• Bulletin of the Korean Mathematical Society
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• v.60 no.6
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• pp.1697-1704
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• 2023

We study a multidimensional nonlinear variational sine-Gordon equation, which can be used to describe long waves on a dipole chain in the continuum limit. By using the method of characteristics, we show that a solution of a nonlinear variational sine-Gordon equation with certain initial data in a multidimensional space has a singularity in finite time.

• Honam Mathematical Journal
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• v.42 no.1
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• pp.17-35
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• 2020

In this paper we introduce and consider a new class of variational inequalities with four operators. This class is called the extended general mixed quasi variational inequality. We show that the extended general mixed quasi variational inequality is equivalent to the fixed point problem. We use this alternative equivalent formulation to discuss the existence of a solution of extended general mixed quasi variational inequality and also develop several iterative methods for solving extended general mixed quasi variational inequality and its variant forms. We consider the convergence analysis of the proposed iterative methods under appropriate conditions. We also introduce a new class of resolvent equation, which is called the extended general implicit resolvent equation and establish an equivalent relation between the extended general implicit resolvent equation and the extended general mixed quasi variational inequality. Some special cases are also discussed.

• Nuclear Engineering and Technology
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• v.54 no.9
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• pp.3181-3204
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• 2022

The variational nodal method for solving the neutron transport equation has evolved over 40 years. Based on a functional form of the Boltzmann neutron transport equation, the method now comprises a complete set of variants that can be employed for different problems. This paper presents an extensive review of the development of the variational nodal method. The emphasis is on summarizing the whole theoretical system rather than validating the methodologies. The paper covers the variational nodal formulation of the Boltzmann neutron transport equation, the Ritz procedure for various application purposes, the derivation of boundary conditions, the extension for adjoint and perturbation calculations, and treatments for anisotropic scattering sources. Acceleration approaches for constructing response matrices and solving the resulting system of algebraic equations are also presented.

• Journal of applied mathematics & informatics
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• v.29 no.1_2
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• pp.381-394
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• 2011

In this paper, a Naveir-Stokes equation is solved by using the Adomian's decomposition method (ADM), modified Adomian's decomposition method (MADM), variational iteration method (VIM), modified variational iteration method (MVIM), modified homotopy perturbation method (MHPM) and homotopy analysis method (HAM). The approximate solution of this equation is calculated in the form of series which its components are computed by applying a recursive relation. The existence and uniqueness of the solution and the convergence of the proposed methods are proved. A numerical example is studied to demonstrate the accuracy of the presented methods.

• Transactions of the Korean Society of Mechanical Engineers
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• v.11 no.6
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• pp.1001-1004
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• 1987

In this study mathematical properties of variational solution and solution of the boundary integral equation of the linear elasticity problem are studied. It is first reviewed that a variational solution for the three-dimensional linear elasticity problem exists in the Sobolev space [ $H^{1}$(.OMEGA.)]$^{3}$ and, then, it is shown that a unique solution of the boundary integral equation is identical to the variational solution in [ $H^{1}$(.OMEGA.)]$^{3}$. To represent the boundary integral equation, the Green's formula in the Sobolev space is utilized on the solution domain excluding a ball, with small radius .rho., centered at the point where the point load is applied. By letting .rho. tend to zero, it is shown that, for the linear elasticity problem, boundary integral equation is valid for the variational solution. From this fact, one can obtain a numerical approximatiion of the variational solution by the boundary element method even when the classical solution does not exist.exist.

• Journal of the Korean Mathematical Society
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• v.41 no.4
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• pp.647-665
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• 2004

The existence of solutions for the nonlinear functional differential equation with nonlocal conditions governed by the variational inequality is studied. The regularity and a variation of solutions of the equation are also given.

• Korean Journal of Mathematics
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• v.17 no.3
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• pp.299-314
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• 2009

We investigate the multiple nontrivial solutions of the asymmetric beam equation $u_{tt}+u_{xxxx}=b_1[{(u + 2)}^+-2]+b_2[{(u + 3)}^+-3]$ with Dirichlet boundary condition and periodic condition on t. We reduce this problem into a two-dimensional problem by using variational reduction method and apply the Mountain Pass theorem to find the nontrivial solutions of the equation.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.23 no.4
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• pp.381-394
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• 2019

In this paper, we are extending fractional partial differential equations to fuzzy fractional partial differential equation under Riemann-Liouville and Caputo fractional derivatives, namely Variational iteration methods, and this method have applied to the fuzzy fractional wave equation with initial conditions as in fuzzy. It is explained by one and two-dimensional wave equations with suitable fuzzy initial conditions.

• Journal of the Korean Operations Research and Management Science Society
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• v.20 no.2
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• pp.125-137
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• 1995

This paper provides a locally convergent algorithm and a globally convergent algorithm for a variational inequality problem over convex polyhedral. The algorithm are based on the B (ouligand)-differentiability of the solution of a nonsmooth equation derived from the variational in-equality problem. Convergences of the algorithms are achieved by the results of Pang[3].

• Journal of the Chungcheong Mathematical Society
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• v.22 no.4
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• pp.675-687
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• 2009

We find the multiple nontrivial solutions of the equation of the form $u_{tt}-u_{xx}=b_1[(u+1)^{+}-1]+b_2[(u+2)^{+}-2]$ with Dirichlet boundary condition. Here we reduce this problem into a two-dimensional problem by using variational reduction method and apply the Mountain Pass theorem to find the nontrivial solutions.