• 대한수학회보
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• 제53권2호
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• pp.335-350
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• 2016

Based on the notion of $q_k$-derivative introduced by the authors in [17], we prove in this paper existence and uniqueness results for nonlinear second-order impulsive $q_k$-difference equations with anti-periodic boundary conditions. Two results are obtained by applying Banach's contraction mapping principle and Krasnoselskii's fixed point theorem. Some examples are presented to illustrate the results.

• International Journal of Naval Architecture and Ocean Engineering
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• 제13권1호
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• pp.526-544
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• 2021

Based on the potential flow theory, a fully nonlinear numerical procedure is developed with boundary element method to analyze the interaction between a fixed semi-submersible platform and incident waves in open water. The incident wave is separated from the scattered wave under fully nonlinear boundary conditions. The mixed Euler-Lagrangian method is used to capture the position of the disturbed wave surface in local coordinate systems. The wave forces exerted on an inverted conical frustum are used to ensure the accuracy of the present method and good agreements with published results are obtained. The hydrodynamic characteristics of the semi-submersible platform interacting with regular waves are analyzed. Pressure distribution with time and space, tension and compression of the platform under wave action are investigated. 3D behaviors of wave run-ups are predicted. Strong nonlinear phenomena such as wave upwelling and wave interference are observed and analyzed.

• 한국해안해양공학회지
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• 제3권3호
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• pp.176-183
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• 1991

본 연구는 경계요소법을 이용하여 비선형 자유표면파을 해석한 것이며, 가상경계처리는 유체 연속성을 고려하여 mass-flux와 energy-flux를 사용하여 유한진폭파동의 해석수법을 제시했다. 유체의 비선형성 때문에 증분법을 적용했으며 경계요소법에 의해 얻어진 결과는 유한요소법의 결과와 실험치와 비교하여 보았으며 좋은 일치가 얻어 졌다. 따라서, 이 방법은 광범위한 파동문제 해석에 유효하게 이용될 수 있으리라 사료된다.

• 한국항공우주학회지
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• 제39권9호
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• pp.805-815
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• 2011

비선형 포물형 안정성 방정식(Nonlinear Parabolized Stability Equations, NPSE)은 보다 전체적인 천이 과정 연구에 효과적으로 사용될 수 있다. NPSE는 천이 과정에서 비선형 구간의 안정성을 직접 수치 모사(Direct Numerical Simulation, DNS)에 비해 적은 계산 비용을 사용하여 효율적으로 해석 할 수 있다. 본 연구에서는 일반 좌표계에서의 NPSE를 구성하고, 수치 계산을 위한 코드를 개발하였다. 코드의 검증을 위해 비압축성 및 압축성 평판 경계층에서의 벤치마크 문제들을 해석하였다. 본 연구의 NSPE 해석 기법이 비선형 안정성 연구에 효율적이고 효과적인 방법임을 확인하였다.

• Korean Journal of Mathematics
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• 제17권1호
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• pp.67-76
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• 2009

We show the existence of at least two nontrivial solutions for a class of the systems of the nonlinear elliptic equations with Dirichlet boundary condition under some conditions for the nonlinear term. We obtain this result by using the variational linking theory in the critical point theory.

• Korean Journal of Mathematics
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• 제23권3호
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• pp.379-391
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• 2015

We investigate the existence of solutions of the nonlinear biharmonic equation with variable coefficient polynomial growth nonlinear term and Dirichlet boundary condition. We get a theorem which shows that there exists a bounded solution and a large norm solution depending on the variable coefficient. We obtain this result by variational method, generalized mountain pass geometry and critical point theory.

• Korean Journal of Mathematics
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• 제18권2호
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• pp.201-211
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• 2010

We investigate the number of the nontrivial solutions of the nonlinear biharmonic equation with Dirichlet boundary condition. We give a theorem that there exist at least three nontrivial solutions for the nonlinear biharmonic problem. We prove this result by the finite dimensional reduction method and the shape of the graph of the corresponding functional on the finite reduction subspace.

• Computers and Concrete
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• 제30권6호
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• pp.433-443
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• 2022

In the current paper, the nonlinear resonance response of functionally graded graphene platelet reinforced (FG-GPLRC) beams by considering different boundary conditions is investigated using the Euler-Bernoulli beam theory. Four different graphene platelets (GPLs) distributions including UD and FG-O, FG-X, and FG-A are considered and the effective material parameters are calculated by Halpin-Tsai model. The nonlinear vibration equations are derived by Euler-Lagrange principle. Then the perturbation method is used to discretize the motion equations, and the loadings and displacement are all expanded, so as to obtain the first to third order perturbation equations, and then the asymptotic solution of the equations can be obtained. Then the nonlinear amplitude-frequency response is obtained with the help of the modified Lindstedt-Poincare method (Chen and Cheung 1996). Finally, the influences of the distribution types of GPLs, total GPLs layers, GPLs weight fraction, elastic foundations and boundary conditions on the resonance problems are comprehensively studied. Results show that the distribution types of GPLs, total GPLs layers, GPLs weight fraction, elastic foundations and boundary conditions have a significant effect on the nonlinear resonance response of FG-GPLRC beams.

• Ocean Systems Engineering
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• 제4권2호
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• pp.83-97
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• 2014

Wave propagation in a three-dimensional (3D) fully nonlinear numerical wave tank (NWT) is studied based on velocity potential theory. The governing Laplace equation with fully nonlinear boundary conditions on the moving free surface is solved using the indirect desingularized boundary integral equation method (DBIEM). The fourth-order predictor-corrector Adams-Bashforth-Moulton scheme (ABM4) and mixed Eulerian-Lagrangian (MEL) method are used for the time-stepping integration of the free surface boundary conditions. A smoothing algorithm, B-spline, is applied to eliminate the possible saw-tooth instabilities. The artificial wave speed employed in MTF (multi-transmitting formula) approach is investigated for fully nonlinear wave problem. The numerical results from incorporating the damping zone (DZ), MTF and MTF coupled DZ (MTF+DZ) methods as radiation condition are compared with analytical solution. An effective MTF+DZ method is finally adopted to simulate the 3D linear wave, second-order wave and irregular wave propagation. It is shown that the MTF+DZ method can be used for simulating fully nonlinear wave propagation very efficiently.

• Korean Journal of Mathematics
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• 제20권4호
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• pp.533-540
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• 2012

We consider the multiplicity of the solutions for the elliptic boundary value problem with $C^1$ nonlinear term decaying at the origin. We get a theorem which shows the existence of the nontrivial solution for the elliptic problem with $C^1$ nonlinear term decaying at the origin. We obtain this result by reducing the elliptic problem with the $C^1$ nonlinear term to the el-liptic problem with bounded nonlinear term and then approaching the variational method and using the mountain pass geometry for the reduced the elliptic problem with bounded nonlinear term.