• 대한수학회보
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• 제60권6호
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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.

• 호남수학학술지
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• 제42권1호
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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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• 제54권9호
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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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• 제29권1_2호
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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.

• 대한기계학회논문집
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• 제11권6호
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• pp.1001-1004
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• 1987

본 연구에서는 우선 선형 탄성문제의 변분해(variational solution)가 Sobol- ev 공간[ $H^{1}$(.OMEGA.)]= $H^{1}$(.OMEGA.)* $H^{1}$(.OMEGA.)* $H^{1}$(.OMEGA.)에서 유일하게 존재함을 재 검토하고 다음으로 경계적분식의 해도 변분해와 같음을 보인다. 이것은 선형 탄 성문제의 경우 고전해(classical solution)가 존재하지 않을 경우에도 BEM을 사용하여 변분해의 수치적 근사치를 구할 수 있다는 수학적 근거가 된다. 이를 위해서 Sobol- ev 공간 내에서의 Green's formula를 적용하는데 점하중해의 특이점(singularity)때문 에 Green's formula를 적용하기가 곤란해진다. 이 문제는 적분영역 .OMEGA.를 .OMEGA.-Ｂ$_{\rho }$로 치환하고 .rho.를 0으로 접근시키는 방법으로 해결한다. 이 때 Ｂ$_{\rho}$는 특이 점에 중심을 두고 매우 작은 변경 .rho.를 갖는 구이다.ho.를 갖는 구이다.

• 대한수학회지
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• 제41권4호
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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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• 제17권3호
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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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• 제23권4호
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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.

• 한국경영과학회지
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• 제20권2호
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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].

• 충청수학회지
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• 제22권4호
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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.