• Journal of the Chungcheong Mathematical Society
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• v.20 no.4
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• pp.551-560
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• 2007

We prove the existence of solutions for the system of the nonlinear biharmonic equations with Dirichlet boundary condition $$\{^{-{\Delta}^2u-c{\Delta}u+{\gamma}(bu^+-av^-)=s{\phi}_1\;in\;{\Omega},\;}_{-{\Delta}^2u-c{\Delta}u+{\delta}(bu^+-av^-)=s{\phi}_1\;in\;{\Omega}}$$, where $u^+$ = max{u, 0}, ${\Delta}^2$ denotes the biharmonic operator and ${\phi}_1$ is the positive eigenfunction of the eigenvalue problem $-{\Delta}$ with Dirichlet boundary condition.

• Korean Journal of Mathematics
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• v.16 no.2
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• pp.135-143
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• 2008

We investigate the existence of nontrivial solutions of the nonlinear biharmonic system with Dirichlet boundary condition $$(0.1)\;\begin{array}{lcr}{\Delta}^2{\xi}+c{\Delta}{\xi}={\mu}h({\xi}+{\eta})\;in{\Omega},\\{\Delta}^2{\eta}+c{\Delta}{\eta}={\nu}h({\xi}+{\eta})\;in{\Omega},\end{array}$$ where $c{\in}R$ and ${\Delta}^2$ denote the biharmonic operator.

• Korean Journal of Mathematics
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• v.21 no.4
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• pp.483-493
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• 2013

We investigate the multiplicity of the solutions for a class of the system of the biharmonic equations with some singular potential nonlinearity. We obtain a theorem which shows the existence of the nontrivial weak solution for a class of the system of the biharmonic equations with singular potential nonlinearity and Dirichlet boundary condition. We obtain this result by using variational method and the generalized mountain pass theorem.

• Korean Journal of Mathematics
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• v.15 no.1
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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.

• Korean Journal of Mathematics
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• v.18 no.4
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• pp.473-487
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• 2010

We investigate the existence of multiple nontrivial solutions (${\xi}$, ${\eta}$) for perturbations $g_1$, $g_2$ of the harmonic system with Dirichlet boundary condition $${\Delta}^2{\xi}+c{\Delta}{\xi}=g_1(2{\xi}+3{\eta})\;in\;{\Omega}\\{\Delta}^2{\eta}+c{\Delta}{\eta}=g_2(2{\xi}+3{\eta})\;in\;{\Omega}$$ where we assume that ${\lambda}_1$ < $c$ < ${\lambda}_2$, $g^{\prime}_1({\infty})$, $g^{\prime}_2({\infty})$ are finite. We prove that the system has at least three solutions under some condition on $g$.

• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1693-1710
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• 2013

In this paper, we consider the biharmonic elliptic systems of the form $$\{{\Delta}^2u=F_u(u,v)+{\lambda}{\mid}u{\mid}^{q-2}u,\;x{\in}{\Omega},\\{\Delta}^2v=F_v(u,v)+{\delta}{\mid}v{\mid}^{q-2}v,\;x{\in}{\Omega},\\u=\frac{{\partial}u}{{\partial}n}=0,\; v=\frac{{\partial}v}{{\partial}n}=0,\;x{\in}{\partial}{\Omega},$$, where ${\Omega}{\subset}\mathbb{R}^N$ is a bounded domain with smooth boundary ${\partial}{\Omega}$, ${\Delta}^2$ is the biharmonic operator, $N{\geq}5$, $2{\leq}q$ < $2^*$, $2^*=\frac{2N}{N-4}$ denotes the critical Sobolev exponent, $F{\in}C^1(\mathbb{R}^2,\mathbb{R}^+)$ is homogeneous function of degree $2^*$. By using the variational methods and the Ljusternik-Schnirelmann theory, we obtain multiplicity result of nontrivial solutions under certain hypotheses on ${\lambda}$ and ${\delta}$.

• Communications of the Korean Mathematical Society
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• v.36 no.2
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• pp.247-256
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• 2021

Using variational methods, we show the existence of a unique weak solution of the following singular biharmonic problems of Kirchhoff type involving critical Sobolev exponent: $$(\mathcal{P}_{\lambda})\;\{\begin{array}{lll}{\Delta}^2u-(a{\int}_{\Omega}{\mid}{\nabla}u{\mid}^2dx+b){\Delta}u+cu=f(x){\mid}u{\mid}^{-{\gamma}}-{\lambda}{\mid}u{\mid}^{p-2}u&&\text{ in }{\Omega},\\{\Delta}u=u=0&&\text{ on }{\partial}{\Omega},\end{array}$$ where Ω is a smooth bounded domain of ℝⁿ (n ≥ 5), ∆² is the biharmonic operator, and ∇u denotes the spatial gradient of u and 0 < γ < 1, λ > 0, 0 < p ≤ 2^# and a, b, c are three positive constants with a + b > 0 and f belongs to a given Lebesgue space.

• Proceedings of the Korean Institute of Navigation and Port Research Conference
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• v.2
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• pp.101-106
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• 2006

The geoidal undulations are needed for determining the orthometric heights from the Global Positioning System GPS-derived ellipsoidal heights. There are several methods for geoidal undulation determination. The paper presents a method employing a simple architecture Two Layer Multiquadric-Biharmonic Artificial Neural Network (TLMB-ANN) to approximate an area of 4200 square kilometres quasigeoid surface with GPS-levelling data. Hardy’s Multiquadric-Biharmonic functions is used as the hidden layer neurons’ activation function and Levenberg-Marquardt algorithm is used to train the artificial neural network. In numerical examples five surfaces were compared: the gravimetric geometry hybrid quasigeoid, Support Vector Machine (SVM) model, Hybrid Fuzzy Neural Network (HFNN) model, Traditional Three Layer Artificial Neural Network (ANN) with tanh activation function and TLMB-ANN surface approximation. The effectiveness of TLMB-ANN surface approximation depends on the number of control points. If the number of well-distributed control points is sufficiently large, the results are similar with those obtained by gravity and geometry hybrid method. Importantly, TLMB-ANN surface approximation model possesses good extrapolation performance with high precision.

• Journal of the Chungcheong Mathematical Society
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• v.9 no.1
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• pp.153-164
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• 1996

An $O(h^4)$ cubic spline collocation method for biharmonic equation with a special boundary conditions is formulated and a fast direct method is proposed for the linear system arising when the cubic spline collocation method is employed. This method requires $O(N^2\;{\log}\;N)$ arithmatic operations over an $N{\times}N$ uniform partition.

• Communications of the Korean Mathematical Society
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• v.20 no.3
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• pp.563-578
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• 2005

In [Z. Cai and B. C. Shin, SIAM J. Numer. Anal. 40 (2002), 307-318], we developed the discrete first-order system least squares method for the second-order elliptic boundary value problem by directly approximating $H(div){\cap}H(curl)-type$ space based on the Helmholtz decomposition. Under general assumptions, error estimates were established in the $L^2\;and\;H^1$ norms for the vector and scalar variables, respectively. Such error estimates are optimal with respect to the required regularity of the solution. In this paper, we study solution methods for solving the system of linear equations arising from the discretization of variational formulation which possesses discrete biharmonic term and focus on numerical results including the performances of multigrid preconditioners and the finite element accuracy.