• 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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• 제18권3호
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• pp.277-288
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• 2010

We consider the nonlinear biharmonic equation with coefficient exponential growth term and Dirichlet boundary condition. We show that the nonlinear equation has at least one bounded solution under the suitable conditions. We obtain this result by the variational method, generalized mountain pass theorem and the critical point theory of the associated functional.

• Korean Journal of Mathematics
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• 제19권4호
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• pp.423-436
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• 2011

We obtain a theorem that shows the existence of multiple solutions for the mixed type nonlinear elliptic equation with Dirichlet boundary condition. Here the nonlinear part contain the jumping nonlinearity and the subcritical growth nonlinearity. We first show the existence of a positive solution and next find the second nontrivial solution by applying the variational method and the mountain pass method in the critical point theory. By investigating that the functional I satisfies the mountain pass geometry we show the existence of at least two nontrivial solutions for the equation.

• 한국전산유체공학회:학술대회논문집
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• 한국전산유체공학회 2007년도 추계 학술대회논문집
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• pp.134-140
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• 2007

Nonlinear parabolized stability equations for compressible flow in general curvilinear coordinate system are derived to deal with a broad range of transition prediction problems on complex geometry. A highly accurate finite difference PSE code has been developed using an implicit marching procedure. Blasius flow is tested. The results of the present computation show good agreement with DNS data. Nonlinear interaction can make the T-S fundamental wave more unstable and the onset of its amplitude decay is shifted downstream relative to linear case. For nonlinear calculations, rather small difference in initial amplitude can produce large change during nonlinear region. Compressible secondary instability at Mach number 1.6 is also simulated and showed that 1.1% initial amplitude for primary mode is enough to trigger the secondary growth.

• 충청수학회지
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• 제23권4호
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• pp.773-784
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• 2010

The existence of solutions in Besov spaces for nonlinear heat equations having transcendental nonlinearity: $$\frac{\partial}{{\partial}t}u-{\Delta}u=F(u)$$ is investigated. In particular, it is proved the local existence and blow-up phenomena of the solutions in Besov spaces for nonlinear heat equations corresponding to two transcendental nonlinear functions $F(u){\equiv}{\mid}u{\mid}e^{u^2}$ and $F(u){\equiv}e^u$ of rapid growth.

• Kyungpook Mathematical Journal
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• 제48권4호
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• pp.637-650
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• 2008

New sufficient conditions for the existence of at least one solution of Neumann type boundary value problems for second order nonlinear differential equations $$\array{\{{p(t)\phi(x'(t)))'=f(t,x(t),\;x(\tau_1(t)),\;{\cdots},\;x(\tau_m(t))),\;t\in[0,T],\\x'(0)=0,\;x'(T)=0,}\,}$$, are established.

• Journal of applied mathematics & informatics
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• 제26권1_2호
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• pp.1-14
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• 2008

In this paper, A Riesz fractional diffusion equation with a nonlinear source term (RFDE-NST) is considered. This equation is commonly used to model the growth and spreading of biological species. According to the equivalent of the Riemann-Liouville(R-L) and $Gr\ddot{u}nwald$-Letnikov(G-L) fractional derivative definitions, an implicit difference approximation (IFDA) for the RFDE-NST is derived. We prove the IFDA is unconditionally stable and convergent. In order to evaluate the efficiency of the IFDA, a comparison with a fractional method of lines (FMOL) is used. Finally, two numerical examples are presented to show that the numerical results are in good agreement with our theoretical analysis.

• Kyungpook Mathematical Journal
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• 제47권4호
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• pp.569-578
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• 2007

Sufficient conditions for the existence of at least one solution of boundary value problems for higher order nonlinear difference equations are established. We allow $f$ to be at most linear, superlinear or sublinear in obtained results.

• 한국정밀공학회지
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• 제8권2호
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• pp.27-35
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• 1991

We propose the crack growth rate equation which applied over three regions (threshold region, stable region, unstable region) of fatigue crack propagation. Constant stress amplitude fatigue tests are conducted for four materials under three stress ratios of R=0.05, R=0.2 and R=0.4. Materials which have different mechanical properties i.e. stainless steel, low carbon steel, medium carbon steel and aluminum alloy are used. The fatigue crack growth rate equation is given by $da/dN={\beta} (1-R)^{\delta}\({\DELTA}K-{\DELTA}K_t)^{\alpha} / (K_{cf}-K_{max})$${\alpha}, {\beta}$ , and ${\delta}$ are constants, and ${\Delta}K_t$ is stress intensity factor range at low ${\Delta}K$ region. The constants are obtained from nonlinear least square method. $K_{ef}$is critical fatigue stress intensity factor. The relation between half crack length and number of cycles obtained by integrating the crack growth rate equation is in agreement with the experimental data. It is also experimented with constant maximum stress and decreasing stress ratios, and the fatigue growth rate of each material is in accord with the proposed equation.

• Korean Journal of Mathematics
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• 제17권1호
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• pp.103-112
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• 2009

We show the existence of the solutions for the following nonlinear elliptic problem under the Dirichlet boundary condition. To show the existence of the solutions we use the variational formulation.