous differential equations : for $t{\geq}t_0$(> 0), $$x^{{\prime}{\prime}}(t)+q(t)x^{\prime}(t)+p(t)x(t)=0,\\x^{{\prime}{\prime}}(t)+q(t)x^{\prime}(t)+F(t,x({\phi}(t)))=0$$, are oscillatory where $0{\leq}{\phi}(t)$, 0 < ${\phi}^{\prime}(t)$, $\lim_{t\to{\infty}}{\phi}(t)={\infty}$. and $F(t,u){\cdot}sgn$ $u{\leq}p(t)|u|$. We obtain comparison theorems.

• Bulletin of the Korean Mathematical Society
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• v.55 no.4
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• pp.1065-1092
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• 2018

This manuscript is involved with a category of second-order impulsive functional integro-differential equations with nonlocal conditions in Banach spaces. Sufficient conditions for existence and approximate controllability of mild solutions are acquired by making use of the theory of cosine family, Banach contraction principle and Leray-Schauder nonlinear alternative fixed point theorem. An illustration is additionally furnished to prove the attained principles.

• Kyungpook Mathematical Journal
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• v.55 no.4
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• pp.997-1030
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• 2015

This paper is concerned with integral type boundary value problems of second order singular differential equations with quasi-Laplacian on whole lines. Sufficient conditions to guarantee the existence and non-existence of positive solutions are established. The emphasis is put on the non-linear term $[{\Phi}({\rho}(t)x^{\prime}(t))]^{\prime}$ involved with the nonnegative singular function and the singular nonlinearity term f in differential equations. Two examples are given to illustrate the main results.

• 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.