• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.18 no.1
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• pp.43-50
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• 2014

In this paper we discussed Euler-Maruyama method for stochastic differential equations with jump diffusion. We give a convergence result for Euler-Maruyama where the coefficients of the stochastic differential equation are locally Lipschitz and the pth moments of the exact and numerical solution are bounded for some p > 2.

• Journal of the Korean Mathematical Society
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• v.55 no.6
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• pp.1285-1303
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• 2018

This note is motivated by gradient estimates of Li-Yau, Hamilton, and Souplet-Zhang for heat equations. In this paper, our aim is to investigate Yamabe equations and a non linear heat equation arising from gradient Ricci soliton. We will apply Bochner technique and maximal principle to derive gradient estimates of the general non-linear heat equation on Riemannian manifolds. As their consequence, we give several applications to study heat equation and Yamabe equation such as Harnack type inequalities, gradient estimates, Liouville type results.

• Honam Mathematical Journal
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• v.39 no.1
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• pp.27-40
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• 2017

In this paper, we construct exact traveling wave solutions of various kind of partial differential equations arising in mathematical science by the system technique. Further, the $Painlev{\acute{e}}$ test is employed to investigate the integrability of the considered equations. In particular, we describe the behaviors of the obtained solutions under certain constraints.

• East Asian mathematical journal
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• v.30 no.3
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• pp.327-334
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• 2014

In this paper, we construct extrapolated expanded mixed finite element approximations to approximate the scalar unknown, its gradient and its flux of semilinear Sobolev equations. To avoid the difficulty of solving the system of nonlinear equations, we use an extrapolated technique in our construction of the approximations. Some numerical examples are used to show the efficiency of our schemes.

• Kyungpook Mathematical Journal
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• v.58 no.4
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• pp.651-675
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• 2018

We present ${\alpha}$-type rational F-contractions in metric-like spaces, and respective fixed and common fixed point results for weakly ${\alpha}$-admissible mappings. Useful examples illustrate the effectiveness of the presented results. As applications, we obtain sufficient conditions for the existence of solutions of a certain type of integral equations followed by examples of nonlinear fractional differential equations that are verified numerically.

• Bulletin of the Korean Mathematical Society
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• v.61 no.2
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• pp.355-379
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• 2024

In this paper, we establish the curvature estimates for a class of curvature equations with general right hand sides depending on the gradient. We show an existence result by using the continuity method based on a priori estimates. We also derive interior curvature bounds for solutions of a class of curvature equations subject to affine Dirichlet data.

• Nonlinear Functional Analysis and Applications
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• v.29 no.1
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• pp.35-45
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• 2024

In this paper, we study the existence and multiplicity of nontrivial solutions for a p-Kirchhoff equation involving critical Sobolev-Hardy exponent by using variational methods and we need to estimate the energy levels.

• Nonlinear Functional Analysis and Applications
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• v.29 no.2
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• pp.419-434
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• 2024

This paper is devoted to the existence of solutions for Kirchhoff type equations with singular nonlinearities, sub-critical and critical exponent. By using the Nehari manifold and Maximum principle theorem, the existence of at least two distinct positive solutions is obtained.

• 한국전산유체공학회:학술대회논문집
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• 2007.10a
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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.

• Journal of computational fluids engineering
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• v.11 no.4 s.35
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• pp.90-100
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• 2006

A digital wave tank (DWT) simulation technique has been developed by authors to investigate the interactions of fully nonlinear waves with 3D marine structures. A finite-difference/volume method and a modified marker-and-cell (MAC) algorithm have been used, which are based on the Navier-Stokes (NS) and continuity equations. The fully nonlinear kinematic free-surface condition is implemented by the marker-density function (MDF) technique or the Level-Set (LS) technique developed for one or two fluid layers. In this paper, some applications for various engineering problems with free-surface are introduced and discussed. It includes numerical simulation of marine environments by simulation equipments, fully nonlinear wave motions around offshore structures, nonlinear ship waves, ship motions in waves and marine flow simulation with free-surface. From the presented simulations, it seems that the developed DWT simulation technique can handle various engineering problems with free-surface and reliably predict hydrodynamic features due to the fully-nonlinear wave motions interacting with such marine structures.