• Korean Journal of Mathematics
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• 제5권1호
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• pp.29-33
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• 1997

In this paper we investigate the existence of positive solutions of a nonlinear biharmonic equation under Dirichlet boundary condition in a bounded open set ${\Omega}$ in $\mathbf{R}^n$, i.e., $${\Delta}^2u+c{\Delta}u=bu^{+}+s\;in\;{\Omega},\\u=0,\;{\Delta}u=0\;on\;{\partial}{\Omega}$$.

• Earthquakes and Structures
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• 제16권2호
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• pp.221-234
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• 2019

Today, many important concrete face rockfill dams (CFRDs) have been built on the world, and some of these important structures are located on the strong seismic regions. In this reason, examination and monitoring of these water construction's seismic behaviour is very important for the safety and future of these dams. In this study, the nonlinear seismic behaviour of Ilısu CFR dam which was built in Turkey in 2017, is investigated for various reservoir water heights taking into account 1995 Kobe near-fault and far-fault ground motions. Three dimensional (3D) finite difference model of the dam is created using the FLAC3D software that is based on the finite difference method. The most suitable mesh range for the 3D model is chosen to achieve the realistic numerical results. Mohr-Coulomb nonlinear material model is used for the rockfill materials and foundation in the seismic analyses. Moreover, Drucker-Prager nonlinear material model is considered for the concrete slab to represent the nonlinearity of the concrete. The dam body, foundation and concrete slab constantly interact during the lifetime of the CFRDs. Therefore, the special interface elements are defined between the dam body-concrete slab and dam body-foundation due to represent the interaction condition in the 3D model. Free field boundary condition that was used rarely for the nonlinear seismic analyses, is considered for the lateral boundaries of the model. In addition, quiet artificial boundary condition that is special boundary condition for the rigid foundation in the earthquake analyses, is used for the bottom of the foundation. The hysteric damping coefficients are separately calculated for all of the materials. These special damping values is defined to the FLAC3D software using the special fish functions to capture the effects of the variation of the modulus and damping ratio with the dynamic shear-strain magnitude. Total 4 different reservoir water heights are taken into account in the seismic analyses. These water heights are empty reservoir, 50 m, 100 m and 130 m (full reservoir), respectively. In the nonlinear seismic analyses, near-fault and far-fault ground motions of 1995 Kobe earthquake are used. According to the numerical analyses, horizontal displacements, vertical displacements and principal stresses for 4 various reservoir water heights are evaluated in detail. Moreover, these results are compared for the near-fault and far-faults earthquakes. The nonlinear seismic analysis results indicate that as the reservoir height increases, the nonlinear seismic behaviour of the dam clearly changes. Each water height has different seismic effects on the earthquake behaviour of Ilısu CFR dam. In addition, it is obviously seen that near-fault earthquakes and far field earthquakes create different nonlinear seismic damages on the nonlinear earthquake behaviour of the dam.

• Korean Journal of Mathematics
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• 제17권2호
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• pp.219-231
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• 2009

We show the existence of at least four solutions for the perturbed parabolic equation with Dirichlet boundary condition and periodic condition when the nonlinear part cross two eigenvalues of the eigenvalue problem of the Laplace operator with boundary condition. We obtain this result by using the Leray-Schauder degree theory, the finite dimensional reduction method and the geometry of the mapping. The main point is that we restrict ourselves to the real Hilbert space instead of the complex space.

• Korean Journal of Mathematics
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• 제15권1호
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• pp.51-57
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• 2007

We prove the existence of a positive solution for the system of the following nonlinear biharmonic equations with Dirichlet boundary condition $$\{{\Delta}^2u+c{\Delta}u+av^+=s_1{\phi}_1+{\epsilon}_1h_1(x)\;in\;{\Omega},\\{\Delta}^2v+c{\Delta}v+bu^+=s_2{\phi}_1+{\epsilon}_2h_2(x)\;in\;{\Omega},$$ where $u^+= max\{u,0\}$, $c{\in}R$, $s{\in}R$, ${\Delta}^2$ denotes the biharmonic operator and ${\phi}_1$ is the positive eigenfunction of the eigenvalue problem $-{\Delta}$ with Dirichlet boundary condition. Here ${\epsilon}_1$, ${\epsilon}_2$ are small numbers and $h_1(x)$, $h_2(x)$ are bounded.

• Ocean Systems Engineering
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• 제7권4호
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• pp.371-398
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• 2017

Calm water wave resistance plays a very important role in ship hull design. Numerical methods are meaningful for this reason. In this study, two prevailing methods, the Neumann-Kelvin and the Rankine source method, were implemented and compared. The Neumann-Kelvin method assumes linearized free surface boundary condition and only needs to mesh the hull surface. The Rankine source method considers nonlinear free surface boundary condition and meshes both the ship hull surface and free surface. Both methods were implemented and the wave resistance of a Wigley III and three Series 60(Cb=0.6, 0.7, 0.8) hulls were analyzed. The results were compared with experimental results and the merits of both numerical techniques were quantified. Based on the results, it is concluded that the Rankine source method is more accurate in the calculation of the wave-making resistance. Using the Neumann-Kelvin method, it is found to be easier to model the hull and can be used for slender ships to solve problems like wave current coupling calculation.

• 대한수학회보
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• 제53권6호
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• pp.1805-1821
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• 2016

In this paper, we are concerned with the nonlinear elliptic equations of the p(x)-Laplace type $$\{\begin{array}{lll}-div(a(x,{\nabla}u))+{\mid}u{\mid}^{p(x)-2}u={\lambda}f(x,u) && in\;{\Omega}\\(a(x,{\nabla}u)\frac{{\partial}u}{{\partial}n}={\lambda}{\theta}g(x,u) && on\;{\partial}{\Omega},\end{array}$$ which is subject to nonlinear Neumann boundary condition. Here the function a(x, v) is of type${\mid}v{\mid}^{p(x)-2}v$ with continuous function $p:{\bar{\Omega}}{\rightarrow}(1,{\infty})$ and the functions f, g satisfy a $Carath{\acute{e}}odory$ condition. The main purpose of this paper is to establish the existence of at least three solutions for the above problem by applying three critical points theory due to Ricceri. Furthermore, we localize three critical points interval for the given problem as applications of the theorem introduced by Arcoya and Carmona.

• 대한수학회지
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• 제34권3호
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• pp.609-622
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• 1997

We investigate the existence of solutions of the nonlinear beam equation under the Dirichlet boundary condition on the interval $-\frac{2}{\pi}, \frac{2}{\pi}$ and periodic condition on the varible t, $Lu + bu^+ -au^- = f(x, t)$, when the jumping nonlinearity crosses the first positive eigenvalue.

• Journal of applied mathematics & informatics
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• 제27권1_2호
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• pp.205-215
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• 2009

Sufficient conditions for the existence of at least one solution of the boundary value problems for higher order nonlinear difference equations $\{{{{{\Delta^n}x(i-1)=f(i,x(i),{\Delta}x(i),{\cdots},\Delta^{n-2}x(i)),i{\in}[1,T+1],\atop%20{\Delta^m}x(0)=0,m{\in}[0,n-3],}\atop%20\Delta^{n-2}x(0)=\phi(\Delta^{n-1}(0)),}\atop%20\Delta^{n-1}x(T+1)=-\psi(\Delta^{n-2}x(T+1))}\$. are established.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제3권1호
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• pp.55-69
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• 1999

In this paper we analyze the error estimates for a single-phase nonlinear Stefan problem with Neumann boundary conditions. We apply the modified Crank-Nicolson method to get the optimal order of error estimates in the temporal direction.

• 대한수학회논문집
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• 제24권1호
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• pp.29-40
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• 2009

Motivated by [Linear Algebra and its Appl. 420(2007), 218-227] and [Linear Algebra and its Appl. 425(2007), 171-183], we, in this paper, study the solvability of periodic boundary value problems of higher order nonlinear functional difference equations with p-Laplacian. Sufficient conditions for the existence of at least one solution of this problem are established.